la15 abstracts · eas of mathematics and its applications, e.g., in matrix...

50 LA15 Abstracts

IP1

Fast Approximation of the Stability Radius and theH∞ Norm for Large-Scale Linear Dynamical Sys-tems

The stability radius and the H∞ norm are well-knownquantities in the robust analysis of linear dynamical sys-tems with output feedback. These two quantities, whichare reciprocals of each other in the simplest interestingcase, respectively measure how much system uncertaintycan be tolerated without losing stability, and how muchan input disturbance may be magnified in the output.The standard method for computing them, the Boyd-Balakrishnan-Bruinsma-Steinbuch algorithm from 1990, isglobally and quadratically convergent, but its cubic costper iteration makes it inapplicable to large-scale dynami-cal systems. We present a new class of efficient methodsfor approximating the stability radius and the H∞ norm,based on iterative methods to find rightmost points of spec-tral value sets, which are generalizations of pseudospectrafor modeling the linear fractional matrix transformationsthat arise naturally in analyzing output feedback. We alsodiscuss a method for approximating the real structured sta-bility radius, which offers additional challenges. Finally, wedescribe our new public-domain MATLAB toolbox for low-order controller synthesis, HIFOOS (H-infinity fixed-orderoptimization — sparse). This offers a possible alternativeto popular model order reduction techniques by applyingfixed-order controller design directly to large-scale dynam-ical systems. This is joint work with Nicola Guglielmi,Mert Gurbuzbalaban and Tim Mitchell. This speaker issupported in cooperation with the International Linear Al-gebra Society.

Michael L. OvertonNew York UniversityCourant Instit. of Math. [email protected]

IP2

Tuned Preconditioners for Inexact Two-Sided In-verse and Rayleigh Quotient Iteration

Computing both right and left eigenvectors of a gener-alised eigenvalue problem simultaneously is of interest inseveral important applications. We provide convergenceresults for inexact two-sided inverse and Rayleigh quotientiteration, which extend the previously established theoryto the generalized non-Hermitian eigenproblem and inex-act solves with a decreasing solve tolerance. Moreover, weconsider the simultaneous solution of the forward and ad-joint problem arising in two-sided methods and extend thesuccessful tuning strategy for preconditioners to two-sidedmethods, creating a novel way of preconditioning two-sidedalgorithms. This is joint work with Patrick Kuerschner(MPI Magdeburg, Germany).

Melina FreitagDept. of Mathematical SciencesUniversity of [email protected]

IP3

Sketching-Based Matrix Computations for Large-Scale Data Analysis

Matrix computations lies at the core of a broad rangeof methods in data analysis and machine learning, andcertain numerical linear algebra primitives (e.g. linear

least-squares regression and principal component analy-sis) are widely and routinely used. Devising scalable al-gorithms that enable large scale computation of the afore-mentioned primitives is crucial for meeting the challengesof Big Data applications. Sketching, which reduces dimen-sionality through randomization, has recently emerged asa powerful technique for scaling-up these primitives in thepresence of massive data, for an extensive class of applica-tions. In this talk, we outline how sketching can be used toaccelerate these core computations, and elaborate on thetradeoffs involved in their use. We will also demonstratethe utility of the presented algorithms for data analysis andmachine learning applications

Haim AvronBusiness Analytics & Mathematical SciencesIBM T.J. Watson Research [email protected]

IP4

Point-Spread Function Reconstruction in Ground-Based Astronomy

Because of atmospheric turbulence, images of objects inouter space acquired via ground-based telescopes are usu-ally blurry. One way to estimate the blurring kernel orpoint spread function (PSF) is to make use of the aberra-tion of wavefronts received at the telescope, i.e., the phase.However only the low-resolution wavefront gradients canbe collected by wavefront sensors. In this talk, I will dis-cuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover thephase and the PSF in high accuracy. I will also address re-lated numerical linear algebra issues such as the estimationof the regularization parameter and the solution of the lin-ear systems arising from the model.

Raymond H. ChanThe Chinese Univ of Hong KongDepartment of [email protected]

IP5

Accelerating Direct Linear Solvers with Hardwareand Algorithmic Advances

Factorization-based algorithms often play a significant rolein developing scalable solvers. The higher fidelity simula-tions and extreme-scale parallel machines present uniquechallenges for designing new parallel algorithms and soft-ware. In this talk, we first present some techniques to en-hance sparse factorization algorihtms to exploit the newerheterogeneous node architectures, such as nodes with GPUaccelerators or Intel Xeon Phi. Secondly, we present a newclass of scalable factorization algorithms that have asymp-totically lower complexity in both flops and storage, forwhich the acceleration power comes form exploiting low-rank submatrices and randomization.

Xiaoye Sherry LiComputational Research DivisionLawrence Berkeley National [email protected]

IP6

Variational Gram Functions: Convex Analysis andOptimization

We propose a class of convex penalty functions, called

LA15 Abstracts 51

”Variational Gram Functions”, that can promote pairwiserelations, such as orthogonality, among a set of vectorsin a vector space. When used as regularizers in convexoptimization problems, these functions of a Gram ma-trix find application in hierarchical classification, multitasklearning, and estimation of vectors with disjoint supports,among other applications. We describe a general conditionfor convexity, which is then used to prove the convexityof a few known functions as well as new ones. We give acharacterization of the associated subdifferential and theproximal operator, and discuss efficient optimization al-gorithms for loss-minimization problems regularized withthese penalty functions. Numerical experiments on a hier-archical classification problem are presented, demonstrat-ing the effectiveness of these penalties and the associatedoptimization algorithms in practice.

Maryam FazelUniversity of WashingtonElectrical [email protected]

IP7

Combinatorial Matrix Theory and Majorization

Majorization theory provides concepts for comparingmathematical objects according to “how spread out’ theirelements are. In particular, two n-vectors may be orderedby comparing the partial sums of the k largest compo-nents for k ≤ n. This is a fruitful concept in many ar-eas of mathematics and its applications, e.g., in matrixtheory, combinatorics, probability theory, mathematical fi-nance and physics. In this talk we discuss some combina-torial problems for classes of matrices where majorizationplays a role. This includes (0, 1)-matrices with line sumand pattern constraints, doubly stochastic matrices andLaplacian matrices of graphs. An extension of majoriza-tion to partially ordered sets is presented. We also discussa problem motivated by mathematical finance which leadsto interesting questions in qualitative matrix theory. Thisspeaker is supported in cooperation with the InternationalLinear Algebra Society.

Geir DahlUniversity of OsloDept. of Mathematics and Dept. of [email protected]

IP8

Numerical Solution of Eigenvalue Problems Arisingin the Analysis of Disc Brake Squeal

We present adaptive numerical methods for the solution ofparametric eigenvalue problems arsing from the discretiza-tion of partial differential equations modeling disc brakesqueal. The eigenvectors are used for model reduction toachieve a low order model that can be used for optimizationand control. The model reduction method is a variation ofthe proper orthogonal decomposition method. Several im-portant challenges arise, some of which can be traced backto the finite element modeling stage. Compared to the cur-rent industrial standard our new approach is more accuratein vibration prediction and achieves a better reduction inmodel size. This comes at the price of an increased com-putational cost, but it still gives useful results when thetraditional method fails to do so. We illustrate the resultswith several numerical experiments, some from real indus-trial models and indicate where improvements of the cur-rent black box industrial codes are advisable. We then also

discuss the use of adaptive methods such as the adaptivefinite element model and the algebraic multilevel substruc-tering method for the discussed problem, and we point outthe challenges and deficiencies in these approaches.

Volker MehrmannTechnische Universitat [email protected]

IP9

Linear Algebra Computations for ParameterizedPartial Differential Equations

The numerical solution of partial differential equations(PDEs) often entails the solution of large linear systemsof equations. This compute-intensive task becomes morechallenging when components of the problem such as coef-ficients of the PDE depend on parameters that are un-certain or variable. In this scenario, there there is aneed to compute many solutions for a single simulation,and for accurate discretizations, costs may be prohibitive.We discuss new computational algorithms designed to im-prove efficiency in this setting, with emphasis on new algo-rithms to handle stochastic problems and new approachesfor reduced-order models.

Howard C. ElmanUniversity of Maryland, College [email protected]

IP10

Accurate Linear Algebra in Computational Meth-ods for System and Control Theory

We discuss the importance of robust and accurate imple-mentation of core numerical linear algebra procedures incomputational methods for system and control theory. Inparticular, we stress the importance of error and pertur-bation analysis that identifies relevant condition numbersand guides computation with noisy data, and careful soft-ware implementation. The themes used as case studies in-clude rational matrix valued least squares fitting (e.g. leastsquares fit to frequency response measurements of an LTIsystem), model order reduction issues (e.g. the DiscreteEmpirical Interpolation Method (DEIM)), accurate com-putation with structured matrices such as scaled Cauchy,Vandermonde and Hankel matrices.

Zlatko DrmacUniversity of ZagrebDepartment of [email protected]

IP11

Low Rank Decompositions of Tensors and Matri-ces: Theory, Applications, Perspectives

Numerical data are frequently organized as d-dimensionalmatrices, also called tensors. However, only small valuesof d are allowed since the computer memory is limited. Inthe case of many dimensions, special representation for-mats are crucial, e.g. so called tensor decompositions. Re-cently, the known tensor decompositions have been con-siderably revisited and the two of them, previously usedonly in theoretical physics, are now recognized as the mostadequate and useful tools for numerical analysis. Thesetwo are the Tensor-Train and Hierarchical-Tucker decom-positions. Both are intrinsically related with low-rank ma-trices associated with a given tensor. We present these

52 LA15 Abstracts

decompositions and the role of low-rank matrices for theconstruction of efficient numerical algorithms.

Eugene TyrtyshnikovInstitute of Numerical Mathematics, Russian Academy [email protected]

IP12

Constrained Low Rank Approximations for Scal-able Data Analytics

Constrained low rank approximations have been widely uti-lized in large-scale data analytics where the applicationsreach far beyond the classical areas of scientific comput-ing. We discuss some fundamental properties of nonneg-ative matrix factorization (NMF) and introduce some ofits variants for clustering, topic discovery in text analy-sis, and community detection in social network analysis.In particular, we show how a simple rank 2 NMF com-bined with a divide-and-conquer framework results in asimple yet significantly more effective and scalable methodfor topic discovery. This simple approach can be furthergeneralized for graph clustering and community detection.Substantial experimental results illustrate significant im-provements both in computational time as well as qualityof solutions obtained.

Haesun ParkGeorgia Institute of [email protected]

SP1

SIAG/Linear Algebra Prize Lecture - LocalizingNonlinear Eigenvalues: Theory and Applications

Vibrations are everywhere, and so are the eigenvalues thatdescribe them. Physical models that include involve damp-ing, delay, or radiation often lead to nonlinear eigenvalueproblems, in which we seek complex values for which an(analytic) matrix-valued function is singular. In this talk,we show how to generalize eigenvalue localization results,such as Gershgorin’s theorem, Bauer-Fike, and pseudospec-tral theorems, to the nonlinear case. We demonstrate theusefulness of our results on examples from delay differentialequations and quantum resonances.

David BindelCornell [email protected]

CP1

An Augmented Hybrid Method for Large Scale In-verse Problems

In this work we use the weighted-GCV hybrid method ofChung/Nagy/Oleary for solving ill-posed inverse problems.This method restricts the solution to a Krylov subspaceand then uses regularization techniques on the projectedproblem in order to stop the typical semiconvergence be-havior of iterative methods. Difficulties arise for large prob-lems where we are required to store the full Krylov space.We have developed a technique to compress the space us-ing harmonic Ritz vectors. Computational examples areprovided.

Geoffrey DillonTexas Tech [email protected]

Julianne Chung, Eric De SturlerVirginia [email protected], [email protected]

CP1

On a Nonlinear Inverse Problem in Electromag-netic Sounding

Electromagnetic induction measurements are often usedfor non-destructive investigation of certain soil properties,which are affected by the electromagnetic features of thesubsurface layers. Starting from electromagnetic data col-lected by a ground conductivity meter, we propose a regu-larized inversion method based on a low-rank approxima-tion of the Jacobian of the nonlinear model. The methoddepends upon a relaxation parameter and a regularizationparameter, both chosen by automatic procedures. The per-formance of the method is investigated by numerical exper-iments both on synthetic and experimental data sets.

Caterina FenuUniversity of Cagliari, [email protected]

Gian Piero DeiddaUniversity of [email protected]

Giuseppe RodriguezUniversity of Cagliari, [email protected]

CP1

Localized-Deim: An Overlapping Cluster Frame-work with Application in Model Reduction forNonlinear Inversion

A numerically efficient application of parametric model re-duction critically depends on affine parametrization of thefull-order state-space quantities so that the online projec-tion step does not depend on the dimension of full model.Discrete Empirical Interpolation Method (DEIM) is com-monly used to construct such affine approximations. In thistalk, we propose overlapping clustering methods to improvethe approximation power of local DEIM and investigate ef-ficient implementations of the underlying greedy selectionprocedure. A nonlinear parametric inversion example aris-ing in diffuse optical tomography is used to illustrate theapproach.

Alexander R. GrimmDepartment of MathematicsVirginia [email protected]

Serkan GugercinVirginia Tech.Department of [email protected]

CP1

Matrix Affine Transformation Algorithm for Rank-Reducing Image Data Informatics Process

Matrix affine transformation T=QR+C is used in a way ofmaximizing informatics content in orthogonal QR process.Sample matrices were taken from images of quadrant pixel-shift patches across evenly spaced 128x128 grid points. Our

LA15 Abstracts 53

proposed algorithm cures the ill-posedness in the matrix in-version. By retrieving a specific rank-one pattern from acorrection matrix C, we can enrich the principal QR fac-torization. We tested the optimal matrix-rank-reducingprocess particularly with the SVD analysis.

Jay Min LeePohang Accelerator Lab, [email protected]

Youngjoo ChungSchool of Info. and [email protected]

Yonghoon KwonDepartment of Mathematics, [email protected]

CP1

Efficiencies in Global Basis Approximation forModel Order Reduction in Diffuse Optical Tomog-raphy

We consider the nonlinear inverse problem of reconstruct-ing parametric images of optical properties from diffuseoptical tomographic data. Recent work shows MOR tech-niques have promise in mitigating the computational bot-tleneck associated with solving for the parameters. In thistalk, we give an algorithm for efficiently computing the ap-proximate global basis needed in MOR by utilizing a newinterpretation of the transfer function and by capitalizingon Krylov recycling in a novel way.

Meghan O’ConnellDepartment of MathematicsTufts [email protected]

Misha E. KilmerTufts [email protected]

Eric De SturlerVirginia [email protected]

Serkan GugercinVirginia Tech.Department of [email protected]

Christopher A. BeattieVirginia Polytechnic Institute and State [email protected]

CP1

Robust Multi-Instance Regression

Multi-instance regression consists in building a regressionmodel that maps sets of instances (bags) to real-valuedoutputs. The Primary-Instance Regression (PIR) methodassumes that there is some primary instance (unknown dur-ing training) which is responsible for the real valued labeland that the rest of the items in the bag are noisy ob-servations of the primary instance. To immunize the pri-mary instance selection to noise, we propose an hyperplanefitting procedure which exploits the algebraic equivalence

between regularization (technique to prevent overfitting)and robustness (technique to immunize against set-induceduncertainty) and finds regularizers without requiring crossvalidation.

Dimitri PapadimitriouBell [email protected]

CP2

Iterative Refinement for Symmetric Eigenvalue De-composition and Singular Value Decomposition

An efficient iterative refinement algorithm is proposed forsymmetric eigenvalue problems. The algorithm is simple,and it mainly consists of matrix multiplications. It con-structs an arbitrarily accurate eigenvalue decomposition,up to the limit of computational precision. Using similartechniques, an iterative refinement algorithm for the singu-lar value decomposition is also derived. Since the proposedalgorithms are based on Newton’s method, they convergequadratically. Numerical results demonstrate the excellentperformance of the proposed algorithm.

Kensuke AishimaGraduate School of Information Science and Technology,University of TokyoKensuke [email protected]

Takeshi OgitaTokyo Woman’s Christian [email protected]

CP2

Some Inverse Numerical Range Problems

We generalize the inverse field of values problem to theq-numerical range and the rank-k numerical range of amatrix. We propose an algorithm for solving the inverseq-numerical range problem. Our algorithms exploits theconvexity of the q-numerical range. Approximating theboundary of the q-numerical requires constructing approx-imation of the Davis-Weilandt shell of a matrix. We notesome connections to computing pseudospectra. For therank-k numerical range, we have found a particular gen-eralized eigenvalue problem whose solution facilitates con-structing subspaces for generating points in the rank-k nu-merical range of a matrix, and also addresses the ques-tion of covering numbers for points in the rank-k numericalrange. The results in this work could have applications incomputing eigenvalues as well as quantum computing.

Russell CardenDepartment of MathematicsUniversity of [email protected]

M JahromiShahid Bahonar [email protected]

Iran Katsouleas, Greece MaroulasNational Technical University of Athensg [email protected], [email protected]

CP2

An Algorithm for Finding a 2-Similarity Transfor-mation from a Numerical Contraction to a Con-

54 LA15 Abstracts

traction

Any matrix A with numerical radius at most 1 can be writ-ten as A = STS−1, where ‖T‖ ≤ 1 and ‖S‖ · ‖S−1‖ ≤ 2.However, no explicit algorithm was given for producingsuch a similarity transformation. In this paper, we givea method for constructing such similarity transformations.As a side benefit, the algorithm indicates if the numericalradius of A is greater than than some given number and socan be used to determine if the numerical radius is greaterthan a given value.

Daeshik ChoiSouthern Illinois University [email protected]

Anne GreenbaumUniversity of [email protected]

CP2

The Markovian Joint Spectral Radius: What It Isand How to Compute It Efficiently

Given a finite set of matrices F = {Ai}Ni=1, with Ai ∈Cd × d, the Joint Spectral Radius (JSR) of F is given bythe generalization of the Gelfand’s formula for the spectralradius of a matrix. In recent works it has been proved thatthe JSR can be computed exactly, under suitable and gen-eral conditions, using polytope norms. In some cases, how-ever, not all the products are allowed, because the matricesin F are multiplied each other following some Markovianlaw. Recently Kozyakin showed in [1] that it is still pos-sibile to compute Joint Spectral Radius in the Markoviancase as the classical JSR of a significantly higher dimen-

sional set of matrices F ={Ai

}N

i=1, with Ai ∈ CNd × Nd.

This implies that the exact evaluation of the MarkovianJSR can be achieved in general using a polytope norm inCNd, which is a challenge task if N is large. In this talk weaddress the question whether it is possible to reduce thecomputational complexity for the calculation of the Marko-vian JSR showing that it is possible to transform the prob-lem into the evaluation of N polytope norms in Cd. Thisapproach is strictly related with the idea of multinormsintroduced by Jungers and Philippe for discrete–time lin-ear constrained switching systems [2]. As an illustrativeapplication we shall consider the zero–stability of variablestepsize 3–step BDF formulas.

[1] V. Kozyakin. The Berger–Wang formula for theMarkovian joint spectral radius. Linear Algebra and itsApplications. 448 (2014), 315–328. [2] M. Philippe andR. Jungers. Converse Lyapunov theorems for discrete-timelinear switching systems with regular switching sequences.(2014) arXiv preprint arXiv:1410.7197.

Antonio CiconeUniversita degli studi dell’[email protected]

Nicola GuglielmiUniversita degli Studi dell’[email protected]

Vladimir Y. ProtasovMoscow State UniversityDepartment of Mechanics and Mathematics

[email protected]

CP2

Two-Level Orthogonal Arnoldi Method for LargeRational Eigenvalue Problems

We propose a two-level orthogonal Arnoldi method for solv-ing large rational eigenvalue problems (REP), which ex-ploits the structure of the Krylov subspace of a Frobenius-like linearization of the REP. We develop such methodby using a different representation of the Krylov vec-tors, which is much more memory efficient than stan-dard Arnoldi applied on the linearization. In addition, wepresent numerical examples that show that the accuracy ofthe new method is comparable to standard Arnoldi.

Javier A. Gonzalez PizarroUniversidad Carlos III de [email protected]

Froilan DopicoDepartment of MathematicsUniversidad Carlos III de Madrid, [email protected]

CP2

A Communication-Avoiding Arnoldi-Type of theComplex Moment-Based Eigensolver

For solving interior eigenvalue problems, complex moment-based eigensolvers have been actively studied because oftheir high parallel efficiency. Recently, we proposed theArnoldi-type complex moment-based eigensolver namedthe block SS–Arnoldi method. In this talk, we proposean improvement of the block SS–Arnoldi method usinga communication-avoiding Arnoldi process (s-step Arnoldiprocess). We evaluate the performance of the proposedmethod and compare with other complex moment-basedeigensolvers.

Akira Imakura, Tetsuya SakuraiDepartment of Computer ScienceUniversity of [email protected], [email protected]

CP3

Block-Smoothers in Multigrid Methods for Struc-tured Matrices

Usually in multigrid methods for structured matrices, likeToeplitz matrices or circulant matrices, as well as in geo-metric multigrid methods point-smoothers are used. Anal-ysis for structured matrices mostly focusses on simplemethods like Richardson, smoothers like multicolor-SORare not considered. In this talk we assess general block-smoothers, where small blocks are inverted instead of singleunknowns. The presented analysis fits in the establishedanalysis framework and results in better converging multi-grid methods.

Matthias BoltenUniversity of WuppertalDepartment of [email protected]

CP3

Multigrid Preconditioners for Boundary Control of

LA15 Abstracts 55

Elliptic-Constrained Optimal Control Problems

Our goal is to design and analyze efficient multigrid pre-conditioners for solving boundary control problems forlinear-quadratic elliptic-constrained optimal control prob-lems. We have considered Dirichlet and Neumann bound-ary conditions. Thus far, we have numerically obtainedan optimal order preconditioner for the Hessian of the re-duced Neumann boundary control problem and suboptimalorder preconditioner for the Hessian of the reduced Dirich-let boundary control problem. Currently we are analyzingthe behavior of the preconditioner in theory.

Mona Hajghassem, Andrei DraganescuDepartment of Mathematics and StatisticsUniversity of Maryland, Baltimore [email protected], [email protected]

Harbir AntilGeorge Mason UniversityFairfax, [email protected]

CP3

A Multigrid Solver for the Tight-Binding Hamilto-nian of Graphene

Since the Nobel prize has been awarded in 2010 forthe isolation of graphene, research on this miraculous 2-dimensional material has flourished. In order to calculatethe electronic properties of graphene structures a tight-binding approach can be used. The resulting discrete eigen-value problem leads to linear systems of equations that aremaximally indefinite and possess a Dirac pseudo-spin struc-ture. In this talk we present a spin-preserving geometricmultigrid approach for these linear systems and show itsscalability with respect to several geometric parameters.

Nils KintscherFachbereich CApplied Computer Science [email protected]

Karsten KahlBergische Universitat WuppertalDepartment of [email protected]

CP3

Adaptive Algebraic Multigrid for Lattice QCD

In this talk, we present a multigrid approach for systems oflinear equations involving the Wilson Dirac operator aris-ing from lattice QCD. It combines components that havealready been used separately in lattice QCD, namely thedomain decomposition method ”Schwarz Alternating Pro-cedure” as a smoother and the aggregation based inter-polation. We give results from a series of numerical testsfrom our parallel MPI-C Code with system sizes of up to200,000,000 unknowns.

Matthias RottmannBergische Universitaet WuppertalDepartment of [email protected]

Andreas J. FrommerBergische Universitaet WuppertalFachbereich Mathematik und Naturwissenschaften

[email protected]


Bjoern LederBergische Universitaet WuppertalDepartment of [email protected]

Stefan KriegForschungszentrum JuelichJuelich Supercomputing [email protected]

CP3

Multigrid for Tensor-Structured Problems

We consider linear systems with tensor-structured matrices

A =∑⊗

Etj .

These problems are found in Markov chains or high-dimensional PDEs. Due to this format, the dimension of Agrows rapidly. The structure has to be exploited for solvingthese systems efficiently. To use multigrid for these mod-els we build a method which keeps the structure intact toguarantee computational savings on all grids. We presenthow to adapt the AMG framework to this setting usingtensor truncation.

Sonja SokolovicBergische Universitaet [email protected]



CP3

Multigrid Preconditioning for the Overlap Opera-tor in Lattice QCD

One of the most important operators in lattice QCD isgiven by the Overlap Dirac Operator. Due to bad condi-tioning solving linear systems with this operator can be-come rather challenging when approaching physically rel-evant parameters and lattice spacings. In this talk wepresent and analyze a novel preconditioning technique thatyields significant speedups. Furthermore we take a closerlook at the matrix sign function and its evaluation as partof the Overlap Dirac Operator.

Artur StrebelBergische Universitaet WuppertalDepartment of [email protected]

James BrannickDepartment of Mathematics

56 LA15 Abstracts

Pennsylvania State [email protected]

Andreas J. FrommerBergische Universitaet WuppertalFachbereich Mathematik und [email protected]


Bjorn Leder, Matthias RottmannBergische Universitaet WuppertalDepartment of [email protected], [email protected]

CP4

Devide-and-conquer Method for Symmetric-definite Generalized Eigenvalue Problems ofBanded Matrices on Manycore Systems

We have recently proposed a divide-and-conquer methodfor banded symmetric-definite generalized eigenvalue prob-lems based on the method for tridiagonal ones (Elsner,1997). The method requires less FLOPs than the con-ventional methods (e.g. DSBGVD in LAPACK) when theband is narrow and has high parallelism. In this presen-tation, we describe the implementation of the proposedmethod for manycore systems and demonstrate the effi-ciency of our solver.

Yusuke HirotaRIKEN [email protected]

Toshiyuki ImamuraRIKEN Advance Institute for Computational [email protected]

CP4

Performance Analysis of the Householder Back-transformation with Asynchronous CollectiveCommunication

Recently, communication avoiding and communication hid-ing technologies are focused on to accelerate the per-formance on parallel supercomputer systems. For denseeigenvalue computation, especially Householder back-transformation of eigenvectors, we observed asynchronouscollective communication is applicable and some specialperformance characteristics and deviations via actual nu-merical experiences, specifically, the peak performance isunevenly distributed. We are going to build a performancemodel and analyze its response when we change some per-formance parameters defined in the performance model.

Toshiyuki ImamuraRIKEN Advance Institute for Computational [email protected]

CP4

Dynamic Parallelization for the Reduction of aBanded Matrix to Tridiagonal Form

In the talk a new dynamic parallelization for the reduction

of a band matrix to tridiagonal form is considered. Wepresent some details of implemented optimizations: dy-namic parallelization of eigenvectors computations, specu-lative computations in QR, dynamic parallelization of thereduction of banded matrix to tridiagonal form and howthese techniques are combined for achieving high perfor-mance.

Nadezhda Mozartova, Sergey V Kuznetsov, AleksandrZotkevichIntel [email protected],[email protected], [email protected]

CP4

Performance of the Block Jacobi-Davidson Methodfor the Solution of Large Eigenvalue Problems onModern Clusters

We investigate a block Jacobi-Davidson method for com-puting a few exterior eigenpairs of a large sparse matrix.The block method typically requires more matrix-vectorand vector-vector operations than the standard algorithm.However, this is more than compensated by the perfor-mance gains through better data reusage on modern CPUs,which we demonstrate by detailed performance engineeringand numerical experiments. The key ingredients to achiev-ing high performance consist in both kernel optimizationsand a careful design of the algorithm that allows usingblocked operations in most parts of the computation. Weshow the performance gains of the block algorithm with ourhybrid parallel implementation for a variety of matrices onup to 5 120 CPU cores. A new development we discuss inthis context is a highly accurate and efficient block orthog-onalization scheme that exploits modern hardware featuresand mixed precision arithmetic.

Melven Roehrig-Zoellner

German Aerospace Center (DLR)Simulation and Software [email protected]

Jonas ThiesGerman Aerospace Center (DLR)[email protected]

Achim BasermannGerman Aerospace Center (DLR)Simulation and Software [email protected]

Florian Fritzen, Patrick AulbachGerman Aerospace Center (DLR)[email protected], [email protected]

CP4

Performance Comparison of Feast and Primme inComputing Many Eigenvalues in Hermitian Prob-lems

Contour integration methods like FEAST are successfullyused to partially solve large eigenproblems arising in Den-sity Functional Theory. These methods are scalable andcan show better performance than restarted Krylov solversin computing many eigenpairs in the interior of the spec-trum. Recently we have used polynomial filters with Gen-eralized Davidson (GD) with similar success under limited

LA15 Abstracts 57

memory. In this talk, we compare FEAST with the GDavailable in PRIMME, solving Hermitian problems fromdifferent applications.

Eloy Romero AlcaldeComputer Science DepartmentCollege of Williams & [email protected]

Andreas StathopoulosCollege of William & MaryDepartment of Computer [email protected]

CP4

The Implicit Hari-Zimmermann Algorithm for theGeneralized Svd

We developed the implicit Hari–Zimmermann method forcomputation of the generalized singular values of matrixpairs, where one of the matrices is of full column rank.The method is backward stable, and, if the matrices per-mit, computes the generalized singular values with smallrelative errors. Moreover, it is easy to parallelize. Un-like the triangular routine DTGSJA from Lapack, the Hari–Zimmermann method needs no preprocessing to make bothmatrices triangular. Even when the matrices are prepro-cessed to a triangular form, the sequential pointwise Hari–Zimmermann method is, for matrices of a moderate size,significantly faster than DTGSJA. A significant speedup isobtained by blocking of the algorithm, to exploit the effi-ciency of BLAS-3 operations.

Sanja SingerFaculty of Mechanical Engineering and NavalArchitecture, University of [email protected]

Vedran NovakovicSTFC, Daresbury LaboratoryWarrington, United [email protected]

Sasa SingerDepartment of Mathematics, University of [email protected]

CP5

Heuristics for Optimizing the Communicability ofDigraphs

In this talk we investigate how to modify the edges of adirected network in order to tune its overall broadcast-ing/receiving capacity. We introduce broadcasting and re-ceiving measures (related to the network’s “hubs” and “au-thorities”, respectively) in terms of the entries of appropri-ate functions of the matrices AAT and ATA, respectively,where A is the adjacency matrix of the network. The largerthese measures, the better the network is at propagating in-formation along its edges. We also investigate the effect ofedge addition, deletion, and rewiring on the overall broad-casting/receiving capacity of a network. In particular, weshow how to add edges so as to increase these quantities asmuch as possible, and how to delete edges so as to decreasethem as little as possible.

Francesca ArrigoUniversita degli Studi dell’[email protected]

Michele BenziDepartment of Mathematics and Computer ScienceEmory [email protected]

CP5

Generalizing Spectral Graph Partitioning to SparseTensors

Spectral graph partitioning is a method for clustering dataorganized as undirected graphs. The method is based onthe computation of eigenvectors of the graph Laplacian. Inmany areas one wants to cluster data from a sequence ofgraphs. Such data can be organized as large sparse tensors.We present a spectral method for tensor partitioning basedon the computation of a best multilinear rank aproximationof the tensor. A few applications are briefly discussed.

Lars EldenLinkoping UniversityDepartment of [email protected]

CP5

Graph Partitioning with Spectral Blends

Spectral partitioning uses eigenvectors and eigenvalues topartition graphs. We show that blends of eigenvectors havebetter pointwise error bounds than the eigenvectors forgraphs with small spectral gaps. We use a model problem,the Ring of Cliques, to demonstrate the utility of spectralblends for graph partitioning. This analysis provides a con-vergence criterion in terms of conductance and the Cheegerinequality.

James P. FairbanksGeorgia Institute of [email protected]

Geoffrey D. SandersCenter for Applied Scientific ComputingLawrence Livermore National [email protected]

CP5

On the Relation Between Modularity and Adja-cency Graph Partitioning Methods

Modularity clustering is a popular method in partitioninggraphs. In this talk we will present the relation betweenthe leading eigenvector b of a modularity matrix and theeigenvectors ui of its corresponding adjacency matrix. Thisrelation allows us to approximate the vector b with someof the ui. The equivalence between normalized versions ofmodularity clustering and adjacency clustering will also bedemonstrated.

Hansi JiangNorth Carolina State [email protected]

Carl MeyerMathematics DepartmentN. C. State University

58 LA15 Abstracts

[email protected]

CP5

Detecting Highly Cyclic Structure with ComplexEigenpairs

Highly 3- and 4-cyclic subgraph topologies are detectablevia calculation of eigenvectors associated with certain com-plex eigenvalues of Markov propagators. We characterizethis phenomenon theoretically to understand the capabil-ities and limitations for utilizing eigenvectors in this ven-ture. We provide algorithms for approximating these eigen-vectors and give numerical results, both for software thatutilizes complex arithmetic and software that is limited toreal arithmetic. Additionally, we discuss the application ofthese techniques to motif detection.

Christine KlymkoLawrence Livermore National [email protected]

Geoffrey D. SandersCenter for Applied Scientific ComputingLawrence Livermore National [email protected]

CP5

Orthogonal Representations, Projective Rank, andFractional Minimum Positive Semidefinite Rank:Connections and New Directions

This paper introduces r-fold orthogonal representations ofgraphs and formalizes the understanding of projective rankas fractional orthogonal rank. Fractional minimum positivesemidefinite rank is defined and it is shown that the pro-jective rank of any graph equals the fractional minimumpositive semidefinite rank of its complement. An r-foldversion of the traditional definition of minimum positivesemidefinite rank of a graph using Hermitian matrices thatfit the graph is also presented.

Leslie HogbenIowa State University,American Institute of [email protected]

Kevin PalmowskiIowa State [email protected]

David RobersonDivision of Mathematical SciencesNanyang Technological [email protected]

Simone SeveriniDepartment of Computer ScienceUniversity College [email protected]

CP6

Inverse Eigenvalue Problems for Totally Nonnega-tive Matrices in Terms of Discrete Integrable Sys-tems

Several discrete integrable systems play key roles in ma-trix eigenvalue algorithms such as the qd algorithm forsymmetric tridiagonal matrices and the dhToda algorithm

for Hessenberg TN matrices whose minors are all nonneg-ative. In this talk, we consider inverse eigenvalue prob-lems for TN matrices including symmetric tridiagonal ma-trices from the viewpoint of discrete integrable systems as-sociated with matrix eigenvalue algorithms. Moreover, wepropose a finite-step construction of TN matrices with pre-scribed eigenvalues.

Kanae Akaiwa, Yoshimasa NakamuraGraduate School of Informatics, Kyoto [email protected], [email protected]

Masashi IwasakiDepartment of Informatics and Environmental Science,Kyoto Prefectural [email protected]

Hisayoshi Tsutsumi, Akira YoshidaDoshisha [email protected],[email protected]

Koichi KondoDoshisha UniversityDept. of [email protected]

CP6

A Kinetic Ising Model and the Spectra of SomeJacobi Matrices

One-dimensional statistical physics models play a funda-mental role in understanding the dynamics of complex sys-tems in a variety of fields, from chemistry and physics,to social sciences, biology, and nanoscience. Due to theirsimplicity, they are amenable to exact solutions that canlead to generalizations in higher dimensions. In 1963,R. Glauber solved exactly a one-dimensional spin model,known in literature as the kinetic Ising chain, KISC, thatled to many applications and two-temperature generaliza-tions. In this talk we consider a case of temperature dis-tributions, extracting information regarding the physicalproperties of the system from the spectrum analysis of ma-trix certain Jacobi matrix. We also analyze the eigenvaluesof some perturbed Jacobi matrices. The results containas particular cases the known spectra of several classes oftridiagonal matrices studied recently. This is a join workwith S. Kouachi, D.A. Mazilu, and I. Mazilu.

Carlos FonsecaDepartment of MathematicsUniversity of [email protected]

Said KouachiQassim University, Al-Gassim, BuraydahSaudi [email protected]

Dan Mazilu, Irina MaziluWashington and Lee University, [email protected], [email protected]

CP6

The Molecular Eigen-Problem and Applications

Referring to the classical eigenproblem of a matrix as the

LA15 Abstracts 59

atomic eigenproblem, we extend this notion to a molecu-lar eigenproblem: Consider the module of matrices M =ICn×m with n ≥ m, over the noncommutative matrix ringR = ICm×m. The m-molecular eigen-problem is the prob-lem of determining Λ ∈ ICm×m and X ∈ ICn×m such thatAX = XΛ. Existence conditions and characterizations ofits solution are given. If (X,Λ) is an m-molecular eigen-pair of A, then so is (XR,Λ), for all R ∈ C(Λ), the cen-tralizer of the set of molecular eigenvalues. This freedommay be exploited to define a canonical molecular eigenvec-tor. We apply this generalized formalism to solve matrix-ODE’s, and give a lattice theoretic representation of thestructure of the solution sets.

Erik VerriestSchool of Electrical and Computer EngineeringGeorgia Institute of [email protected]

Nak-Seung HyunSchool of Electrical and Computer EngineeringGeorgia Institute of [email protected]

CP6

Spectral Properties of the Boundary Value Prob-lems for the Discrete Beam Equation

In this talk, we will consider the boundary-value problemfor the fourth-order discrete beam equation

Δ4yi + bi+2yi+2 = λai+2yi+2, −1 ≤ i ≤ n− 2,

Δ2y−1 = Δ3y−1 = Δ2yn−1 = Δ3yn−1 = 0,

which is a discrete analogy to the following boundary-valueproblem for the fourth-order linear beam equation:

y(4)(t)+b(t)y(t) = λa(t)y(t), y′(0) = y′(0) = y′(1) = y′(1) = 0.

For the ordinary differential equation boundary-value prob-lem, the monotonicity of the smallest positive eigenvaluewas studied in the literature. The special structure of thematrices associated with the discrete problem allows us toanalyze the spectral properties of the problem and to es-tablish the monotonicity of all eigenvalues of the discreteproblem as the sequences {ai}ni=1 and {bi}ni=1 change.

Jun Ji, Bo YangKennesaw State [email protected], [email protected]

CP6

Matrix Nearness Problems for General Lyapunov-Type Stability Domains

In many applications asymptotic instability of the mathe-matical (dynamical) system leads to the loss of the struc-tural integrity of the real physical system due to the am-plification of the perturbations in the initial conditions.However, in some cases, although dynamical system isasymptotically stable, the corresponding physical systemcan loose its structural integrity due to transitional insta-bility (due to too large amplitude or too high frequency)typical for dynamics governed by nonnormal matrices. Toconsider such applications, we review general Lyapunov-type domains and formulate two matrix nearness problems- the distance to delocalization and distance to localizationto generalize the distance to instability and the distanceto stability, in both, discrete and continuous sense. Then,

we present numerical algorithms for their solution. Finally,present computations some medium size and large sparsematrices arising in different scientific and industrial appli-cations.

Vladimir KosticUniversity of Novi Sad, Faculty of ScienceDepartment of Mathematics and [email protected]

Agnieszka MiedlarTechnische Universitat [email protected]

CP6

A Multigrid Krylov Method for Eigenvalue Prob-lems

We propose a new multigrid Krylov method for eigenvalueproblems of differential operators. Arnoldi methods areused on multiple grids. Approximate eigenvectors from acoarse grid can be improved on a fine grid. We compare thenew method with other approaches, and we also give anal-ysis of the convergence. This multigrid Arnoldi method ismore robust than standard multigrid and has potential fordramatic improvement compared to regular Arnoldi.

Zhao YangDepartment of MathematicsOklahoma State UniversityA [email protected]

Ronald MorganDepartment of MathematicsBaylor Universityronald [email protected]

CP7

Block-Asynchronous Jacobi Iterations with Over-lapping Domains

Block-asynchronous Jacobi is an iteration method wherea locally synchronous iteration is embedded in an asyn-chronous global iteration. The unknowns are partitionedinto small subsets, and while the components within thesame subset are iterated in Jacobi fashion, no update orderin-between the subsets is enforced. The values of the non-local entries remain constant during the local iterations,which can result in slow inter-subset information propa-gation and slow convergence. Interpreting of the subsetsas subdomains allows to transfer the concept of domainoverlap typically enhancing the information propagationto block-asynchronous solvers. In this talk we explore theimpact of overlapping domains to convergence and perfor-mance of block-asynchronous Jacobi iterations, and presentresults obtained by running this solver class on state-of-the-art HPC systems.

Hartwig AnztInnovate Computing LabUniversity of [email protected]

Edmond ChowSchool of Computational Science and EngineeringGeorgia Institute of [email protected]

Daniel B. Szyld

60 LA15 Abstracts

Temple UniversityDepartment of [email protected]

Jack J. DongarraDepartment of Computer ScienceThe University of [email protected]

CP7

Performance Evaluation of the Choleskyqr2 Algo-rithm

Cholesky QR computes the QR factorization through theCholesky factorization. It has excellent suitability for HPCbut is rarely practical due to its numerical instability. Re-cently, we have pointed out that an algorithm that repeatsCholesky QR twice, which we call CholeskyQR2, has muchimproved stability. In this talk, we present the performanceresults of CholeskyQR2 and show its practicality. We alsodiscuss its application to the block Gram-Schmidt orthog-onalization and the block Householder QR algorithm.

Takeshi FukayaHokkaido [email protected]

Yuji NakatsukasaDepartment of MathematicsUniversity of [email protected]

Yuka YanagisawaWaseda [email protected]

Yusaku YamamotoThe University of Electro-Communications, [email protected]

CP7

High Performance Resolution of Dense Linear Sys-tems Using Compression Techniques and Applica-tion to 3D Electromagnetics Problems

Solving large 3-D electromagnetic problems is challenging.Currently, accurate numerical methods are used to solveMaxwells equations in the frequency domain, which leadsto solve dense linear systems. Thanks to recent advances onfast direct solvers by the means of compression techniques,we have developed a solver capable of handling systemswith millions of complex unknowns and thousands of righthand sides suited for our Petascale machine.

David Goudin, Cedric Augonnet, Agnes Pujols, MurielSesquesCEA/[email protected], [email protected],[email protected], [email protected]

CP7

Batched Matrix-Matrix Multiplication Operationsfor IntelR© XeonR© Processor and IntelR© XeonPhi

TM

Co-Processor

Many numerical algorithms such as sparse solvers and finiteelement method rely on a number of matrix-matrix multi-plication operations that can be performed independently.

In this talk, we present the new interfaces and implemen-tation details of the batched matrix-matrix multiplicationroutines in IntelR© Math Kernel Library 11.3. Compared tothe optimized non-batched counterparts, batched routinesprovide 8× and 15× speedups on average for IntelR© XeonR©

processor and IntelR© Xeon PhiTM

coprocessor respectively.

Murat E. Guney, Sarah Knepper, Kazushige Goto, VamsiSripathi, Greg Henry, Shane StoryIntel [email protected], [email protected],[email protected], [email protected],[email protected], [email protected]

CP7

Data Sparse Technique

In this talk we will describe how H-matrix data sparse tech-niques can be implemented in a parallel hybrid sparse linearsolver based on algebraic non overlapping domain decom-position approach. Strong-hierarchical matrix arithmeticand various clustering techniques to approximate the localSchur complements will be investigated, aiming at reduc-ing workload and memory consumption while complyingwith structures of the local interfaces of the sub-domains.Utilization of these techniques to form effective global pre-conditioner will be presented.

Yuval [email protected]

Luc GiraudInriaJoint Inria-CERFACS lab on [email protected]

Emmanuel [email protected]

Eric DraveStanford [email protected]

CP7

A Newly Proposed BLAS Extension (xGEMMT)to Update a Symmetric Matrix Efficiently

We propose a complement to the Level 3 BLAS xGEMMroutine that computes C := α× A× B + β × C, where Cremains symmetric for general A and B matrices. For in-stance, A may be the product of BT and a symmetric or di-agonal matrix. This new xGEMMT routine provides func-tionality used in numerous algorithms and within Hessian-based optimization methods. In this talk, we discuss thesubtleties of implementing and optimizing xGEMMT inthe IntelR© Math Kernel Library.

Sarah Knepper, Kazushige Goto, Murat E. Guney, GregHenry, Shane StoryIntel [email protected], [email protected],[email protected], [email protected],

LA15 Abstracts 61

[email protected]

CP8

Eigenvalue Condition Numbers of PolynomialEigenvalue Problems under Mobius Transforma-tions

We study the effect of Mobius transformations on the sen-sitivity of polynomial eigenvalues problems (PEP). Moreprecisely, we compare eigenvalue condition numbers for aPEP and the correspondent eigenvalue condition numbersfor the same PEP modified with a Mobius transforma-tion. We bound this relationship with factors that dependson the condition number of the matrix that induces theMobius transformation and on the eigenvalue whose norm-wise condition number we consider, and establish sufficientconditions where Mobius transformations do not alter sig-nificantly the condition numbers.

Luis Miguel Anguas MarquezUniversidad Carlos III de [email protected]


CP8

Computation of All the Eigenpairs for a ParticularKind of Banded Matrix

For a particular kind of banded matrix which is character-ized by the band width, we propose a new algorithm forcomputing all the eigenpairs effectively. Though the in-tended matrix has complex eigenvalues, our algorithm cancompute all the complex eigenpairs only by the arithmeticof real numbers. We also present an error analysis andnumerical examples for the proposed algorithm.

Hiroshi TakeuchiTokyo University of [email protected]

Kensuke AiharaDepartment of Mathematical Information Science,Tokyo University of [email protected]

Akiko FukudaShibaura Institute of [email protected]

Emiko IshiwataDepartment of Mathematical Information Science,Tokyo University of Science, [email protected]

CP8

Fixed-Point Singular Value Decomposition Algo-rithms with Tight Analytical Bounds

This work presents an analytical approach for finding theranges of the variables in fixed-point singular value decom-position algorithm based on upper bound for the spectralnorm of input matrix. We show that if each element of a

matrix is divided by the upper bound for spectral norm,then unvarying and tight ranges for the variables in thealgorithm are obtained. Thus overflow is avoided for allrange of input matrices with reduced hardware cost.

Bibek Kabi, West Bengal, [email protected]

Aurobinda Routray, Ramanarayan MohantyIndian Institute of Technology KharagpurWest Bengal, [email protected], [email protected]

CP8

A Novel Numerical Algorithm for TriangularizingQuadratic Matrix Polynomials

For any monic linearization λI + A of a quadratic matrixpolynomial, Tisseur and Zaballa [SIAM J. Matrix Anal.Appl., 34-2 (2013), pp. 312-337] show that there exists anonsingular matrix [U AU ] that transforms A to a compan-ion linearization of a (quasi)-triangular quadratic matrixpolynomial. We observe that the matrix [U AU ] may beill-conditioned while U is perfectly-conditioned. To con-quer the ill conditioning challenge, we design a numericalalgorithm without the orthonormal characteristic of U .

Yung-Ta LiFu Jen Catholic University, [email protected]

CP8

A Fiedler-Like Approach to Spectral Equivalenceof Matrix Polynomials

In this talk we extend the notion of Fiedler pencilsto square matrix polynomials of the form P (λ) =∑k

i=0 Aiφi(λ) where {φi(λ)}ki=0 is either a Bernstein, New-ton, or Lagrange basis. We use this new notion to pro-vide a systematic way to easily generate large new familiesof matrix pencils that are spectrally equivalent to P (λ),and consequently, for solving the polynomial eigenproblemP (λ)x = 0, x �= 0. Time permitting, we will discuss somenumerical properties of these matrix pencils.

Vasilije PerovicWestern Michigan UniversityDepartment of [email protected]

D. Steven MackeyDepartment of MathematicsWestern Michigan [email protected]

CP8

On First Order Expansions for Multiplicative Per-turbation of Eigenvalues

Let A be a matrix with any Jordan structure, and λ aneigenvalue of A whose largest Jordan block has size n.We present first order eigenvalue expansions under mul-

tiplicative perturbations A = (I + εC)A (I + εB) usingNewton Polygon. Explicit formulas for the leading coef-ficients are obtained, involving the perturbation matrices

62 LA15 Abstracts

and appropriately normalized eigenvectors of A. If λ �= 0,

the perturbation in the eigenvalue is of order of ε1n , while

if λ = 0, it is generically of order ε1

n−1 .

Fredy E. SosaDepartamento de MatrmaticasUniversidad Carlos III de Madrid, [email protected]

Julio MoroUniversidad Carlos III de MadridDepartamento de [email protected]

CP9

Approximating the Leading Singular Triplets of aLarge Matrix Function

Given a large square matrix A and a sufficiently regularfunction f , we are interested in the approximation of theleading singular values and corresponding vectors of f(A),and in particular of ‖f(A)‖, where ‖·‖ is the induced matrix2-norm. Since neither f(A) nor f(A)v can be computed ex-actly, we introduce and analyze an inexact Golub-Kahan-Lanczos bidiagonalization procedure. Particular outer andinner stopping criteria are devised to cope with the lack ofa true residual.

Sarah W. GaafEindhoven University of [email protected]

Valeria SimonciniUniversita’ di [email protected]

CP9

Inverse Probing for Estimating diag(f(A))

Computing diag(f(A)) where f(A) is a function of a largesparse matrix can be computationally difficult. Probingattempts to solve this by using matrix polynomials to de-termine the structure of f(A). However, these matrix poly-nomials can converge slowly, causing probing to fail. Toavoid this, we propose a new method which we term In-verse Probing, that directly approximates the structure off(A) based on a small sample of columns of f(A). We showthat this is more effective than probing in most situations.

Jesse LaeuchliCollege of William and [email protected]


CP9

The Waveguide Eigenvalue Problem and the Ten-sor Infinite Arnoldi Method

We present a new iterative algorithm for nonlinear eigen-value problems (NEPs), the tensor infinite Arnoldi method,which is applicable to a general class of NEPs. More-over we show how to specialize the algorithm to a spe-

cific NEP: the waveguide eigenvalue problem, which arisesfrom a finite-element discretization of a partial-differentialequation used in the study waves propagating in periodicmedium. The algorithm is successfully applied to solvebenchmark problems as well as complicated waveguides.

Giampaolo MeleKTH Royal Institute of TechnologyDepartment of numerical [email protected]

Elias JarlebringKTH [email protected]

Olof RunborgKTH, Stockholm, [email protected]

CP9

Verified Solutions of Delay Eigenvalue Problemswith Multiple Eigenvalues

Consider computing error bounds for numerical solu-tions of nonlinear eigenvalue problems arising from delay-differential equations: given A,B ∈ Cn×n and τ ≥ 0, findλ ∈ C and x ∈ Cn \ {0} such that

(λI − A−Be−τλ)x = 0,

where I is the identity matrix, for giving reliability of thesolutions. The author previously proposed an algorithm forcomputing the error bounds. However, this is not applica-ble when λ is multiple. We hence propose an algorithmwhich is applicable even in this case.

Shinya MiyajimaGifu [email protected]

CP9

Taylor’s Theorem for Matrix Functions and Pseu-dospectral Bounds on the Condition Number

We generalize Taylor’s theorem from scalar functions tomatrix functions, obtaining an explicit expression for theremainder term. Consequently we derive pseudospectralbounds on the remainder which can be used to obtain anupper bound on the condition number of the matrix func-tion. Numerical experiments show that this upper boundcan be calculated very quickly for f(A) = At, almost threeorders of magnitude faster than the current state-of-the-artmethod.

Samuel ReltonUniversity of Manchester, [email protected]

Edvin DeadmanUniversity of [email protected]

CP9

Contour Integration Via Rational Krylov for Solv-ing Nonlinear Eigenvalue Problems

The Cauchy integral reformulation of the nonlinear eigen-value problem A(λ)x = 0 has led to subspace methods

LA15 Abstracts 63

for nonlinear eigenvalue problems, where approximationsof contour integration by numerical quadrature play therole of rational filters of the subspace. We show that insome cases this filtering of the subspace by rational func-tions can be efficiently performed by applying a restartedrational Krylov method. We illustrate that this approachincreases computational efficiency. Furthermore, locking ofconverged eigenvalues can in the rational Krylov methodbe performed in a robust way.

Roel Van BeeumenKU [email protected]

Karl MeerbergenK. U. [email protected]

Wim MichielsKU [email protected]@

CP10

A Block Gram-Schmidt Algorithm with Reorthog-onalization and Conditions on the Tall, Skinny Qr

The talk considers block the Gram-Schmidt with reorthog-onalization (BCGS2) algorithm discussed in [J. Barlow andA. Smoktunowicz, Reorthogonalized Block Classical Gram-Schmidt, Num. Math. ,123:398–423, 2013.] for producingthe QR factorization of a matrix X that is partitioned intoblocks. A building block operation for BCGS2 is the “tall-skinny QR’ factorization (TSQR) which is assumed to be abackward stable factorization. However, that assumptionexcludes some possible TSQR algorithms, in particular, arecent TSQR algorithm in [I.Yamazaki, S. Tomov, and J.Dongarra. Mixed-Precision Cholesky QR factorization andIts Case Studies on Multicore CPU with mulitiple GPUS.to appear, SIAM J. Sci. Computing, 2015]. It is shownthat the weaker stability conditions satisfied by the Ya-mazaki et al. algorithm are sufficient for BCGS2 to pro-duce a conditionally backward stable factorization.

Jesse L. BarlowPenn State UniversityDept of Computer Science & [email protected]

CP10

Block Methods for Solving Banded Symmetric Lin-ear Systems

Several years ago we discovered an algorithm for factoringbanded symmetric systems of equations which required halfthe number of multiplications and 2/3 the space of theexisting algorithms that ignored symmetry. The algorithmreduced that matrix to a sequence of 1 x 1 and 2 x 2 pivots.To prevent fillin and to promote stability a sequence of 2x 2 planar transformations are necessary when using 2 x 2pivots which greatly complicates a block approach.

Linda KaufmanWilliam Paterson [email protected]

CP10

On the Factorization of Symmetric Indefinite Ma-

trices into Anti-Triangular Ones

A factorization of a symmetric indefinite matrix, A =QMQT , with Q orthogonal and M symmetric block an-titriangular (BAT) is described in [1], relying only on or-thogonal transformations and revealing the inertia of thematrix. In [2] a block algorithm performing all operationsalmost entirely in level 3 BLAS is developed, featuring amore favorable memory access pattern. In the same papera lack of reliability is noticed in computing the inertia. Inthis talk we describe a new implementation of the BATfactorization and compare it to the other implementationsin terms of stability and reliability.

1 N. Mastronardi, P. Van Dooren, The antitriangularfactorization of symmetric matrices, SIMAX 34 2013173-196.

2 Z. Bujanovic, D. Kressner, A block algorithm for com-puting antitriangular factorizations of symmetric ma-trices, Numer Algor, to appear.

Nicola MastronardiIstituto per le Applicazioni del CalcoloNational Research Council of Italy, [email protected]

Paul Van DoorenUniversite Catholique de [email protected]

CP10

Gram-Schmidt Process with Respect to BilinearForms

Gram-Schmidt orthogonalization process is probably themost popular and frequently used scheme to obtain mutu-ally orthogonal vectors. In this contribution we considerorthogonalization schemes with respect to the symmetricbilinear forms and skew-symmetric bilinear forms.We ana-lyze their behavior in finite precision arithmetic and givebounds for the loss of orthogonality between the computedvectors.

Miro RozloznikCzech Academy of SciencesPrague, Czech [email protected]

CP10

Roundoff Error Analysis of the Choleskyqr2 andRelated Algorithms

Cholesky QR is an ideal QR factorization algorithm fromthe viewpoint of high performance computing, but it hasrarely been used in practice due to numerical instability.Recently, we showed that by repeating Cholesky QR twice,we can greatly improve the stability. In this talk, wepresent a detailed error analysis of the algorithm, whichwe call CholeskyQR2. Numerical stability of related al-gorithms, such as the block Gram-Schmidt method usingCholeskyQR2, is also discussed.

Yusaku YamamotoThe University of Electro-Communications, [email protected]

Yuji NakatsukasaDepartment of MathematicsUniversity of Manchester

64 LA15 Abstracts

[email protected]

Yuuka YanagisawaWaseda [email protected]

Takeshi FukayaHokkaido [email protected]

CP10

Mixed-Precision Orthogonalization Processes

Orthogonalizing a set of dense vectors is an important com-putational kernel in subspace projection methods for solv-ing large-scale problems. In this talk, we discuss our ef-forts to improve the performance of the kernel, while main-taining its numerical accuracy. Our experimental resultsdemonstrate the effectiveness of our approaches.

Ichitaro YamazakiUniversity of Tennessee, [email protected]

Jesse L. BarlowPenn State UniversityDept of Computer Science & [email protected]

Stanimire TomovInnovative Computing Laboratory, Computer ScienceDeptUniversity of Tennessee, [email protected]

Jakub KurzakUniversity of Tennessee, [email protected]


CP11

A New Asynchronous Solver for Banded LinearSystems

Banded linear systems occur frequently in mathematicsand physics. A solution approach is discussed that decom-poses the original linear system into q-subsystems, whereq is the number of superdiagonals. Each system is solvedasynchronously, followed by a q×q constraint matrix prob-lem, and a final superposition. Reduction to a lower trian-gular system is never required, so the method can be fastwhen q-processors are available. Numerical experimentsusing Matlab and Fortran are also discussed.

Michael A. JandronNaval Undersea Warfare [email protected]

Anthony RuffaNaval Undersea Warfare Center, Newport, RI [email protected]

James BaglamaDepartment of Mathematics

University of Rhode Island, Kingston, RI [email protected]

CP11

The Inverse of Two Level Topelitz Operator Ma-trices

The celebrated Gohberg-Semencul formula provides a for-mula for the inverse of a Toeplitz matrix based on theentries in the first and last columns of the inverse, un-der certain nonsingularity conditions. In this talk we willpresent similar formulas for two-level Toeplitz matrices andwill provide a two variable generalization of the Gohberg-Semencul formula in the case of a positive definite two-level Toeplitz matrix with a symbol of the form f(z1, z2) =P (z1, z2)

∗−1P (z1, z2)−1 = R(z1, z2)

−1R(z1, z2)∗−1 for

z1, z2 ∈ T, where P (z1, z2) and R(z1, z2) are stable op-erator valued polynomials of two variables. In addition,we propose an approximation of the inverse of a multilevelToeplitz matrix with a positive symbol, and use it as theinitial value for a Hotelling iteration to compute the in-verse. Numerical results are included.

Selcuk KoyuncuUniversity of North [email protected]

Hugo WoerdemanDrexel [email protected]

CP11

Recursive, Orthogonal, Radix-2, Stable Dct-DstAlgorithms and Applications

Discrete Fourier Transformation is engaged in image pro-cessing, signal processing, speech processing, feature ex-traction, convolution etc. In this talk, we address com-pletely recursive, radix-2, stable, and solely based discretecosine transformation (DCT) and discrete sine transfor-mation (DST) algorithms having sparse, orthogonal, ro-tation, rotation-reflection, and butterfly matrices. We alsopresent image compression results and signal transform de-signs based on the completely recursive DCT and DST al-gorithms having sparse and orthogonal factors.

Sirani M. PereraEmbry Riddle Aeronautical [email protected]

CP11

Structured Condition Numbers for the Solution ofParameterized Quasiseparable Linear Systems

We derive relative condition numbers for the solution ofquasiseparable linear systems with respect to relative per-turbations of the parameters in the quasiseparable and inthe Givens-vector representations of the coefficient matrixof the system. We compare these condition numbers withthe unstructured one and provide numerical experimentsshowing that the structured condition numbers we presentcan be small in situations where the unstructured one ishuge.

Kenet J. Pomes PortalUniversidad Carlos III de MadridDepartment of [email protected]

LA15 Abstracts 65

Froilan M. DopicoDepartment of MathematicsUniversidad Carlos III de [email protected]

CP11

Local Fourier Analysis of Pattern Structured Op-erators

Local Fourier Analysis (LFA) is a well known tool foranalysing and predicting the convergence behavior ofmultigrid methods. When first introduced, LFA requiredoperators with constant coefficients. This requirement hasbeen relaxed in different ways, e.g., to analyze the completetwo-grid method. We continue in this manner by extend-ing the applicability of LFA by introducing a frameworkfor analyzing pattern structured operators.

Hannah RittichUniversity of [email protected]



CP11

A Fast Structured Eigensolver for Toeplitz Matri-ces

Abstract not available at time of publication.

Xin YePurdue [email protected]

Jianlin XiaDepartment of MathematicsPurdue [email protected]

Raymond H. ChanThe Chinese Univ of Hong KongDepartment of [email protected]

CP12

Solving Dense Symmetric Indefinite Linear Sys-tems on GPU Accelerated Architectures

We present new implementations for dense symmetric in-definite factorizations on multicore processors acceleratedby a GPU. Though such algorithms are needed in manyscientific applications, obtaining high performance of thefactorization on the GPU is difficult because of the cost ofsymmetric pivoting. To improve performance, we exploredifferent techniques that reduce communication and syn-chronization between the CPU and GPU. We present per-formance results using Bunch-Kaufmann, Aasen and RBTmethods.

Marc Baboulin

University of Paris-Sud/[email protected]


Adrien RemyUniversity of [email protected]



CP12

A Parallel Divide-and-Conquer Algorithm forComputing the Moore-Penrose Inverses on Gpu

The massive parallelism of the platforms known as Com-pute Unified Device Architecture (CUDA) makes it pos-sible to achieve dramatic runtime reductions over a stan-dard CPU in many applications at a very affordable cost.However, traditional algorithms are serial in nature andcan’t take advantage of the multiprocessors that CUDAplatforms provide. In this paper, we present a divide-and-conquer algorithm for computing the Moore-Penrose in-verse A† of a general matrix A. Our algorithm consists oftwo phases. The first phase uses a parallel algorithm to or-thogonally reduce a general matrix A to a bi-diagonal ma-trix B while the second one computes the Moore-Penrose ofB using a divide-and-conquer approach. Then the Moore-Penrose inverse of A is finally constructed from the resultsof these two phases. Numerical tests show the dramaticspeedup of our new approach over the traditional algo-rithms.

Xuzhou ChenFitchburg State [email protected]

Jun JiKennesaw State [email protected]

CP12

Exploring Qr Factorization on Gpu for QuantumMonte Carlo Simulation

Evaluation of probability computed as determinant of adense matrix of wave functions is a computational kernelin QMCPack. This matrix undergoes a rank-one update ifthe event is accepted. Sherman-Morrison formula is usedto update the inverse of this matrix. The explicit inverse isrecomputed occasionally to maintain numerical stability.QR factorization maintains stability without refactoriza-tion. This effort explores low-rank updates to QR factor-ization on GPU to replace this computational kernel.

Eduardo F. D’AzevedoOak Ridge National Laboratory

66 LA15 Abstracts

Computer Science and Mathematics [email protected]

Paul Kent, Ying Wai LiOak Ridge National [email protected], [email protected]

CP12

Comparing Hybrid and Native GPU Accelerationfor Linear Algebra

Accelerating dense linear algebra using GPUs admits twomodels: hybrid CPU-GPU and GPU-only. The hybridmodel factors the panel on the CPU while updating thetrailing matrix on the GPU, concentrating the GPU onhigh-performance matrix multiplies. The GPU-only modelperforms the entire computation on the GPU, avoidingcostly data transfers to the CPU. We compare these twoapproaches for three QR-based algorithms: QR factoriza-tion, rank revealing QR, and reduction to Hessenberg.

Mark GatesComputer Science DepartmentUniversity of [email protected]

Azzam HaidarDepartment of Electrical Engineering and ComputerScienceUniversity of Tennessee, [email protected]


CP12

Efficient Eigensolver Algorithm on AcceleratorBased Architecture

The enormous gap between the high-performance capabil-ities of GPUs and the slow interconnect between themhas made the development of numerical software that isscalable across multiple GPUs extremely challenging. Wedescribe a successful methodology on how to address thechallenges -starting from our algorithm design, kernel opti-mization and tuning, to our programming model- in the de-velopment of a scalable high-performance symmetric eigen-value and singular value solver.


Piotr LuszczekUniversity of [email protected]

Stanimire TomovUniversity of Tennessee, [email protected]

Jack J. DongarraDepartment of Computer Science

The University of [email protected]

CP12

Batched Matrix Computations on Hardware Accel-erators Based on GPUs.

We will present techniques for small matrix computa-tions on GPUs and their use for energy efficient, high-performance solvers. Work on small problems delivers highperformance through improved data reuse. Many numeri-cal libraries and applications need this functionality furtherdeveloped. We describe the main factorizations LU,QR,and Cholesky for a set of small dense matrices in paral-lel. We achieve significant acceleration and reduced energyconsumption against other solutions. Our techniques areof interest to GPU application developers in general.


Ahmad AhmadUniversity of [email protected]



CP13

Recycling Krylov Subspace Methods for Sequencesof Linear Systems with An Application to LatticeQcd

Many problems in engineering, numerical simulations inphysics etc. require the solution of long sequences of slowlychanging linear systems. As an alternative to GCRO-DR,we propose an algorithm that uses an oblique projection,in the spirit of truly deflated GMRES, for deflating theeigenvalues of smallest magnitude. This oblique projectionrequires approximations to right and left eigenvectors, andwe propose a way of getting the approximations to the lefteigenvectors, without having to build a Krylov subspacewith respect to AH , which saves a lot of work. We will showthat the new algorithm is numerically comparable to theGCRO-DR algorithm and in addition, we further improveour results for the lattice QCD application by exploitingnon-trivial symmetries.

Nemanja BozovicUniversity of Wuppertal, [email protected]


Matthias BoltenUniversity of Wuppertal

LA15 Abstracts 67

Department of [email protected]

CP13

Generalized Jacobi Matrices and Band Algorithms

Jacobi matrices, i.e. symmetric tridiagonal matrices withpositive subdiagonal entries, represent thoroughly stud-ied objects connected to the Lanczos tridiagonalization,the Golub-Kahan bidiagonalization, the Gauss quadrature,etc. In this presentation, we study recently introduced ρ-wedge-shaped matrices that can be viewed as a generaliza-tion of Jacobi matrices. The definition is motivated by thestructure of output matrices in the band generalization ofthe Golub-Kahan bidiagonalization and the band Lanczosalgorithm.

Iveta HnetynkovaCharles University, [email protected]

Martin PlesingerTechnical University of [email protected]

CP13

Symmetric Inner-Iteration Preconditioning forRank-Deficient Least Squares Problems

Several steps of stationary iterative methods with a sym-metric splitting matrix serve as inner-iteration precondi-tioning. We give a necessary and sufficient condition suchthat the inner-iteration preconditioning matrix is definite,show that short recurrence type Krylov subspace methodspreconditioned by the inner iterations determine a leastsquares solution and the minimum-norm solution of linearsystems of equations, whose coefficient matrices may berank-deficient, and give bounds for these methods.

Keiichi MorikuniInstitute of Computer ScienceAcademy of Sciences of the Czech [email protected]

CP13

On Solving Linear Systems Using Adaptive Strate-gies for Block Lanczos Method

Block Methods based on conjugate directions improvearithmetic intensity and rate of convergence. When thesemethods are used for the resolution of algebraic linear sys-tems Ax = b (A symmetric positive definite) a drawbackis the need to find an adequate block size. In this work wediscuss and present results of several strategies for adap-tively updating the size of the block based on Ritz valuesand a threshold for determining when the rule applies.

Christian E. SchaererInstitute for Pure and Applied Mathematics - [email protected]

Pedro TorresPolytechnic SchoolNational University of [email protected]

Amit BhayaFederal University of Rio de Janeiro

[email protected]

CP13

On Rotations and Approximate Rational KrylovSubspaces

Rational Krylov subspaces are useful in many applications.These subspaces are built by not only matrix vector prod-ucts but also by inverses of a matrix times a vector. Thisresults in significant faster convergence, but on the otherhand often creates numerical issues. It is shown that ratio-nal Krylov subspaces under some assumptions are retrievedwithout any explicit inversion or system solves involved.Instead the necessary computations are done implicitly us-ing information from an enlarged Krylov subspace. In thistalk the generic building blocks underlying rational Krylovsubspaces: rotations, twisted QR-factorizations, turnovers,fusions, are introduced. We will illustrate how to shrinka large Krylov subspace by unitary similarity transforma-tions to a smaller subspace without -if all goes well- essen-tial data loss. Numerical experiments support our claimsthat this approximation can be very good and as such canlead to time savings when solving particular ODE’s.

Raf VandebrilKatholieke Universiteit LeuvenDepartment of Computer [email protected]

Thomas MachTU [email protected]

Miroslav PranicFaculty of ScienceUniversity of Banja [email protected]

CP13

Improving Thick-Restarting Lanczos Method bySubspace Optimization For Large Sparse Eigen-value Problems

The Thick-Restart Lanczos method is an effective methodfor large sparse eigenvalue problems. In this work, we pro-pose and study a subspace optimization technique to ac-celerate this method. The proposed method augments theKrylov subspace with a certain number of vectors from theprevious restart cycle and then apply the Rayleigh-Ritzprocedure in the new subspace. Numerical experimentsshow that our method can converges two or three timesfaster than the Lanczos method.

Lingfei WuDepartment of Computer ScienceCollege of William & [email protected]

Andreas StathopoulosCollege of William & MaryDepartment of Computer Science

68 LA15 Abstracts

[email protected]

CP14

Ca-Ilu Preconditioner

In this talk, we present a new parallel preconditioner calledCA-ILU. It performs ILU(k) factorization in parallel and isapplied as preconditioner without communication. BlockJacobi and Restrictive Additive Schwartz preconditionersare compared to this new one. Numerical results showthat CA-ILU(0) outperforms BJacobi and is close to RASin term of iterations whereas CA-ILU, with complete LUon each block, does for both of them but uses a lot of mem-ory. Thus CA-ILU(k) takes advantages of both in term ofmemory consumption and iterations.

Sebastien [email protected]

Laura [email protected]

Sophie M. MoufawadLRI - INRIA Saclay, [email protected]

CP14

The Truncation of Low-Rank Preconditioner Up-dates with Applications

Rather than recomputing preconditioners for a sequence oflinear systems, it is often cheaper to update the precondi-tioner. Here we consider low-rank preconditioner updatesand truncating those updates to keep them cheap. Weconsider applications from nonlinear PDEs and ElectronicStructure Calculations.


Ming LiDept. MathematicsVirginia Polytechnic Institute and State [email protected]

Vishwas RaoDepartment of Computer ScienceVirginia [email protected]

CP14

Interpretation of Algebraic Preconditioning AsTransformation of the Discretization Basis

In numerical solution of PDEs it is convenient to con-sider discretization and preconditioning closely linked to-gether. As suggested by several techniques used through-out decades, preconditioning can always be linked withtransformation of the discretization basis. It has beenshown recently that effective algebraic preconditioners canbe interpreted as the transformation of the discretizationbasis and, simultaneously, as the change of inner prod-uct on corresponding Hilbert space. This contribution will

present recent work in this direction.

Toma GergelitsFaculty of Mathematics and Physics,Charles University in [email protected]

Jan PapezCharles University in PragueDept Numerical [email protected]

Zdenek StrakosFaculty of Mathematics and Physics,Charles University in [email protected]

CP14

Reusing and Recycling Preconditioners for Se-quences of Matrices

For sequences of related linear systems, computing a newpreconditioner for each system can be expensive. We canreuse a fixed preconditioner, but as the matrix changesthis approach can become ineffective. We can alterna-tively apply cheap updates to the preconditioner. Thispresentation discusses the benefits of reusing and recy-cling preconditioners and proposes an update scheme werefer to as a Sparse Approximate Map. Applications in-clude the QMCmethod, model reduction, tomography, andHelmholtz-type problems.

Arielle K. Grim McnallyVirginia TechEric de [email protected]


CP14

Lu Preconditioners for Non-Symmetric SaddlePoint Matrices with Application to the Incompress-ible Navier-Stokes Equations

The talk focuses on threshold incomplete LU factorizationsfor non-symmetric saddle point matrices. The research ismotivated by the numerical solution of the linearized in-compressible Navier–Stokes equations. The resulting pre-conditioners are used to accelerate the convergence of aKrylov subspace method applied to finite element dis-cretizations of fluid dynamics problems in three space di-mensions. We shall discuss the stability of the factoriza-tion for generalized saddle point matrices and consider anextension for non-symmetric matrices of the Tismenetsky–Kaporin incomplete factorization. We demonstrate that innumerically challenging cases of higher Reynolds numberflows one benefits from using this two-parameter modifica-tion of a standard threshold ILU preconditioner. The per-formance of the ILU preconditioners is studied numericallyfor a wide range of flow and discretization parameters, andthe efficiency of the approach is shown if threshold parame-ters are chosen suitably. The practical utility of the methodis further demonstrated for the haemodynamic problem ofsimulating a blood flow in a right coronary artery of a realpatient.

Maxium Olshanskii

LA15 Abstracts 69

University of [email protected]

CP14

Right Preconditioned MINRES using Eisenstat-SSOR for Positive Semidefinite Systems

Consider solving symmetric positive semidefinite systems.We use MINRES because CG does not necessarily convergefor inconsistent systems. We use right preconditioning be-cause it is easier to preserve the equivalence of the prob-lem for inconsistent systems. We apply Eisenstats trick toeconomize the right SSOR preconditioned MINRES. Nu-merical experiments show that the method is more efficientand robust compared to no-preconditioned and right scal-ing preconditioned MINRES.

Kota SugiharaDepartment of InformaticsSOKENDAI (The Graduate University for AdvancedStudies)[email protected]

Ken HayamiNational Institute of [email protected]

CP15

Ky Fan Theorem Applied to Randic Energy

Let G be an undirected simple graph of order n with vertexset V = {v1, . . . , vn}. Let di be the degree of the vertexvi. The Randic matrix R = (ri,j) of G is the square ma-

trix of order n whose (i, j)-entry is equal to 1/√

di dj ifthe vertices vi and vj are adjacent, and zero otherwise.The Randic energy is the sum of the absolute values of theeigenvalues of R. Let X, Y, and Z be matrices, such thatX+Y = Z. Ky Fan established an inequality between thesum of singular values of X, Y, and Z. We apply this in-equality to obtain bounds on Randic energy. Some resultsare presented considering the energy of a symmetric parti-tioned matrix, as well as an application to the coalescenceof graphs.

Enide C. AndradeCenter for Research & Development in Mathematics andApplications University of [email protected]

Ivan GutmanFaculty of Science, University of Kragujevac, Serbiaand State University of Novi Pazar, [email protected]

Marıa Robbiano, Bernardo MartınDepartamento de Matematicas, Universidad Catolicadel Norte, Antofagasta, [email protected], [email protected]

CP15

The Linear Transformation that Relates theCanonical and Coefficient Embeddings of Ideals inCyclotomic Integer Rings

The geometric embedding of an ideal in the algebraic inte-ger ring of some number field is called an ideal lattice. Ideallattices and the shortest vector problem (SVP) are at the

core of many recent developments in lattice-based cryptog-raphy. We utilize the matrix of the linear transformationthat relates two commonly used geometric embeddings toprovide novel results concerning the equivalence of the SVPin these ideal lattices arising from rings of cyclotomic inte-gers.

Scott C. BatsonNorth Carolina State [email protected]

CP15

Coupled Sylvester-Type Matrix Equations andBlock Diagonalization

We prove Roth’s type theorems for systems of matrixequations including an arbitrary mix of Sylvester and -Sylvester equations, in which also the transpose or con-jugate transpose of the unknown matrices appear. In fullgenerality, we derive consistency conditions by proving thatsuch a system has a solution if and only if the associatedset of 2 × 2 block matrix representations of the equationsare block diagonalizable by (linked) equivalence transfor-mations. Various applications leading to several particularcases have already been investigated in the literature, somerecently and some long ago. Solvability of these cases followimmediately from our general consistency theory. We alsoshow how to apply our main result to systems of Stein-typematrix equations.

Andrii DmytryshynUmea UniversityDepartment of Computing [email protected]

Bo T. KagstromUmea UniversityComputing Science and [email protected]

CP15

Generalized Inverses of Copositive Matrices, Self-Conditional Positive Semidefinite Matrices and In-heritance Properties

A symmetric matrix A ∈ Rn×n is called copositive ifxTAx ≥ 0 for all x ≥ 0, where the ordering of vectors is thestandard component wise ordering. A ∈ Rn×n is said to becopositive of order n−1 if all the (n−1)× (n−1) principalminors of A are copositive. Let A ∈ Rn×n be a copositivematrix of order n − 1. A rather well known result statesthat A is not copositive if and only if A is invertible andthat all the entries of the inverse of A are nonpositive. Re-cently, we have proved a version of this result for singularmatrices, using the group generalized inverse of A. Amongother things, this talk will showcase this result.

Sivakumar K.C.Indian Institute of Technology [email protected]

Kavita BishtIndian Institute of Technology Madras, [email protected]

Ravindran G.Indian Statistical Institute, Chennai

70 LA15 Abstracts

[email protected]

CP15

The Principal Rank Characteristic Sequence andthe Enhanced Principal Rank Characteristic Se-quence

The enhanced principal rank characteristic sequence of ann× n symmetric matrix B is 12 · · · n, where k is A (re-spectively, N) if all (respectively, none) the principal minorsof order k are nonzero; if some but not all are nonzero,then k = S. Results regarding the attainability of cer-tain classes of sequences are introduced, and restrictionsfor some subsequences to appear in an attainable sequenceare discussed.

Xavier Martinez-RiveraIowa State [email protected]

CP15

A Characterization of Minimal Surfaces in theLorentz Group L3

In this talk we establish the equation for the Gaussian Cur-vature of a minimal surface in the Lorentz Group L3. Us-ing the Gauss equation we prove that minimal surfaces inL3 with constant contact angle have non-positive Gaus-sian curvature. Also, we provide a congruence theorem forminimal surfaces immersed in the Lorentz space L3.

Rodrigo R. MontesFederal University of Parana [email protected]

CP16

Linear Algebra Provides a Basis for ElasticityWithout Stress Or Strain

Linear algebra can describe the energy of deformation ofhyper-elastic materials in terms of a single deformation gra-dient tensor without the need to define stress or strain.Positions of points replace strain and force replaces stress.The model is appropriate for both infinitesimal and finitedeformations, isotropic and anisotropic materials, quasi-static and dynamic elastic responses.

Humphrey H. HardyPiedmont [email protected]

CP16

Bounds on Algebraic, Discretization, and Total Nu-merical Approximation Errors for Linear DiffusionPDEs

We present a posteriori error estimates that allow to givea guaranteed bound on the algebraic and discretizationerrors for conforming finite element approximations ofthe Laplace equation. This extends the results of [Ern,Vohralık 2013] which allow to estimate the different compo-nents of the error and to guarantee an upper bound on thetotal approximation error. The efficiency of the bounds isdiscussed and extensive numerical results for higher-orderfinite element approximations are presented.

Jan PapeCharles University in Prague

[email protected]

Martin VohralıkINRIA [email protected]

CP16

Iterative Method for Mildly OverdeterminedLeast-Squares Problems with a Helmholtz Block

We present a matrix-free iterative method to solve least-squares problems, which contain a Helmholtz discretiza-tion. This type of problems arises in quadratic-penaltymethods for PDE-constrained optimization. The problemsof interest have multiple-right hand sides. The proposedstrategy includes preconditioning to induce and exploit anidentity + low-rank system matrix structure. Random-ization and subsampling of one of the blocks in the least-squares problem is used to reduce the computational cost.

Bas [email protected]

Chen GreifDepartment of Computer ScienceThe University of British [email protected]

Felix J. HerrmannSeismic Laboratory for Imaging and ModelingThe University of British [email protected]

CP16

On the Backward Stability of Chebyshev Rootfind-ing Via Colleague Matrix Eigenvalues

In this work, we analyze the backward stability of the poly-nomial rootfinding problem solved with colleague matrices.In other words, given a scalar polynomial p(x) or a ma-trix polynomial P (x) expressed in the Chebyshev basis,the question is to determine whether the whole set of com-puted eigenvalues of the colleague matrix, obtained with abackward stable algorithm are the set of roots of a nearbypolynomial or not. We derive a first order backward erroranalysis of the polynomial rootfinding algorithm using col-league matrices adapting the geometric arguments in [A.Edelman and H. Murakami, Polynomial roots for compan-ion matrix eigenvalues, Math. Comp., 1995] to the Cheby-shev basis. We show that, if the norm of the coefficients ofthe polynomial are bounded by a moderate number, com-puting its roots via the eigenvalues of its colleague matrixusing a backward stable eigenvalue algorithm is backwardstable.

Javier Perez lvaroSchool of Mathematics, The University of [email protected]

Vanni NoferiniUniversity of [email protected]

CP16

Companion Matrices of Hermite-Birkhoff Inter-

LA15 Abstracts 71

polants

In computational mathematics, we sometimes have to dealwith polynomials that are not represented in the monomialbasis. They are given by values of a function and/or itsderivatives of various orders at some points called nodes orsample points. This case often leads to solving a Hermite-Birkhoff interpolation problem. The present work unifiesvarious aspects of the Newton, Hermite and Birkhoff bases.We study the companion pencils as an efficient tool forcomputing roots of such polynomials.

Nikta ShayanfarTechnische Universitat [email protected]

Amir AmiraslaniSTEM Department, University of Hawaii-Maui [email protected]

CP16

Fixing Gauge and Rank Deficiency

Gauge fixing is a powerful concept for handling redundantdegrees of freedom in Lagrangian mechanics. The challengeis finding logically consistent and mathematically tractableprocedures for fixing the gauge. Working in the context ofgradients of scalar potentials demonstrated how fixing thegauge also fixes rank deficient linear systems by restoringthem to full rank. Also presented is an analytic derivationof gauge conditions and simplistic geometric methods foridentifying gauge conditions beyond one dimension.

Daniel TopaLos Alamos National LaboratoryU.S. Army Engineer Research and Development [email protected]

Pedro EmbidUniversity of New MexicoDepartment of Mathematics and [email protected]

CP17

Matrix-Free Krylov Subspace Methods for Solvinga Riemannian Newton Equation

We consider an optimization problem on the Stiefel man-ifold which is equivalent to the singular value decomposi-tion. The Riemannian Newton method for the problem hasbeen proposed, but the Newton equation is expressed bya system of matrix equations which is difficult to solve di-rectly. In this talk, we apply matrix-free Krylov subspacemethods to a sparse linear system into which the Newtonequation is rewritten. Numerical experiments show the ef-fectiveness of our proposed method.

Kensuke AiharaDepartment of Mathematical Information Science,Tokyo University of [email protected]

Hiroyuki SatoDepartment of Management Science,Tokyo University of Science

[email protected]

CP17

Approximating the Cardinality of a Vector to Han-dle Cardinality Constraints in Optimization

An obvious, often loose, lower bound of cardinality of avector, i.e. the number of nonzero elements, is the squaredratio of norm one over Euclidian norm. From this bound,we derive an estimate of the cardinality by maximizing aRayleigh ratio involving a positive definite matrix with asimple structure, leading to an analytical formula for esti-mating the cardinality. Numerical experiments show esti-mation errors systematically smaller than those obtainedwith the initial bound.

Agnes BialeckiEDF R&[email protected]

Laurent El GhaouiUC [email protected]

Riadh ZorgatiEDF R&DDepartment [email protected]

CP17

Primal and Dual Algorithms for the Ball CoveringProblem

Consider the convex optimization problem of finding a Eu-clidean ball of minimum radius that covers m Euclideanballs with fixed centers and radii in �n. We propose twodirectional search methods in which search paths (rays ortwo-dimensional hyperbolas in �n) are constructed by in-tersecting bisectors (hyperboloids in �n) and the step sizeis determined explicitly. We prove interesting propertiesabout pairwise intersection of n-dimensional hyperboloidsto achieve finite convergence.

Akshay Gupte, Lin DearingDepartment of Mathematical SciencesClemson [email protected], [email protected]

CP17

Finding the Nearest Valid Covariance Matrix: a FxMarket Case

We consider the problem of finding a valid covariance ma-trix in the FX market given an initial non-PSD estimate ofsuch a matrix. The standard no-arbitrage assumption im-plies additional linear constraints on such matrices, whichautomatically makes them singular. As a result, such aproblem is not well-posed while the PSD-solution is notstrictly feasible. In order to deal with this issue, we de-scribed a low-dimensional face of the PSD cone that con-tains the feasible set. After projecting the initial problemonto this face, we come out with a reduced problem, whichturns out to be well posed and of a smaller scale. We showthat after solving the reduced problem the solution to theinitial problem can be uniquely recovered in one step.

Aleksei MinabutdinovNational Research University Higher School of Economics

72 LA15 Abstracts

[email protected]

CP17

Convergence of Newton Iterations for Order-Convex Matrix Functions

In stochastic problems and some physical problems, itis important that solving the elementwise minimal non-negative solvent of a nonlinear matrix equation. A lotof such equations have properties of differentiable order-convex functions. Using the properties of differentiableorder-convex functions, We will show that the Newton it-erations for the equations are well-defined and convergeto the elementwise minimal nonnegative solvent. Finally,it is given numerical experiments of the iterations for theequations.

Sang-Hyup SeoDepartment of MathematicsPusan National [email protected]

Jong-Hyeon SeoBasic [email protected]

Hyun-Min KimDepartment of mathematicsPusan National [email protected]

CP17

Modulus-Type Inner Outer Iterative Methods forNonnegative Constrained Least Squares Problems

For the solution of large sparse nonnegative constrainedleast squares (NNLS) problems, a new iterative method isproposed by using conjugate gradient least squares methodfor inner iterations and the modulus iterative method in theouter iterations for solving linear complementarity problemresulting from Karush-Kuhn-Tucker conditions of NNLS.Convergence analysis is presented and numerical exper-iments show the proposed methods outperform the pro-jected gradient methods.

Ning Zheng

SOKENDAI (The Graduate University for AdvancedStudies)[email protected]


Junfeng YinDepartment of Mathematics, Tongji University, [email protected]

CP18

Tropical Bounds for the Eigenvalues of Block Struc-tured Matrices

We establish a log-majorization inequality, which relatesthe moduli of the eigenvalues of a block structured ma-trix with the tropical eigenvalues of the matrix obtainedby replacing every block entry of the original matrix by

its norm. This inequality involves combinatorial constantsdepending on the size and pattern of the matrix. Its proofrelies on diagonal scalings, constructed from the optimaldual variables of a parametric optimal assignment prob-lem.

Marianne Akian, Stephane GaubertINRIA and CMAP, Ecole [email protected], [email protected]

Andrea MarchesiniINRIA andCMAP, Ecole [email protected]

CP18

Coherency Preservers

We will present a description of continuous coherency pre-servers on hermitian matrices. Applications in mathemat-ical physics will be discussed.

Peter SemrlUniversity of [email protected]

CP18

A Connection Between Comrade Matrices andReachability Matrices

In this paper is explained a connection between comradematrices and reachability matrices. This work is doneby orthogonal polynomials, companion matrices and theirconnections with reachability matrices. Firstly an alge-braic structure is introduced by these matrices. Then someproperties of these matrices are described such as deter-minant, eigenvalues and eigenvectors. The fields are re-stricted to that F = R or C, namely real and complex num-bers respectively. Characteristic polynomial of the com-panion matrix C is related by

p(x) = xn − cn−1xn−1 − . . .− c1x− c0 ∈ F[x]. (1)

Let A ∈ Fn×n, b ∈ Fn, the matrix

R(A,b) = [b, Ab, . . . , An−1b] ∈ Fn×n,

is the reachability matrix of the pair (A, b). Also, A is thecomrade matrix similar as C by an invertible matrix Qn

i.e.A = QnCQ−1

n .

Easily, it is shown

R(C,b) = Q−1n R(A,Qnb).

The generating polynomial is defined of b =[b0, b1, . . . , bn−1]

T ∈ Fn×1 by b(x) =∑n−1

k=0 bkxk then

R(C,b) =

n−1∑k=0

bkR(C,ek) =

n−1∑k=0

bkCk = b(C) = Q−1

n R(A,Qnb)

(2)where ek stands for the (k+1)th unit vector in the standardbasis of Fn×1. Now, an algebraic structure is introduced of

S(A) = {Q−1n R(A,Qnb)|b ∈ F

n×1},that is proven S(A) � F[x]/ ≺ d(x) . Determinants,eigenvalues and eigenvectors are described of this connec-tion. Also the colleague matrices and Chebyshev polyno-mials are investigated of this connection.

Maryam Shams Solary

LA15 Abstracts 73

Payame Noor [email protected]

CP18

Maximal Lower Bounds in Loewner Order

We show that the set of maximal lower bounds of two sym-metric matrices with respect to Loewner order can be iden-tified to the quotient set O(p, q)/(O(p)×O(q)). Here, (p, q)denotes the inertia of the difference of the two matrices,O(p) is the p-th orthogonal group, and O(p, q) is the indef-inite orthogonal group arising from a quadratic form withinertia (p, q). We discuss the application of this result tothe synthesis of ellipsoidal invariants of hybrid dynamicalsystems.

Nikolas StottINRIA - CMAP, Ecole [email protected]

Xavier AllamigeonINRIA and CMAP, Ecole Polytechnique, [email protected]

Stephane GaubertINRIA-Saclay & CMAPEcole [email protected]

CP19

Coupled Preconditioners for the IncompressibleNavier-Stokes Equations

After linearization and discretization of the incompressibleNavier Stokes equations one has to solve block-structuredindefinite linear systems. In this talk we consider SIMPLE-type preconditioners as proposed in [1]. It appears thatthis type of preconditioners can lead to a considerable ac-celeration of the solver and are weakly depending on thenumber of grid points and Reynolds number. Recently, theaugmented Lagrangian method becomes popular [2]. Themethod is more expensive per iteration but independent ofthe grid size and Reynolds number for academic problems.The third preconditioner is the so-called ’grad-div’ precon-ditioner. We also define modified versions of the SIMPLEand ’grad-div’ preconditioners. In this talk we investigatetheir performance for academic problems [3].

Xin HeDelft University of Technology, [email protected]

Kees VuikDelft University of [email protected]

CP19

On Some Algebraic Issues of Domain Decomposi-tion Approaches

Various domain decomposition approaches are consideredwith regard to parallel solution of very large sparse sys-tems of linear algebraic equations arising from some fi-nite element or finite volume approximations of compli-cated boundary value problems on non-structured grids.The proposed methods include automatic construction ofthe balanced subdomains with or without a parametrizedoverlapping, different type of interface conditions on in-

ternal boundaries of subdomains, augmented iterativeadditive Schwarz algorithm in Krylov subspaces, multi-preconditioning techniques and aggregation. The work fo-cuses on experimental analysis of performance of the afore-mentioned methods. The implementation of the proposedapproaches was done on the basis of hybrid parallel pro-gramming and taking into account sparse matrix structureand with using the compressed sparse row format of thematrix representation. The experiments have been per-formed for a representative set of test problems.

Valery P. Il’inInstitute of Computational Mathematicsand Mathematical Geophysics SB [email protected]

CP19

Robust Incomplete Factorization Preconditionerwith Mixed-Precision for Parallel Finite ElementAnalysis

In our previous study, localized IRIF(0) and IRIF(1) pre-conditioners were proposed to solve the equation systemin the frame work of the parallel FEM. This research aimsto apply the mixed precision algorithm to this precondi-tioning method. As the numerical results, matrices givenfrom shell structures were solved. It has been shown thatthe mixed-precision IRIF(0) and IRIF(1) are effective inreducing the processing time by 40% while preserving thenumber of iterations.

Naoki Morita, Gaku Hashimoto, Hiroshi OkudaGraduate School of Frontier SciencesThe University of [email protected], [email protected], [email protected]

CP19

Efficient Simulation of Fluid-Structure InteractionsUsing Fast Multipole Method

Regularized Stokes formulation has been shown to be veryeffective at modeling fluid-structure interactions when thefluid is highly viscous. However, its computational costgrows quadratically with the number of particles immersedin the fluid. We demonstrate how kernel-independent fastmultipole method can be applied to significantly improvethe efficiency of this method, and present numerical resultsfor simulating the dynamics of a large number of elasticrods immersed in 3D stokes flows.

Minghao W. Rostami, Sarah D. OlsonWorcester Polytechnic [email protected], [email protected]

CP19

Block Preconditioners for An Incompressible Mag-netohydrodynamics Problem

We focus on preconditioning techniques for a mixed fi-nite element discretization of an incompressible magneto-hydrodynamics (MHD) problem. Upon discretization andlinearization, a 4x4 non-symmetric block-structured linearsystem needs to be (repetitively) solved. One of the prin-cipal challenges is the presence of a skew-symmetric termthat couples the fluid velocity with the electric field. Ourproposed technique exploits the block structure of the un-derlying linear system, utilizing and combining effectivepreconditioners for the mixed Maxwell and Navier-Stokes

74 LA15 Abstracts

subproblems. The preconditioner is based on dual and pri-mal Schur complement approximations to yield a scalablesolution method. Large scale numerical results demon-strate the effectiveness of our approach.

Michael P. WathenThe University of British [email protected]

Chen GreifDepartment of Computer ScienceThe University of British [email protected]

Dominik SchotzauThe University of British [email protected]

CP19

The WR-HSS Method and Its Subspace Accelera-tion for the Unsteady Elliptic Problem

We consider the numerical methods for the unsteady ellip-tic problem with Dirichlet boundary condition. Taking intoaccount the idea of Hermitian/skew-Hermitian (HS) split-ting, a class of waveform relaxation methods is establishedbased on the HS splitting of the linear differential operatorin the corresponding differential equations, i.e., the WR-HSS method. Furthermore, the subspace acceleration ofthe WR-HSS method is also considered. Finally, the theo-retical and numerical behaviors of the above methods arecarefully analyzed.

XI YangDepartment of MathematicsNanjing University of Aeronautics and [email protected]

MS1

Functions of Matrices with Kronecker Sum Struc-ture

Functions of large and sparse matrices with tensor struc-ture arise in a number of scientific and engineering appli-cations. In this talk we focus on functions of matrices inKronecker sum form. After discussing some recent resultson the decay behavior of such matrix functions, we willdescribe efficient Krylov subspace techniques for evaluat-ing a function of a matrix times a vector. The results ofnumerical experiments will be discussed.


Valeria SimonciniUniversita’ di [email protected]

MS1

First-Order Riemannian Optimization Techniquesfor the Karcher Mean

The Karcher mean is a matrix function of several positivedefinite matrices which represents their geometric mean. Itis the unique minimum over the set of positive definite ma-trices of a real function related to a Riemannian structure

of this set. We explore first-order optimization algorithms,showing that exploiting the geometric structure leads toa considerable acceleration. Moreover, we present a newstrategy which appears to be faster than the existing algo-rithms.

Bruno IannazzoUniversita‘ di Perugia, [email protected]

Margherita PorcelliUniversity of [email protected]

MS1

Error Estimation in Krylov Subspace Methods forMatrix Functions

Using the Lanczos method to approximate f(A)b, there isno straightforward way to measure the error of the currentiterate. Therefore, different error estimators have been sug-gested, all of them specific to certain classes of functions.We add a technique to compute error bounds for Stielt-jes functions, using an integral representation of the error.These bounds can be computed essentially for free, withcost independent of the iteration number and the dimen-sion of A.

Marcel SchweitzerBergische Universitat [email protected]


MS1

A High Performance Algorithm for the Matrix SignFunction

The talk will described a blocked (partitioned) algorithmfor the matrix sign function, along with performance re-sults that compare the existing algorithm to the new one.The algorithm is a blocked version of the Higham’s stabi-lized version of the Parlett-Schur matrix-sign algorithm fortriangular matrices.

Sivan A. ToledoTel Aviv [email protected]

MS2

Mpgmres-Sh : Multipreconditioned GMRES forShifted Systems

We propose using the Multipreconditioned GeneralizedMinimal Residual (MPGMRES) method to solve shiftedlinear systems (A+σjI)xj = b for j = 1, . . . , Nσ with mul-tiple shift-and-invert preconditioners. The multiprecondi-tioned space is obtained by applying the preconditionersto all search directions and searching for a minimum normsolution over a larger subspace. We show that for thie par-ticular problem this space grows linearly and discuss ourimplementation on systems arising from hydraulic tomog-raphy and matrix function computations.

Tania Bakhos

LA15 Abstracts 75

Institute for Computational and MathematicalEngineeringStanford [email protected]

Peter K. KitanidisDept. of Civil and Environmental EngineeringStanford [email protected]

Scott LadenheimTemple [email protected]

Arvind SaibabaDepartment of Electrical and Computer EngineeringTufts [email protected]

Daniel B. SzyldTemple UniversityDepartment of [email protected]

MS2

Nested Krylov Methods for Shifted Linear Systems

Most algorithms for the simultaneous solution of shiftedlinear systems make use of the shift-invariance property ofthe underlying Krylov spaces. This particular comes intoplay when preconditioning is taken into account. We pro-pose a new iterative framework for the solution of shiftedsystems that uses an inner multi-shift Krylov method asa preconditioner within a flexible outer Krylov method.Shift-invariance is preserved if the inner method yieldscollinear residuals.

Manuel BaumannDelft University of TechnologyFaculty of Electrical Engineering, Mathematics [email protected]

Martin B. van GijzenDelft University of [email protected]

MS2

One Matrix, Several Right Hand Sides: EfficientAlternatives to Block Krylov Subspace Methods

Matrix-block vector products are more efficient than a se-quence of matrix-vector products on current architectures.This is why methods for linear systems with several, ssay, right hand sides gained renewed interest recently. Weinvestigate short recurrence alternatives to known blockKrylov subspace methods and, in particular, variants ofBiCG and QMR which need just one (not s) multiplica-tions with the transposed matrix.


Somaiyeeh RashediDepartment of Mathematical Sciences, University ofTabriz, 5

s rashedi [email protected]

MS2

Iterative Methods for Solving Shifted Linear Sys-tems Built Upon a Block Matrix-vector Product

We discuss methods for simultaneous solving a family ofshifted linear systems of the form

(A+ σjI)xσj = bσj j = 1, 2, . . . , L with A ∈ Cn×n.

We explore methods built upon Krylov subspace iterationswhich employ a block matrix-vector product as their coreoperation. We demonstrate that by carefully designingthese block-iterative methods, we can overcome some re-strictions present in many existing techniques, i.e., the re-quirement that right-hand sides be collinear, incompatibil-ity with many preconditioners, and difficulty of integrationwith subspace recycling techniques.

Kirk M. SoodhalterTemple UniversityDepartment of [email protected]

MS3

Linearizations of Hermitian Matrix PolynomialsPreserving the Sign Characteristic

The sign characteristic of a Hermitian matrix polynomialP (λ) is a set of signs attached to the real eigenvaluesof P (λ), which is crucial for determining the behavior ofsystems described by Hermitian matrix polynomials. Inthis talk we present a characterization of the Hermitianstrong linearizations that preserve the sign characteristicof a given Hermitian matrix polynomial and identify sev-eral families of such linearizations that can be easily con-structed from the coefficients of the matrix polynomial.

Maria BuenoDepartment of MathematicsUniversity of California, Santa [email protected]


Susana FurtadoFaculdade de Economia do [email protected]

MS3

Low Rank Perturbations of Canonical Forms

In this talk, we will review some know results that describethe change of the following canonical forms under low rankperturbations:

• The Jordan canonical form of a square matrix.

• The Weierstrass canonical form of a regular matrixpencil.

• The Kronecker canonical of a singular matrix pencilwithout full rank.

• The Smith form of a regular matrix polynomial.

76 LA15 Abstracts

We will also present some recent results on the change ofthe Weierstrass canonical form of regular matrix pencilsunder low rank perturbations.

Fernando De Teran, Froilan M. DopicoDepartment of MathematicsUniversidad Carlos III de [email protected], [email protected]

Julio MoroUniversidad Carlos III de MadridDepartamento de [email protected]

MS3

Matrix Polynomials in Non-Standard Form

Matrix polynomials P (λ) and their associated eigenprob-lems are fundamental for a variety of applications. Cer-tainly the standard (and apparently most natural) way toexpress such a polynomial has been

P (λ) = λkAk + λk−1Ak−1 + · · ·+ λA1 +A0 ,

where Ai ∈ Fm×n. However, it is becoming increasinglyimportant to be able to work directly and effectively withpolynomials in the non-standard form

Q(λ) = φk(λ)Ak+φk−1(λ)Ak−1+· · ·+φ1(λ)A1+φ0(λ)A0 ,

where {φi(λ)}ki=0 is some other basis for the space of allscalar polynomials of degree at most k. This talk will de-scribe some new approaches to the systematic construc-tion of families of linearizations for matrix polynomials likeQ(λ), with emphasis on the classical bases associated withthe names Newton, Hermite, Bernstein, and Lagrange.


MS3

A Diagonal Plus Low Rank Family of Linearizationsfor Matrix Polynomials

We present a family of linearizations for an m × m ma-trix polynomial P (x) =

∑di=0 Pix

i that have the form

A(x) = D(x) + UV t with D(x) diagonal. The diagonalterm D(x) is a dm× dm matrix and U and V are dm×m.Moreover, D(x) can be chosen almost arbitrarily and U andV can be computed from P (x) and D(x). We show thatappropriate choices of D(x) lead to linearizations with verygood numerical properties. We provide a cheap strategy todetermine good choices for D(x).


Leonardo RobolScuola Normale [email protected]

MS4

Undergraduate Research in Linear Algebra 4


Charles Johnson

William & [email protected]

MS4

Student Research on Infinite Toeplitz Matrices

A Toeplitz matrix A has constant entries down each diag-onal. So the whole matrix is determined by those numbersak. Then those numbers are the coefficients in the func-tion a(θ) = Σake

ikθ. This symbol function is the super-convenient way to study the infinite matrix. The studentproject started by factoring a(θ) into lower times upper(negative and positive powers), leading to A = LU . (HugoWoerdeman has studied the much harder 2-level problemwhen a(θ, φ).) Then the student Liang Wang moved to theRadon Transform Matrix – with neat results that are stillto be published.

Gil [email protected]

MS4

Research in Linear Algebra with Undergraduates

I will discuss the benefits and difficulties and general is-sues in working with undergraduate students on researchprojects in Linear Algebra. I will share results fromprojects with students from the past few years, includinga Sinkhorn-Knopp fixed point problem

x =(AT (A x )(−1)

)(−1)

(where for x = (x1, . . . , xn), x(−1) = (1/x1, . . . , 1/xn)), an

approximation of derivatives using the LU factorization ofVandermonde Matrices, and mosaicking with digital im-ages.

David M. StrongDepartment of MathematicsPepperdine [email protected]

MS4

Undergraduate Research Projects in Linear Alge-bra

Linear Algebra lends itself well for undergraduate researchprojects as with one or two terms of Linear Algebra classesa variety of research projects are accessible. In addition,problems can often be explored by numerical and symboliccomputations. In this talks we discuss some successfulprojects from the past, some ongoing ones, as well as somefuture possibilities. The focus will be on matrix completionproblems and questions regarding determinantal represen-tations of multivariable polynomials.

Hugo J. WoerdemanDrexel Universityhugo @ math.drexel.edu

MS5

Exploiting Sparsity in Parallel Sparse Matrix-Matrix Multiplication

In this talk, we discuss graph and hypergraph partitioningbased models and methods proposed to exploit the spar-sity of SpGEMM computations for achieving efficiency on

LA15 Abstracts 77

distributed memory architectures for different paralleliza-tion schemes. All models rely on one-dimensional parti-tioning of both input matrices. In these models, parti-tioning constraint encodes balancing computational loadsof processors, whereas partitioning objective of minimiz-ing cutsize corresponds to increasing locality within eachprocessor and reducing volume of interprocessor communi-cation. We further address latency issues in these models.The performance of these models will be discussed throughan MPI-based parallel SpGEMM library developed for thispurpose.

Kadir Akbudak, Oguz Selvitopi, Cevdet AykanatBilkent [email protected], [email protected],[email protected]

MS5

Generalized Sparse Matrix-Matrix Multiplicationand Its Use in Parallel Graph Algorithms

We present experimental results for optimized implemen-tations of various (1D, 2D, 3D) parallel SpGEMM algo-rithms and their applications to graph problems such astriangle counting, betweenness centrality, graph contract-ing, and Markov clustering. We quantify the effects ofin-node multithreading and the implementation trade-offsinvolved. We will also present a new primitive, maskedSpGEMM, that avoids communication when the outputstructure (or an upper bound on it) is known beforehand.Masked SpGEMM has applications to matrix-based trian-gle counting and enumeration.

Ariful AzadPurdue [email protected]

Aydin BulucLawrence Berkeley National [email protected]

John R. GilbertDept of Computer ScienceUniversity of California, Santa [email protected]

MS5

Hypergraph Partitioning for Sparse Matrix-MatrixMultiplication

We model sparse matrix-matrix multiplication using aninput-specific hypergraph, where nets of the hypergraphcorrespond to nonzero entries and vertices correspond tononzero scalar multiplications. The communication costof a parallel algorithm corresponds to the size of the cutof a multi-way partition of the hypergraph. We will dis-cuss the efficiency of various algorithms for input matricesfrom several applications in the context of this and relatedhypergraph models.

Grey BallardSandia National [email protected]

Alex DruinskyComputational Research DivisionLawrence Berkeley National [email protected]

Nicholas KnightNew York [email protected]

Oded SchwartzHebrew University of [email protected]

MS5

The Input/Output Complexity of Sparse MatrixMultiplication

We consider the problem of multiplying sparse matrices(over a semiring) where the number of non-zero entries islarger than main memory. In this paper we generalize theupper and lower bounds of Hong and Kung (STOC’81)to the sparse case. Our bounds depend of the numberN of nonzero entries in A and C, as well as the num-ber Z of nonzero entries in AC. We show that AC

can be computed using O

(NBmin

(√ZM, NM

))I/Os, with

high probability. This is tight (up to polylogarithmic fac-tors) when only semiring operations are allowed, even fordense rectangular matrices: We show a lower bound of

Ω

(NBmin

(√ZM, NM

))I/Os.

Morten Stockel, Rasmus PaghIT University of [email protected], [email protected]

MS6

Eigenvector Norms Matter in Spectral Graph The-ory

We investigate the role of eigenvector norms in spectralgraph theory to various combinatorial problems includ-ing the densest subgraph problem, the Cheeger constant,among others. We introduce randomized spectral algo-rithms that produce guarantees which, in some cases, arebetter than the classical spectral techniques. In particular,we will give an alternative Cheeger sweep (graph partition-ing) algorithm which provides a linear spectral bound forthe Cheeger constant at the expense of an additional factordetermined by eigenvector norms. Finally, we apply theseideas and techniques to problems and concepts unique todirected graphs.

Franklin KenterRice UniversityDept. Computational and Applied [email protected]

MS6

Local Clustering with Graph Diffusions and Spec-tral Solution Paths

Eigenvectors of graph-related matrices are intimately con-nected to diffusion processes that model the spread ofinformation across a graph. Local Cheeger inequalitiesguarantee that localized diffusions like the personalizedPageRank eigenvector and heat kernel identify small, good-conductance clusters. We provide constant-time algo-rithms for computing a generalized class of such diffusions,apply their solution paths to produce refined clusters, andprove that these diffusions always identify clusters of con-

78 LA15 Abstracts

stant size.

Kyle KlosterPurdue University, Department of Mathematics150 N. University Street West Lafayette, IN [email protected]

David F. GleichPurdue [email protected]

MS6

Signed Laplacians and Spectral Clustering ViaQuotient Space Metrics

Signed Laplacians represent the structure of graphs whoseedges possess signatures, i.e., certain group actions. Theyare quite useful for many purposes, e.g., modelling so-cial networks with friend/enemy relationships between itsmembers. We investigate the spectral clustering methodfor signed Laplacians. Like in the traditional spectral al-gorithms, the eigenvectors are used to map the graph ver-tices into unit spheres. We show that, by using the metricsof certain quotient spaces of the spheres, e.g., projectivespaces, one can cluster the graph vertices to find inter-esting signature-related substructures. In particular, thisprovides a spectral approach to Harary’s structural balancetheory. The approach also unify the (higher order) Cheegertype inequalities for the small and large eigenvalues of theusual unsigned Laplacians, which encode the connectivityand bipartiteness properties of the underlying graphs, re-spectively.

Shiping LiuUniversity of DurhamDept. of Mathematical [email protected]

Fatihcan M. AtayMax Planck Institute for Mathematics in the Sciences,[email protected]

MS6

Eigenvectors of the Nonlinear Graph p-Laplacianand Application in Graph Clustering

We investigate the discrete p-Laplacian operator Lp on con-nected graphs. We show that many relevant properties ofthe spectrum of the linear Laplacian can be extended to thenonlinear Lp. In particular we provide a variational charac-terization of a set of n points of the spectrum of Lp, hencewe prove a p-Laplacian nodal domain theorem and somehigher-order Cheeger-type inequalities. We interpret theobtained results in the context of graph clustering, propos-ing few examples and discussing some computational is-sues.

Francesco TudiscoDept. of Mathematics and Computer ScienceSaarland [email protected]

Matthias HeinSaarland UniversityDept. of Mathematics and Computer Science

[email protected]

MS7

Recovering Planted Subgraphs via Convex GraphInvariants

Extracting planted subgraphs from large graphs is a fun-damental question that arises in a range of application do-mains. In this talk, we outline a computationally tractableapproach based on convex optimization to recovery struc-tured planted graphs that are embedded in larger graphscontaining spurious edges. (Joint work with Utkan Cando-gan.)

Venkat ChandrasekaranCalifornia Institute of [email protected]

MS7

Computationally-Efficient Approximations to Ar-bitrary Linear Dimensionality Reduction Opera-tors

We examine a fundamental matrix approximation problem,motivated by computational considerations in large-scaledata processing: how well can an arbitrary linear dimen-sionality reduction (LDR) operator (a matrix A ∈ Rm×n

with m < n) be approximated by a computationally effi-cient surrogate? As one illustrative example, we establisha fundamental result for partial circulant approximationsof general matrices. We also examine approximations em-ploying other computationally-efficient “primitives,’ anddiscuss how low-dimensional structure may be exploitedin these approximation tasks.

Jarvis HauptUniversity of [email protected]

MS7

More Data, Less Work: Sharp DataComputationTradeoffs for Linear Inverse Problems

Often one wishes to estimate an unknown but structuredsignal from linear measurements where the number of mea-surements is far less that the dimension of the signal.We present a unified theoretical framework for conver-gence rates of various optimization schemes for such prob-lems. Our framework applies to both convex and non-convex objectives, providing precise tradeoffs between run-ning time, data, and structure complexity for many opti-mization problems. Joint with B. Recht and S. Oymak.

Mahdi SoltanolkotabiUniversity of California, [email protected]

MS7

A Data-dependent Weighted LASSO under PoissonNoise

Sparse linear inverse problems appear in a variety of set-tings, but often the noise contaminating observations can-not accurately be described as bounded or arising froma Gaussian distribution. Poisson observations in particu-lar are a characteristic feature of several real-world appli-cations. Previous work on sparse Poisson inverse prob-lems encountered several limiting technical hurdles. In

LA15 Abstracts 79

this talk I will describe an alternative, streamlined analy-sis approach for sparse Poisson inverse problems which (a)sidesteps the technical challenges present in previous work,(b) admits estimators that can readily be computed usingoff-the-shelf LASSO algorithms, and (c) hints at a generalweighted LASSO framework for broader classes of prob-lems. At the heart of this new approach lies a weightedLASSO estimator for which data-dependent weights arebased on Poisson concentration inequalities. Unlike previ-ous analyses of the weighted LASSO, the proposed analysisdepends on conditions which can be checked or shown tohold in general settings with high probability.

Xin JiangDuke [email protected]

Patricia Reynaud-BouretUniversity of [email protected]

Vincent RivoirardUniversity of Paris - [email protected]

Laure SansonnetArgo [email protected]

Rebecca WillettUniversity of [email protected]

MS8

Estimating the Condition Number of f(A)b

I will present some new algorithms for estimating the con-dition number of f(A)b. Standard condition number es-timation algorithms for f(A) require explicit computationof matrix functions and their Frechet derivatives and aretherefore unsuitable for the large, sparse A typically en-countered in f(A)b problems. The algorithms proposedhere use only matrix-vector multiplications. The numberof matrix-vector multiplications required to estimate thecondition number is proportional to the square of the num-ber required by the underlying f(A)b algorithm. Numericalexperiments demonstrate that the condition estimates arereliable and of reasonable cost.

Edvin DeadmanUniversity of [email protected]

MS8

An Algorithm for the Lambert W Function on Ma-trices

Despite being implicitly defined by the simple equationz = W (z) exp(W (z)), the so-called “Lambert W’ functionis not straightforward to compute, mainly because of thelack of regularity and logarithm-like properties. The New-ton method proves itself worthy for the scalar case, but atrivial extension to matrices suffers from numerical insta-bility and difficulty in choosing a suitable starting matrix.These issues are fixed by deriving a numerically stable,coupled form of the Newton iteration and by employing ablocked Schur decomposition, with suitable starting ma-trices chosen for each diagonal block. We obtain a robust

algorithm that works well in practice.

Massimiliano FasiUniversita di [email protected]

Nicholas HighamSchool of MathematicsThe University of [email protected]

Bruno IannazzoUniversita‘ di Perugia, [email protected]

MS8

The Leja Method: Backward Error Analysis andImplementation

The Leja method is a well established scheme for comput-ing the action of the matrix exponential. We present a newbackward error analysis allowing a more efficient method.From a scalar computation in high precision we predict thenecessary number of scaling steps based only on a roughestimate of the field of values or norm of the matrix andthe desired backward error. The efficiency of the approachis shown in numerical experiments.

Marco CaliariUniveristy of [email protected]

Peter KandolfUniversitat [email protected]

Alexander OstermannUniversity of [email protected]

Stefan RainerUniversity of InnsbruckDepartment of [email protected],at

MS8

An Exponential Integrator for Polynomially Per-turbed Linear ODEs

Consider the initial value problem u′(t) =

(∑N

�=0 ε�A�)u(t), u(0) = u0, where A0, A1, . . . , AN ∈ Cn×n

and u0 ∈ Cn. We present results that allow us to evaluateu(t) efficiently for several values of ε and t.We propose a new Krylov subspace method which uses theapproximation of the product of the matrix exponentialand a vector. The approach is based on the Taylorexpansion of the exact solution u(t) with respect to ε.We show that the coefficient vectors of this expansionare given by the matrix exponential of a block Toeplitzmatrix. This result can be seen as a generalization of aresult given by Najfeld and Havel (1995).A priori error bounds are derived for the approximationwhich show a superlinear convergence for any N .

Antti KoskelaUniversity of [email protected]

80 LA15 Abstracts

Elias JarlebringKTH [email protected]

M.E. HochstenbachEindhoven University of [email protected]

MS9

Interpolatory Techniques for Model Reduction ofMultivariate Linear Systems using Error Estima-tion

We consider interpolatory techniques for model reductionof multivariate linear systems as they appear in the inputoutput representation of quadratic bilinear systems. Exist-ing interpolatory techniques [C.Gu, QLMOR: a projection-based nonlinear model order reduction approach usingquadratic-linear representation of nonlinear systems, 2011],[P.Benner and T.Breiten, Two-sided projection methodsfor nonlinear model order reduction 2015] for model reduc-tion of quadratic bilinear systems interpolate these gener-alized transfer functions at some random set of interpola-tion points or by using the corresponding linear iterativerational Krylov algorithm. The goal here is to proposean approach that identifies a good choice of interpolationpoints based on the error bound expressions derived re-cently in [L. Feng, A. C. Antoulas and P. Benner, Somea posteriori error bounds for reduced order modelling of(non-)parameterized linear systems, 2015]. The approachiteratively updates the interpolation points in a predefinedsample space by computing the maximum error bound cor-responding to the previous set of interpolation points. Thisresults in a greedy type algorithm for model reduction ofthe generalized transfer functions. Numerical results showthe importance of choosing the interpolation points forsome benchmark examples.

Mian Ilyas AhmadMax Planck Institute - MagdeburgComputational Methods in Systems and Control [email protected]

Lihong Feng, Peter BennerMax Planck Institute, Magdeburg, [email protected], [email protected]

MS9

Parallelization of the Rational Arnoldi Method

The rational Arnoldi algorithm is a popular method in sci-entific computing used to construct an orthonormal basisof a rational Krylov space. Each basis vector is a ratio-nal matrix function times the starting vector. Rationalfunctions possess a partial fraction expansion which of-ten allows to compute several basis vectors simultaneously.However, this parallelism may cause instability due to theorthogonalization of ill-conditioned bases. We present andcompare continuation strategies to minimize these effects.

Mario BerljafaSchool of MathematicsThe University of [email protected]

Stefan GuettelThe University of Manchester

[email protected]

MS9

Solving Linear Systems with Nonlinear ParameterDependency

We present the CORK method for solving the paramet-ric linear system A(σ)x = b. The idea is to approxi-mate A(σ) by a (rational) matrix polynomial and thenextract x from the solution of a large scale problem withaffine parameter. Previous work was reported using mono-mial (GAWE-method) and Chebyshev polynomials (infi-nite Arnoldi). The CORK framework is a general one thatallows for much more freedom in the selection of polyno-mial or rational polynomial basis.

Karl MeerbergenK. U. [email protected]

Roel Van Beeumen, Wim MichielsKU [email protected],[email protected]

MS9

Perfectly Matched Layers and Rational KrylovSubspaces with Adaptive Shifts for Maxwell Sys-tems

The efficient and accurate modeling of waves in inhomoge-neous media on unbounded domains is of paramount im-portance in many different areas in science and engineer-ing. We show that optimal complex scaling and stability-corrected wave functions allow for the efficient computationof wave fields via Krylov subspace techniques. Polynomialand rational Krylov methods are discussed and numericalexperiments will illustrate the performance of the proposedKrylov methods.

Vladimir L. DruskinSchlumberger-Doll [email protected]

Rob RemisCircuits and Systems GroupDelft University of [email protected]

Mikhail ZaslavskySchlumberger-Doll [email protected]

Joern ZimmerlingCircuits and Systems GroupDelft University of [email protected]

MS10

Recent Advances on Inverse Problems for MatrixPolynomials

We summarize several results on inverse eigenstructureproblems for matrix polynomials that have been obtainedrecently and discuss how they complete other results previ-ously known in the literature. These new results are closelyconnected to the Index Sum Theorem and to the new classof Polynomial Zigzag matrices. In particular, we solve the

LA15 Abstracts 81

most general possible version of complete inverse eigen-structure problem for matrix polynomials.

Froilan M. Dopico, Fernando De TeranDepartment of MathematicsUniversidad Carlos III de [email protected], [email protected]


Paul Van DoorenUniversite Catholique de [email protected]

MS10

Matrix Functions: The Contributions of LeibaRodman

In this presentation I will try to review the voluminouscontributions to linear algebra made by Leiba Rodman. Iwill combine this with some new insights into canonicalstructures for selfadjoint quadratic matrix polynomials.

Peter LancasterUniversity of CalgaryDept of Math and [email protected]

MS10

Generic Low Rank Perturbations of StructuredMatrices

Recently, the theory of generic structure-preserving rank-one perturbations of matrices that have symmetry struc-tures with respect to some indefinite inner product hasbeen developed. Concerning the case of arbitrary rank k,it was conjectured, but is not immediate, that a genericrank k perturbations can be interpreted as sequence of kgeneric rank-one perturbations. In this talk, we close thisgap by giving an affirmative answer.

Leonhard BatzkeTU [email protected]

Christian MehlTU BerlinInstitut fuer [email protected]

Andre C. RanVrije [email protected]

Leiba RodmanCollege of William and MaryDepartment of [email protected]

MS10

A Factorized Form of the Inverse Polynomial Ma-trix Problem

In this talk we consider the problem of constructing a poly-

nomial matrix of degree d with prescribed sets of left andright minimal indices and with prescribed sets of finite ele-mentary divisors, satisfying the degree sum theorem. Theuse of a factorized form has the advantage that the differ-ent structural elements are easily recovered from each ofthe three factors. The left factor gives the left null spacestructure, the right factor gives the right null space struc-ture and the middle factor reveals the finite elementarydivisors.

Paul M. Van DoorenCatholic University of [email protected]

MS11

When Life is Linear Worldwide

In this spring of 2015, Tim Chartier taught a MOOCthrough Davidson College and edX. The course came intwo parts with both teaching applications of linear algebrain computer graphics and data mining. Part 1 taught ba-sics of linear algebra and was targeted to a broad audience.Part 2 taught more advanced concepts such as eigenvectors,Markov Chains, and the SVD. Both parts had activities in-tended to encourage exploration, discovery, and creativity.

Tim ChartierDavidson CollegeDept. of [email protected]

MS11

Coding the Matrix: Linear Algebra through Com-puter Science Applications

Starting in 2008, I have been developing and teaching a lin-ear algebra course, Coding the Matrix, aimed at computerscience students. Concepts are motivated and illustratedusing applications from computer science. I view program-ming as a learning modality; students write programs todevelop and reinforce their understanding of concepts andproofs. In the MOOC version, taught twice through Cours-era, student programs are submitted to a grader and au-tomatically checked using example data. I’ll discuss myexperiences.

Philip KleinBrown [email protected]

MS11

LAFF Long and Prosper?

Linear Algebra: Foundations to Frontiers (LAFF) is acourse developed specifically for the edX platform. It en-compasses more than 850 pages of notes, 250+ videos, on-line activities, homeworks with answers, and programmingexercises. We took on this activity to a large degree be-cause of the MOOC movement’s promise to educate themasses, including those who don’t necessarily have accessto high quality classes. What is the intended audience?Who showed up to take the course? What is the shortterm and long term prognosis? Will MOOCs change edu-cation as we know it? The fact is that it is too early to tell.We discuss design decisions that underlie our course andearly experiences from offering this course in Spring 2014

82 LA15 Abstracts

and Spring 2015.

Maggie E. MyersThe University of Texas at [email protected]

Robert A. van de GeijnThe University of Texas at AustinDepartment of Computer [email protected]

MS11

Experience with OpenCourseWare Online VideoLectures

I will share my experience in preparing video lecturesfor basic mathematics courses. Linear Algebra and alsoComputational Science came directly from the MIT class-room. Calculus and the new Differential Equations videoswere filmed without an audience—I bought a camera andstudents helped. All are uploaded to OpenCourseWareocw.mit.edu and to YouTube (where an individual can cre-ate a Channel, but most viewers will find videos from es-tablished websites). Using a blackboard can still be moreeffective than slides (but there is a place for slides). Yourvoice and movements make a human connection that canchange lives.

Gilbert StrangMassachusetts Institute of [email protected]

MS12

Strong Scaling and Stability: SpAMM Accelera-tion for the Matrix Square Root Inverse and theHeavyside Function

For matrices with decay, incomplete/inexact approxima-tions based on the dropping of small elements leads to asparse approximation and fast solutions for preconditioners(and even close to full solutions in well-conditioned circum-stances) via the SpMM kernel. In this talk, I will developan alternative strategy for the N-Body multiplication ofdata local decay matrices, the Sparse Approximate MatrixMultiply (SpAMM), based on recursive Cauchy-Swartz oc-clusion of sub-multiplicative norms (metric query). TheSpAMM kernel is sparse in the multiplication (task) space,via occlusion and culling of small norms (rather than inthe vector (data) space), yielding bounds for the productthat are multiplicative rather than additive. I will developa stability analysis for SpAMM accelerated Newton Schulziterations towards the inverse square root, and show howthe convergence of Frechet channels towards idempotenceand nilpotence determine stability. I will demonstrate ex-treme stability under SpAMM approximation for NS itera-tion with congruential transformation, and show dramaticerror cancellation between the nilpotent Frechet derivativeand the corresponding skew displacement, which grows as

cond(Z−1/2k ), withZk− > A−1/2. Finally, I will describe

how N-Body structure of the SpAMM algorithm leads toencouraging results for distributed memory implementa-tions, with recent results for a hybrid Charm++/OpenMPapproach to task parallelism, yielding strong scaling upto 500 cores per heavy atom for calculation of the matrixHeavyside function in the context of electronic structuretheory.

Matt ChallacombeLos Alamos National Laboratory

Theoretical [email protected]

MS12

Analyzing Spgemm on Gpu Architectures

Mapping sparse matrix-sparse matrix multiplication(SpGEMM) to modern throughput-oriented architectures,such as GPUs, presents a unique set challenges in com-parison with traditional CPUs implementations. In thispresentation we will present an overview of the architec-tural issues that must be addressed by SpGEMM oper-ations on GPUs and discuss recent work to address theseareas of concern. Specifically we will compare and contrastsegmented and flat processing schemes and the impact ofeach scheme on the expected SpGEMM performance.

Steven DaltonUniversity of Illinois at [email protected]

Luke OlsonDepartment of Computer ScienceUniversity of Illinois at [email protected]

MS12

The Distributed Block-Compressed Sparse Row Li-brary: Large Scale and GPU Accelerated SparseMatrix Multiplication

The Distributed Block-Compressed Sparse Row (DBCSR)library is designed to efficiently perform block-sparsematrix-matrix multiplication, among other operations. Itis MPI and OpenMP parallel, and can exploit accelerators.It is developed as part of CP2K, where it provides corefunctionality for linear scaling electronic structure theory.The deployed algorithm takes in account the sparsity of theproblem in order to reduce the MPI communication. Wewill discuss performance results and development insights.

Alfio Lazzaro, Ole SchuettETH [email protected], [email protected]

Joost VandeVondeleNanoscale SimulationsETH [email protected]

MS12

A Framework for SpGEMM on GPUs and Hetero-geneous Processors

This talk will focus on our framework and correspond-ing algorithms for solving three main challenges (un-known number of nonzeros of the resulting matrix, expen-sive insertion and load imbalance) of executing SpGEMMon GPUs and emerging heterogeneous processors. Seehttp://arxiv.org/abs/1504.05022 for a preprint paper ofthis talk.

Weifeng LiuCopenhagen [email protected]

Brian Vinter

LA15 Abstracts 83

Niels Bohr Institute, University of [email protected]

MS13

Active Subspaces in Theory and Practice

Active subspaces are an emerging set of tools for workingwith functions of several variables. I will motivate dimen-sion reduction in scalar-valued functions of several vari-ables, define the active subspace, and analyze methods forestimating it based on the function’s gradient.

Paul ConstantineColorado School of MinesApplied Mathematics and [email protected]

MS13

Sketching Active Subspaces

When gradients are not available, estimating the eigenvec-tors that define a multivariate function’s active subspaceis especially challenging. We present a matrix sketchingapproach that exploits the connection between directionalderivatives and linear projections to estimate the eigenvec-tors with fewer function calls than standard finite differ-ences.

Armin Eftekhari, First Name EftehariColorado School of [email protected], [email protected]

Paul ConstantineColorado School of MinesApplied Mathematics and [email protected]

Michael B. WakinDept. of Electrical Engineering and Computer ScienceColorado School of [email protected]

MS13

Adaptive Morris Techniques for Active SubspaceConstruction

In this presentation, we discuss the use of adaptive Morristechniques for adaptive subspace construction. In appli-cations ranging from neutronics models for nuclear powerplant design to partial differential equations with dis-cretized random fields, the number of inputs can rangefrom the thousands to millions. In many cases, however,the dimension of the active subspace of influential param-eters is moderate in the sense that it is less than one hun-dred. For codes in which adjoints are available, gradientapproximations can be used to construct these active sub-spaces. In this presentation, we will discuss techniques,which are applicable when adjoints are not available orare prohibitively expensive. Specifically, we will employMorris indices with adaptive stepsizes and step-directionsto approximate the active subspace. We illustrate thesetechniques using examples from nuclear and aerospace en-gineering.

Allison LewisNorth Carolina State [email protected]

Ralph C. SmithNorth Carolina State UnivDept of Mathematics, [email protected]

Brian WilliamsLos Alamos National [email protected]

MS13

Recovery of Structured Multivariate Functions

Approximation of multivariate functions typically suffersfrom curse of dimension, i.e. the number of sampling pointsgrows exponentially with the dimension. We pose a struc-tural assumption on the functions to be approximated -namely that they take a form of a ridge, f(x) = g(Ax).We present recovery algorithms for this kind of a problem,including tractability results.

Jan VybiralTechnical University [email protected]

Massimo FornasierTechnical University [email protected]

Karin SchnassUni [email protected]

MS14

Fractional Tikhonov Regularization and the Dis-crepancy Principle for Deterministic and Stochas-tic Noise Models

Recently, two different regularization methods called frac-tional Tikhonov regularization have been introduced,in particular to reduce the over-smoothing of classicalTikhonov regularization. In this talk, we discuss conver-gence properties of these methods in the classical deter-ministic setting and show how one can lift these results tothe case of stochastic noise models. We compare the re-construction quality of fractional Tikhonov methods withthe classical approach for the discrepancy principle.

Daniel GerthFaculty for MathematicsTechnical University [email protected]

Esther KlannUniversity of Linz, [email protected]

Lothar ReichelKent State [email protected]

Ronny RamlauJohannes Kepler Universitat Linz, [email protected]

MS14

A Singular Value Decomposition for Atmospheric

84 LA15 Abstracts

Tomography

The image quality of ground based astronomical telescopessuffers from turbulences in the atmosphere. Adaptive Op-tics (AO) systems use wavefront sensor measurements ofincoming light from guide stars to determine an optimalshape of deformable mirrors (DM) such that the image ofthe scientific object is corrected after reflection on the DM.An important step in the computation of the mirror shapesis Atmospheric Tomography, where the turbulence profileof the atmosphere is reconstructed from the incoming wave-fronts. We present several reconstruction approaches andwill in particular focus on reconstructions based on a sin-gular value decomposition of the underlying operator.

Ronny RamlauJohannes Kepler Universitat Linz, [email protected]

MS14

Unbiased Predictive Risk Estimator for Regular-ization Parameter Estimation in the Context of It-eratively Reweighted Lsqr Algorithms for Ill-PosedProblems

Determination of the regularization parameter using themethod of unbiased predictive risk estimation (UPRE) forthe LSQR Tikhonov regularization of under determinedill-posed problems is considered and contrasted with thegeneralized cross validation and discrepancy principle tech-niques. Examining the UPRE for the projected problem itis shown that the obtained regularized parameter providesa good estimate for that to be used for the full problemwith the solution found on the projected space. The resultsare independent of whether systems are over or underde-termined. Numerical simulations support the analysis anda large scale problem of image restoration validates the useof these techniques.

Rosemary A. RenautArizona State UniversitySchool of Mathematical and Statistical [email protected]

Saeed VatankhahUniversity of Tehran, [email protected]

MS14

Shape Reconstruction by Photometric Stereo withUnknown Lighting

Photometric stereo is a typical technique in computer vi-sion to extract shape and color information from an objectwhich is observed under different lighting conditions, froma fixed point of view. We will describe a fast and accuratealgorithm to approximate the framed object, treating, inparticular, the case when the position of the light sourcesis not known. Numerical experiments concerning an appli-cation in archaeology will be illustrated.

Giuseppe RodriguezUniversity of Cagliari, [email protected]

Riccardo DessıPolitecnico di [email protected]

Gabriele Stocchino, Massimo VanziUniversity of [email protected], [email protected]

MS15

Sparse Approximate Inverse Preconditioners, Re-visited

Preconditioners that are sparse approximations to the in-verse of a matrix can be applied very efficiently in paral-lel. A disadvantage, however, is that they are local–thepreconditioning only propagates information from one gridpoint to nearby grid points, resulting in slow convergence.We investigate applying sparse approximate inverses in anhierarchical basis. The hierarchical basis transformationsare also sparse matrix vector products. The method isrelated to using sparse approximate inverses as multigridsmoothers and to using wavelet bases, which have been in-vestigated in the past. Our goal, however, is to develop asimple method that is easily parallelizable.


MS15

Iterative Solver for Linear Systems Arising in Inte-rior Point Methods for Semidefinite Programming

Interior point methods for Semidefinite Programming(SDP) face a difficult linear algebra subproblem. We pro-pose an iterative scheme to solve the Newton equation sys-tem arising in SDP. It relies on a new preconditioner whichexploits well the sparsity of matrices. Theoretical insightsinto the method will be provided. The computational re-sults for the method applied to large MaxCut and matrix-norm minimization problems will be reported.

Jacek GondzioThe University of Edinburgh, United [email protected]

Stefania BellaviaUniversita di [email protected]

Margherita PorcelliUniversity of [email protected]

MS15

On Nonsingular Saddle-Point Systems with a Max-imally Rank Deficient Leading Block

We consider nonsingular saddle-point matrices whose lead-ing block is maximally rank deficient, and show that theinverse in this case has unique mathematical properties.We then develop a class of indefinite block precondition-ers that rely on approximating the null space of the leadingblock. The preconditioned matrix is a product of two indef-inite matrices but under certain conditions the conjugategradient method can be applied and is rapidly convergent.Spectral properties of the preconditioners are observed andvalidated by numerical experiments.

Chen Greif

LA15 Abstracts 85

Department of Computer ScienceThe University of British [email protected]

MS15

A Tridiagonalization Method for Saddle-Point andQuasi-Definite Systems

The tridiagonalization process of Simon, Saunders and Yip(1988) applied to a rectangular operator A gives rise toUSYMQR, which solves a least-squares problem with A,and USYMLQ, which solves a least-norm problem withAT . Symmetric saddle-point systems may be viewed as apair of least-squares/least-norm problems. This allows usto merge USYMQR and USYMLQ into a single methodthat solves both problems in one pass. We present precon-ditioned and regularized variants that apply to symmetricand quasi-definite linear systems and illustrate the perfor-mance of our implementation on systems coming from op-timization and fluid flow.

Dominique OrbanGERAD and Dept. Mathematics and IndustrialEngineeringEcole Polytechnique de [email protected]

MS16

Averaging Block-Toeplitz Matrices with Preserva-tion of Toeplitz Block Structure

In many applications, data measurements are transformedinto matrices. To find a trend in repeated measurements,an average of the corresponding matrices is required. Weuse the barycenter, or minimizer of the sum of squared in-trinsic distances, as our averaging operation. This intrin-sic distance depends on the chosen geometry for the setof interest. We present an application-inspired geometryfor positive definite Toeplitz-Block Block-Toeplitz matri-ces. Both the corresponding barycenter and an efficientapproximation are discussed.

Ben JeurisKU [email protected]


MS16

Theory and Algorithms of Operator Means

We will consider the theory of matrix and operator meansbased on the Kubo-Ando and Loenwer theories. We willconsider theoretical methods for extending, defining andapproximating means of more than two positive operators,in fact means of probability measures supported on thecone of positive operators. The techniques will be moti-vated by the geometric mean of positive definite matricesand discrete-time gradient flows for geodescally convex po-tential functions.

Miklos PalfiaKyoto University

[email protected]

MS16

Using Inverse-free Arithmetic in Large-scale Ma-trix Equations

Inverse-free arithmetic and permuted Riccati bases havebeen used recently in the solution of small-scale algebraicRiccati equations, with excellent stability properties. Weexplore their use to solve large-scale matrix equations: ourtarget is being able to deal with problems where the ma-trix X has large norm, and hence the columns of [I ;X] arean ill-conditioned basis for its column space. For simplic-ity, we focus on solving Lyapunov equations with the ADIalgorithm.

Volker MehrmannTechnische Universitat [email protected]

Federico PoloniUniversity of [email protected]

MS16

Preconditioned Riemannian Optimization for Low-Rank Tensor Equations

Solving partial differential equations on high-dimensionaldomains leads to very large linear systems. In these cases,the degrees of freedom in the linear system grow expo-nentially with the number of dimensions, making classicapproaches unfeasible. Approximation of the solution bylow-rank tensor formats often allows us to avoid this curseof dimensionality by exploiting the underlying structureof the linear operator. We propose a truncated Newtonmethod on the manifold of tensors of fixed rank, in partic-ular, Tensor Train (TT) / Matrix Product States (MPS)tensors. We demonstrate the flexibility of our algorithmby comparing different approximations of the RiemannianHessian as preconditioners for the gradient directions. Fi-nally, we compare the efficiency of our algorithm with othertensor-based approaches such as the Alternating LinearScheme (ALS).

Bart VandereyckenETH [email protected]

Daniel KressnerEPFL [email protected]

Michael SteinlechnerEPF [email protected]

MS17

Tensors and Structured Matrices of Low Rank

We present an interesting connection between higher-ordertensors and structured matrices, in the context of low-rankapproximation. In particular, we show that the tensor lowmultilinear rank approximation problem can be reformu-lated as a structured matrix low-rank approximation, thelatter being an extensively studied and well understoodproblem. For simplicity, we consider symmetric tensors.

86 LA15 Abstracts

By imposing simple constraints in the optimization prob-lem, the proposed approach is applicable to general tensors,as well as to affinely structured tensors to find (locally) bestlow multilinear rank approximation with the same struc-ture.

Mariya IshtevaVrije Universiteit [email protected]

Ivan MarkovskyVrije Universiteit Brussel (VUB)[email protected]

MS17

Alternating Least-Squares Variants for Tensor Ap-proximation

The Alternating Least-Squares (ALS) algorithm for CP ap-proximation of tensors has spawned several variants, thebest-known of these being ALS with (Tikhonov) regular-ization. We present a unified description of these vari-ants of ALS, and further generalize it to produce novelvariants of the algorithm. One variant, tentatively titled”greased ALS”, converges significantly more rapidly thanALS for several test cases. We seek to understand the cir-cumstances under which these variants provide more rapidconvergence, and fully understand the reasons for the im-provement.

Nathaniel McClatcheyOhio UniversityDepartment of [email protected]

Martin J. MohlenkampOhio [email protected]

MS17

Generating Polynomials and Symmetric TensorDecomposition

For symmetric tensors of a given rank, there exist linear re-lations of recursive patterns among their entries. Such re-lations can be represented by polynomials, which are calledgenerating polynomials. The homogenization of a generat-ing polynomial belongs to the apolar ideal of the tensor.A set of generating polynomials can be represented by amatrix, which is called a generating matrix. Generally,a symmetric tensor decomposition can be uniquely deter-mined by a generating matrix satisfying certain conditions.We characterize the sets of such generating matrices andinvestigate their properties (e.g., the existence, dimensions,nondefectiveness). Using these properties, we propose com-putational methods for symmetric tensor decompositions.Extensive examples are shown to demonstrate their effi-ciency.

Jiawang NieUniversity of California, San [email protected]

MS17

On the Convergence of Higher-order OrthogonalityIteration and Its Extension

The higher-order orthogonality iteration (HOOI) method

has been popularly used to find a best low multilinear-rank approximation of a tensor. It often performs well inpractice to learn a multilinear subspace. However, little ofits convergence behavior has been known. In this talk, Iwill present a greedy way to implement the HOOI methodand show its subsequence convergence to a stationary pointwithout any assumption. Assuming a nondegeneracy con-dition, I will further show its global sequence convergenceresult, which guarantees global optimality if the startingpoint is sufficiently close to a globally optimal solution.The results are then extended to the multilinear SVD withmissing values.

Yangyang XuRice UniversityUniversity of [email protected]

MS18

Communication-optimal Loop Nests

Scheduling an algorithm’s operations to minimize commu-nication can reduce its runtime and energy costs. Westudy communication-efficient schedules for a class of al-gorithms including many-body and matrix/tensor compu-tations and, more generally, loop nests operating on arrayvariables subscripted by linear functions of the loop iter-ation vector. We derive lower bounds on communicationbetween levels in a memory hierarchy and between parallelprocessors, and present communication-optimal schedulesthat attain these bounds in many cases.


MS18

A Computation and Communication-Optimal Par-allel Direct 3-Body Algorithm

We present a new 3-way interactions N-body (3-body) al-gorithm that is both computation and communication op-timal. Its optional replication factor, c, saves c3 in commu-nication latency (number of messages) and c2 in bandwidth(volume). We also extend the algorithm to support k-wayinteractions with cutoff distance. The 3-body algorithmdemonstrates 99% efficiency on tens of thousands of cores,showing strong scaling properties with order of magnitudespeedups over the naıve algorithm.

Penporn KoanantakoolUC [email protected]

MS18

Communication Lower Bounds for Distributed-Memory Computations

We present a new approach to the study of the commu-nication requirements of distributed computations. Thisapproach advocates for the removal of the restrictive as-sumptions under which earlier lower bounds on communi-cation complexity were derived, and prescribes that lowerbounds rely only on a mild assumption on work distributionin order to be significant and applicable. This approach isillustrated by providing tight communication lower boundsfor several fundamental problems, such as matrix multipli-

LA15 Abstracts 87

cation.

Michele ScquizzatoUniversity of [email protected]

Francesco SilvestriUniversity of [email protected]

MS18

Minimizing Communication in Tensor ContractionAlgorithms

Accurate numerical models of electronic correlation inmolecules are derived in terms of tensors that representmulti-orbital functions and consequently contain permu-tational symmetries. Traditionally, contractions of suchsymmetric tensors exploit only the symmetries in the setof tensor element products. I will present an algebraically-restructured method that requires fewer operations formany such contractions of symmetric tensors. Addition-ally, I will present communication lower bounds that con-tain surprising new insights for both approaches.

Edgar SolomonikETH [email protected]

MS19

Numerical Tensor Algebra in Modern QuantumChemistry and Physics

I will review the role of tensor operations in quantum simu-lations, and in particular, the challenges of numerical com-putation in the matrix product state and tensor networkrepresentations. Issues of parallelization and numerical sta-bility will be discussed, especially for tensor networks withcycles.

Garnet ChanPrinceton [email protected]

MS19

Efficient Algorithms for Transition State Calcula-tions

Transition states are fundamental to understanding the re-action dynamics qualitatively. To date various methods offirst principle location of transition states have been devel-oped. In the absence of the knowledge of the final struc-ture, the minimal-mode following method climbs up to atransition state without calculating the Hessian matrix. Inthis talk, we introduce a locally optimal search directionfinding algorithm and an iterative minimizing method forthe translation which improve the rotational step by a fac-tor and the translational step a quantitative scale. Numeri-cal experiments demonstrate the efficiency of our proposedalgorithms. This is joint work with Jing Leng, Zhi-PanLiu, Cheng Shang and Xiang Zhou.

Weiguo GaoFudan University, [email protected]

MS19

Compression of the Electron Repulsion Integral

Tensor

The electron repulsion integral tensor has ubiquitous appli-cation in quantum chemistry calculation. In this talk, wediscuss some recent progress on compressing the electronrepulsion tensor into the tensor hypercontraction formatwith cubic scaling computational cost. (joint work withLexing Ying)

Jianfeng LuMathematics DepartmentDuke [email protected]

MS19

Reduced Density Matrix Methods in TheoreticalChemistry and Physics

The energy of an N-electron quantum system can be ex-pressed as a functional of the two-electron reduced densitymatrix (2-RDM). In 2-RDM methods the 2-RDM of anN-electron system is directly computed without the wavefunction. In this lecture recent theoretical and computa-tional advances as well as examples from chemistry andcondensed-matter physics will be discussed.

David A. MazziottiUniversity of [email protected]

MS20

On Powers of Certain Positive Matrices

A matrix is called totally positive (resp. nonnegative) ifall of its minors are positive (resp. nonnegative). It isknown that such matrices are closed under conventionalmultiplication, but not necessarily closed under entry-wiseor Hadamard multiplication. On the other hand, a realmatrix is called positive definite (resp. semidefinite) if itis symmetric and has positive (resp. nonnegative) prin-cipal minors. In this case, such matrices need not beclosed under matrix multiplication, but are closed underHadamard multiplication. In this talk, we will survey ex-isting work and discuss some current advances on contin-uous Hadamard and conventional powers of both totallypositive and positive definite matrices (along with theirclosures), in the spirit of identifying a critical exponent ineither situation, should one exist.

Shaun M. FallatUniversity of [email protected]

MS20

Perron-Frobenius Theory of Generalizations ofNonnegative Matrices

An eventual property is one that is attained by all powersof a matrix beyond some power, so a matrix is eventuallypositive if Ak is entrywise positive for all k ≥ k0, Thistalk will survey Perron-Frobenius theory of positive andnonnegative matrices and and how it extends to general-izations of nonnegative matrices, especially those definedby eventual properties.

Leslie HogbenIowa State University,American Institute of Mathematics

88 LA15 Abstracts

[email protected]

MS20

Inverses of Acyclic Matrices and Parter Sets

An important result of Gantmacher and Krein describesthe inverses of symmetric, irreducible and invertible tridi-agonal matrices. In the early 2000s, Nabben extended thisby describing the inverses of invertible matrices, M , whosegraph is a given tree T under the assumption that eachdiagonal entry of M−1 is positive. These are closely re-lated to the important class of ultrametric matrices. Weestablish a similar description for arbitrary trees T withoutthe condition on the diagonal entries, and relate this to theParter-sets of M , i.e. subsets α of vertices such that thenullity of the matrix obtained from M by deleting the rowsand columns indexed by α is the cardinality of α.

Bryan L. ShaderDepartment of MathematicsUniversity of [email protected]

Curtis NelsonDepartment of MathematicsBrigham Young [email protected]

MS20

Geometric Mapping Properties of SemipositiveMatrices

Semipositive matrices map a positive vector to a positivevector. The geometric mapping properties of semipositivematrices exhibit parallels to the theory of cone preservingand cone mapping matrices: For a semipositive matrix A,there exist a positive proper polyhedral cone K1 and a pos-itive polyhedral cone K2 such that AK1 = K2. The set ofall nonnegative vectors mapped by A to the nonnegativeorthant is a proper polyhedral cone; as a consequence, Abelongs to a proper polyhedral cone comprising semiposi-tive matrices. When the powers Ak have a common semi-positivity vector, then A has a positive eigenvalue. Whenthe spectral radius of A is the sole peripheral eigenvalueand the powers of A are semipositive, A leaves a propercone invariant.

Michael TsatsomerosWashington State [email protected]

MS21

Vector Etrapolation for Image Restoration


Hassane SadokUniversite du LittoralCalais Cedex, [email protected]

MS21

Psf Reconstruction for Extremely Large Telescopes

Objects on the sky appear blurred when observed by aground based telescope due to diffraction of light and tur-bulences in the atmosphere. Mathematically the relation

between the real object and its observed image can be de-scribed as a convolution with the so called point spreadfunction (PSF). Even though the to-be-built generation ofextremely large telescopes (ELTs) use adaptive optics (AO)systems to reduce the effects of the atmosphere, still someresidual blurring remains. The PSF serves as quality mea-sure for the AO system and thus needs to be very accurate.As the PSF cannot be measured directly, a reconstructionform the data of the AO system is necessary. In this talk,an efficient algorithm for PSF reconstruction for the Euro-pean ELT (E-ELT) is presented. The reconstruction pro-cess is based on splitting the PSF in one part calculated onthe fly from observation data and a second part that canonly be obtained from simulation. Especially for the latterone, a good model of the atmosphere is needed and due tothe size of the E-ELT all calculations have to be made asefficient as possible even though simulations can be donebeforehand. On the fly computations need to be fast andusing little memory as the measurements are usually takenat a frequency of 500 Hz, leading to a huge amount of datajust for one single night. In addition, astronomers wantto evaluate the current observations conditions using thePSF and thus need it almost on time. We show how touse an efficient representation, leading to a sparse matrix.The reconstructed PSF can be used for improvement ofthe observed images. We opt for a blind deconvolutionscheme to also improve the quality of the reconstructedPSF. Again the choice of an efficient representation, nowfor the observed image, is crucial to get a sparse linear sys-tem. Finally, a blind deconvolution algorithm taking careof the features of ground based astronomy is presented.

Roland WagnerRadon Institute for Computational and AppliedMathematicsAustrian Academy of [email protected]

MS21

Randomized Tensor Singular Value Decomposition

The tensor Singular Value Decomposition (t-SVD) pro-posed by Kilmer and Martin [2011] has been applied suc-cessfully in many fields, such as computed tomography, fa-cial recognition, and video completion. In this talk, I willpresent a probabilistic method that can produce a factor-ization with similar properties to the t-SVD, but is stableand more computationally efficient on very large datasets.This method is an extension of a well-known randomizedmatrix method. I will present the details of the algorithm,theoretical results, and provide experimental results for twospecific applications.

Jiani Zhang, Shuchin AeronTufts [email protected], [email protected]

Arvind SaibabaDepartment of Electrical and Computer EngineeringTufts [email protected]


MS21

Multidirectional Subspace Expansion for Single-Parameter and Multi-Parameter Tikhonov Regu-

LA15 Abstracts 89

larization

Tikhonov regularization is often used to approximate so-lutions of linear discrete ill-posed problems when the ob-served or measured data is contaminated by noise. Multi-parameter Tikhonov regularization may improve the qual-ity of the computed approximate solutions. We proposea new iterative method for large-scale multi-parameterTikhonov regularization with general regularization oper-ators based on a multidirectional subspace expansion.

Ian ZwaanDepartment of Mathematics and Computer ScienceTechnical University [email protected]

M.E. HochstenbachEindhoven University of [email protected]

MS22

Fast Iterative Solvers for a Coupled Cahn-Hilliard/Navier-Stokes System

We consider a diffuse interface model that governs the hy-drodynamics of two-phase flows. Our research builds onthe work of Garcke, Hinze and Kahle. Our contributionconcerns the fully iterative solution of the arising largeand sparse linear systems. This is based on precondition-ing techniques we have developed for Cahn–Hilliard varia-tional inequalities. Block preconditioners using an effectiveSchur complement approximation are presented. Numeri-cal results illustrate the efficiency of the approach.

Jessica BoschMax Planck Institute [email protected]

Martin StollMax Planck Institute, [email protected]

MS22

Preconditioning a Mass-Conserving DG Discretiza-tion of the Stokes Equations

While there is a long history of work on conforming finite-element discretizations of the Stokes and Navier-Stokesequations, using H(div)-conforming elements for the ve-locity has recently emerged as a way to ensure strong en-forcement of the incompressibility condition at the elementscale. In this talk, we explore the applicability of clas-sic block preconditioning and monolithic multigrid meth-ods for Stokes-like saddle-point problems to one such dis-cretization, using BDM elements for the velocities.

Scott MaclachlanDepartment of MathematicsTufts [email protected]

MS22

A Multilevel Preconditioner for Data Assimilationwith 4D-Var

Large-scale variational data assimilation problems are com-mon in numerical weather prediction. The 4D-Var methodis frequently used to calculate a forecast model trajectory

that best fits the available observations to within the obser-vational error over a period of time. This results in a large-scale nonlinear weighted least-squares problem, typicallysolved by a Gauss-Newton method. We will discuss usinga multilevel preconditioner within 4D-Var to reduce mem-ory requirements and increase computational efficiency.

Kirsty BrownUniversity of [email protected]

Igor [email protected]

Alison RamageUniversity of [email protected]

MS22

Asynchronous Optimized Schwarz Methods

Optimized Schwarz Methods are domain decompositionmethods, where one imposes Robin conditions on the arti-ficial interfaces. The Robin parameter can be optimized toobtain very fast convergence. We present an asynchronousversion of this method, where the problem in each sub-domain is solved using whatever boundary data is locallyavailable and with no synchronizations with other pro-cesses. We prove convergence of the method, and illustrateits efficiency on large three-dimensional problems. (Joint

work with Frederic Magoules and Cedric Venet, Ecole Cen-trale Paris).


MS23

Title Not Available at Time of Publication


Xin HeDelft University of Technology, [email protected]

MS23

Preconditioners for Linear Systems Arising FromThe Eddy Current Problem

We consider the preconditioning techniques for thestructued systems of linear equations arising from the fi-nite element discretization of the eddy current problem. Aclass of positive definite and positive semidefinite splittingsis proposed to solve such linear systems. This kind of ma-trix splitting, as well as its relaxed variants, can be used asthe preconditoners for Krylov subspace methods. The per-formance of the preconditioners is illustrated by numericalexamples.

Jianyu PanEast China Normal University, [email protected]

Xin Guan

90 LA15 Abstracts

Department of mathematics, East China [email protected]

MS23

An Alternating Positive Semidefinite Splitting Pre-conditioner for Saddle Point Problems from Time-harmonic Eddy Current Models

For the saddle point problem arising from the finite ele-ment discretization of the hybrid formulation of the time-harmonic eddy current problem, we propose an alternat-ing positive semidefinite splitting preconditioner which isbased on two positive semidefinite splittings of the saddlepoint matrix. It is proved that the corresponding alter-nating positive semidefinite splitting iteration method isunconditionally convergent. We show that the new precon-ditioner is much easier to implement than the block alter-nating splitting implicit preconditioner proposed in [Z.-Z.Bai, Numer. Linear Algebra Appl., 19 (2012), 914–936]when they are used to accelerate the convergence rate ofKrylov subspace methods such as GMRES. Numerical ex-amples are given to show the effectiveness of our proposedpreconditioner.

Zhiru RenChinese Academy of [email protected]

Yang CaoSchool of TransportationNantong [email protected]

MS23


In this talk, we focus on solving a class of complex linearsystems arising from distributed optimal control with time-periodic parabolic equations. A block alternating splitting(BAS) iteration method and preconditioner are presented.The convergence theory of the BAS iteration and the spec-tral properties of the BAS preconditioned matrix are dis-cussed. Numerical experiments are presented to illustratethe efficiency of the BAS iteration as a solver as well as apreconditioner for Krylov subspace methods.

Guo-Feng Zhang, Zhong ZhengLanzhou University, Chinagf [email protected], [email protected]

MS24

Localization and Reconstruction of Brain SourcesUsing a Constrained Tensor-based Approach

This talk addresses the localization and reconstructionof spatially distributed sources from ElectroEncephalo-Graphic (EEG) signals. The occurrence of several ampli-tude modulated spikes originating from the same epilepticregion are used to build a space-time-spike tensor from theEEG data. A Canonical Polyadic (CP) decomposition ofthis tensor is computed such that the column vectors ofone loading matrix are linear combinations of a physics-driven dictionary. A performance study on realistic simu-lated EEG data of the proposed tensor-based approach isprovided.

Laurent Albera

Universite de Rennes [email protected]

Ahmad KarfoulAl-Baath [email protected]

Hanna BeckerGIPSA-Lab [email protected]

Remi GribonvalProjet METISS, IRISA-INRIA35042 Rennes cedex, [email protected]

Amar Kachenoura, Lofti SenhadjiUniversite de Rennes [email protected],[email protected]

Guillotel PhilippeTechnicolor R&D France, [email protected]

Isabelle MerletInserm & Universite de Rennes 1, [email protected]

MS24

On the Uniqueness of Coupled Matrix Block Di-agonalization in the Joint Analysis of MultipleDatasets

Uniqueness of matrix (or tensor) factorizations is neces-sary in order to achieve interpretability and attribute phys-ical meaning to factors. In this work, we discuss newuniqueness results on a coupled matrix block diagonaliza-tion model. This model is associated with the joint blindsource separation of multiple datasets, where each datasetsis a linear mixture of statistically independent multidimen-sional components.

Dana Lahat, Christian [email protected],[email protected]

MS24

The Decomposition of Matrices

There are well-known matrix decompositions:LU-decomposition, SVD-decomopostion and QR-decomposition. In this talk we will discuss new types ofmatrix decompositions. We will use algebraic geometryto show that every matrix can be decomposed into theproduct of finitely many Toeplitz or Hankel matrices.Our method can also recover LU-decomposition andQR-decompositon for generic matrices. This is a jointwork with Lim Lek-Heng.

Ke Ye, Lek-Heng LimUniversity of [email protected], [email protected]

MS24

”Sequentially Drilled” Joint Congruence Decom-

LA15 Abstracts 91

positions (SeDJoCo): The Simple Case and theExtended Case

We present the formulation and several associated prop-erties of a particular congruence transformations, whichwe term ”Sequentially Drilled” Joint Congruence (SeD-JoCo) decompositions and Extended SeDJoCo decompo-sitions. In its basic form, SeDJoCo is defined as follows:Given an ordered set of K (conjugate-)symmetric target-matrices of dimensions KxK, find a KxK transformationmatrix, such that when the k-th target-matrix is multi-plied by the transformation matrix on the left, and byits (conjugate) transpose on the right, the resulting ma-trix is drilled on its k-th column (and k-th row), namely,all elements in that column are zeros, except for the di-agonal ((k,k)-th) element. We show that this problem isclosely related to Maximum Likelihood (ML) IndependentComponent Analysis (ICA) in some contexts, as well as tothe problem of Coordinated Beamforming (CBF) in multi-user Multiple-Inputs Multiple-Outputs (MIMO) communi-cations. We show that a sufficient condition for existenceof a solution to this problem is that all the target matri-ces be positive definite. We also present some iterativesolutions. We then proceed to describe the Extended SeD-JoCo, in which, for some M, an ordered set of KM2 targetmatrices (each of dimensions KxK) is considered, and Mtransformation matrices are sought, satisfying a more com-plicated set of congruence transformations. This extendedversion is closely related to ML Independent Vector Anal-ysis (IVA), an emerging extension of ICA which considersM mixtures, with correlations between sources participat-ing in different mixtures. It is also related to CBF in thecontext of multicast Multi-Users MIMO.

Yao ChengIlmenau University of [email protected]

Arie yeredorTel-Aviv [email protected]

Martin HaardtIlmenau University of [email protected]

MS25

Network Oblivious Algorithms

Network obliviousness is a framework for design of algo-rithms that can run efficiently on machines with differentparallelism and communication capabilities. Algorithmsare specified only in terms of the input size and evaluatedon a model with two parameters, capturing parallelism andcommunication latency. Under broad conditions, optimal-ity in the latter model implies optimality in DecomposableBSP. Optimal network-oblivious algorithms are given for afew key problems. Limitations of non-oblivious broadcastare established.

Gianfranco Bilardi, Andrea Pietracaprina, Geppino PucciUniversity of [email protected], [email protected],[email protected]

Michele ScquizzatoUniversity of [email protected]

Francesco SilvestriIT University of [email protected]

MS25

Characterizing the Communication Costs of SparseMatrix Multiplication

Parallel sparse matrix-matrix multiplication is a key com-putational kernel in scientific computing. Although its run-ning time is typically dominated by the cost of interpro-cessor communication, little is known about these commu-nication costs. We propose a hypergraph model for thisproblem, in which vertices represent nonzero scalar mul-tiplications, hyper-edges represent data, and the optimalvertex partition represents a communication-minimizing,load-balanced distribution of work. We demonstrate thepotential of this model and similar coarser ones for deriv-ing communication lower bounds.

Grey BallardSandia National [email protected]

Alex DruinskyComputational Research DivisionLawrence Berkeley National [email protected]


Oded SchwartzHebrew University of [email protected]

MS25

Data Movement Lower Bounds for ComputationalDirected Acyclic Graphs


Saday SadayappanOhio State [email protected]

MS25

Using Symmetry to Schedule Algorithms

Presented with a machine with a new interconnect topol-ogy, designers use intuition about the geometry of the algo-rithm and the machine to design time and communication-efficient schedules that map the algorithm onto the ma-chine. Is there a systematic procedure for designing suchschedules? We explore a new algebraic approach with aview towards automating schedule design. Specifically, wedescribe how the ”symmetry” of matrix multiplication canbe used to derive schedules for different network topologies.

Harsha Vardhan SimhadriLawrence Berkeley National [email protected]

MS26

O(N) Density Functional Theory Calculations: The

92 LA15 Abstracts

Challenge of Going Sparse

To reduce computational complexity in Density FunctionalTheory calculations with wave functions, a special rep-resentation of the solution is required. Sparse solutions,which retain the accuracy of traditional approaches arepossible but present many challenges. Nonetheless, theyalso offer many opportunities to solve much larger andmore realistic problems in material sciences, and to scaleon much larger high performance computing platforms. Inthis talk, we will discuss our recent developments usingreal-space discretizations. This work was performed underthe auspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract DE-AC52-07NA27344.

Jean-Luc FattebertLawrence Livermore National [email protected]

Daniel Osei-KuffuorLawrence Livermore National [email protected]

MS26

Randomized Estimation of Spectral Densities ofLarge Matrices Made Accurate

In physics, it is sometimes desirable to compute the spec-tral density of a Hermitian matrix A. In many cases, theonly affordable operation is matrix-vector multiplication,and stochastic based methods can be used to estimate theDOS. The accuracy of stochastic algorithms converge asO(1/

√Nv), where Nv is the number of stochastic vectors.

I will introduce a new method called spectrum sweeping,which is a stochastic method and the convergence rate canbe much faster than O(1/

√Nv) for modest choice of Nv .

Lin LinUniversity of California, BerkeleyLawrence Berkeley National [email protected]

MS26

Challenges and Opportunities for Solving Large-scale Eigenvalue Problems in Electronic StructureCalculations

Realistic first-principle quantum simulations applied tolarge-scale atomistic systems, pose new challenges in thedesign of numerical algorithms that are both capable ofprocessing a considerable amount of generated data, andachieving significant parallel scalability on modern highend computing architectures. In this presentation, we ex-plore variations of the FEAST eigensolver that can consid-erably broaden the perspectives for addressing such diffi-culties.

Eric PolizziUniversity of Massachusetts, Amherst, [email protected]

MS26

Towards Large and Fast Density Functional TheoryCalculations

Electronic structure calculations based on Density Func-tional Theory (DFT) have been remarkably successful indescribing material properties and behavior. However, the

large computational cost associated with these simulationshas severely restricted the size of systems that can be rou-tinely studied. In this talk, previous and current effortsof the speaker to develop efficient real-space formulationsand parallel implementations for DFT will be discussed.These include linear scaling methods applicable to bothinsulating and metallic systems.

Phanish SuryanarayanaGeorgia Institute of [email protected]

MS27

Arrangements of Equal Minors in the PositiveGrassmannian

We investigate the structure of equalities and inequalitiesbetween the minors of totally positive matrices. We showthat collections of equal minors of largest value are in bijec-tion with sorted sets, which earlier appeared in the contextof alcoved polytopes. We also prove in many cases that col-lections of equal minors of smallest value are exactly weaklyseparated sets. Such sets are closely related to the positiveGrassmannian and the associated cluster algebra.

Miriam [email protected]

Alexander PostnikovMassachusetts Institute of [email protected]

MS27

Preserving Positivity for Matrices with SparsityConstraints

Functions preserving positivity have been investigated byanalysts and linear algebraists for much of the last fewdecades. It is well known from the work of Rudin andSchoenberg that functions (applied entrywise) which areanalytic and absolutely monontonic preserve positivity forall positive definite matrices of any size. The emergence ofdata science has raised new questions in the field of posi-tivity. In this talk we investigate maps which preserve pos-itivity when various sparsity, rank and other constraintsare imposed. As a consequence, we extend the results ofRudin and Schoenberg with modern motivations in mind.(Joint with B. Rajaratnam and A. Khare)

Bala RajaratnamStanford [email protected]

Dominique GuillotUniversity of [email protected]

Apoorva KhareStanford [email protected]

MS27

Critical Exponents and Positivity

We classify the powers that preserve positivity, when ap-plied entrywise to positive semidefinite (psd) matrices withrank and sparsity constraints. This is part of a broad pro-

LA15 Abstracts 93

gram to study positive entrywise functions on distinguishedsubmanifolds of the cone. In our first main result, we com-pletely classify the powers preserving Loewner propertieson psd matrices with fixed dimension and rank. This in-cludes the case where the matrices have negative entries.Our second main result characterizes powers preservingpositivity on matrices with zeros according to a chordalgraph. We show how preserving positivity relates to thegeometry of the graph, thus providing interesting connec-tions between combinatorics and analysis. The work hasapplications in regularizing covariance/correlation matri-ces, where entrywise powers are used to separate signalfrom noise, while minimally modifying the entries of theoriginal matrix. (Based on joint work with D. Guillot andB. Rajaratnam.)

Apoorva KhareStanford [email protected]

Dominique GuillotUniversity of [email protected]

Bala RajaratnamStanford [email protected]

MS27

The Jordan Form of An Irreducible EventuallyNonnegative Matrix

A square complex matrix A is eventually nonnegative ifthere exists a positive integer k0 such that for all k ≥ k0,Ak ≥ 0. We say a matrix is strongly eventually nonnega-tive if it is eventually nonnegative and has an irreduciblenonnegative power. We present some results about theJordan forms of strongly eventually nonnegative matricesand the Jordan forms of irreducible, eventually nonnega-tive matrices.

Ulrica WilsonMorehouse [email protected]

MS28

Dynamic Network Centrality with Edge Multi-damping

Dynamic centrality measures provide a numerical tool forthe identification of key players in a network whose edgesare evolving over time. In this talk we consider a Katz-typecentrality measure modified to accommodate the dynami-cal aspect of the network. We study the significance of anedge attenuation parameter to the ordinal rankings at agiven time. We introduce a concept of edge multidampingallowing for a time-dependent variant of the standard Katzparameter.

Mary AprahamianSchool of MathematicsThe University of Manchester

[email protected]

MS28

Solving Graph Laplacians for Complex Networks

The graph Laplacian is often used in network analysis.Therefore, solving linear systems and eigensystems forgraph Laplacians is of great practical interest. For largeproblems, iterative solvers such as preconditioned conju-gate gradients are needed. We show that Jacobi and Gauss-Seidel preconditioning are remarkably effective for power-law graphs. Support-graph (combinatorial) precondition-ers can reduce the number of iterations further but maynot be faster in practice. Finally, we discuss and show pre-liminary results for a recent theoretically almost-optimalalgorithm by Kelner et al.

Erik G. BomanCenter for Computing ResearchSandia National [email protected]

Kevin DeweeseUC Santa [email protected]

MS28

On Growth and Form of Networks

A new spectral scaling method to detecting topological ir-regularities in networks is introduced. It is based on thecommunicability function and the principal eigenvector ofthe adjacency matrix of a network. The relation betweenthe spatial efficiency of a network, measured by means ofthe communicability angles, and the topological hetero-geneities is found. We then analyse the growing of Erdos-Renyi random graphs, networks growing with modularity,fractal networks, and networks growing under spatial con-straints.

Ernesto EstradaUniversity of [email protected]

MS28

Using the Heat Kernel of a Graph for Local Algo-rithms

The heat kernel is a fundamental tool in extracting local in-formation about large graphs. In fact, the heat kernel canbe restricted to a few entries which are enough to summa-rize local structure. In this talk, I will present an efficientalgorithm for approximating the heat kernel of a graph. Iwill also present two local algorithms using the approxima-tion procedure; a local cluster detection algorithm, and alocal Laplacian linear solver.

Olivia SimpsonUniversity of California, San [email protected]

MS29

Constraint Preconditioning for the CoupledStokes-Darcy System

We propose the use of constraint preconditioned GMRESfor the solution of the linear system arising from the finiteelement discretization of coupled Stokes-Darcy flow. We

94 LA15 Abstracts

provide spectral and field-of-values bounds for the precon-ditioned system which are independent of the underlyingmesh size. We present several numerical experiments in 3Dthat illustrate the effectiveness of our approach.

Scott Ladenheim, Daniel B. SzyldTemple [email protected], [email protected]

Prince ChidyagwaiLoyola University MarylandDepartment of Mathematics and [email protected]

MS29

Krylov Methods in Adaptive Finite Element Com-putations

Since decades modern technological applications lead tochallenging PDE eigenvalue problems, e.g., vibrations ofstructures, modeling of photonic gap materials, analysis ofthe hydrodynamic stability, or calculations of energy lev-els in quantum mechanics. Recently, a lot of research isdevoted to the so-called Adaptive Finite Element Meth-ods (AFEM). In most AFEM approaches the underlyingalgebraic problems, i.e., linear systems or eigenvalue prob-lems, are of large dimension which makes the Krylov sub-space methods of particular importance. The goal of this

work is to analyze the influence of the accuracy of the al-gebraic approximation, e.g. inexact Krylov method on theadaptivity process. Moreover, we discuss how to reducethe computational effort of the iterative solver by adapt-ing the size of the Krylov subspace. Based on perturba-tion results in H1(Ω)- and H−1(Ω)-norm derived in [1] anda standard residual a posteriori error estimator a balanc-ing AFEM algorithm is proposed. [1] U. L. Hetmaniuk

and R. B. Lehoucq. Uniform accuracy of eigenpairs from ashift-invert Lanczos method. SIAM J. Matrix Anal. Appl.,28:927–948, 2006.

Agnieszka MiedlarTechnische Universitat [email protected]

MS29

Near-Krylov for Eigenvalues and Linear EquationsIncluding Multigrid and Rank-1

We look at several eigenvalue and linear equations com-putations that involve near-Krylov subspaces. Includedare two-grid methods that are more robust than standardmultigrid and methods for Rank-1 updates of large matri-ces. Recycling methods for changing matrices are also con-sidered. We analyze some of the properties of near-Krylovsubspaces that are used by these methods.

Ron MorganBaylor UniversityRonald [email protected]

MS29

Efficient Iterative Algorithms for Linear StabilityAnalysis of Incompressible Flows

Linear stability analysis of a dynamical system entails find-ing the rightmost eigenvalue for a series of eigenvalue prob-lems. For large-scale systems, it is known that conventional

iterative eigenvalue solvers are not reliable for computingthis eigenvalue. A recently developed and more robustmethod, Lyapunov inverse iteraiton, involves solving large-scale Lyapunov equations, which in turn requires the solu-tion of large, sparse linear systems analogous to those aris-ing from solving the underlying partial differential equa-tions. This study explores the efficient implementation ofLyapunov inverse iteration when it is used for linear sta-bility analysis of incompressible flows. Efficiencies are ob-tained from effective solution strategies for the Lyapunovequations and for the underlying partial differential equa-tions. Solution strategies based on effective precondition-ing methods and on recycling Krylov subspace methodsare tested and compared, and a modified version of a Lya-punov solver is proposed that achieves significant savingsin computational cost.

Minghao W. RostamiWorcester Polytechnic [email protected]


MS30

Squish Scalings

Given non-negative matrices A,L,U ∈ Rn×n we show howto compute a scalar α ∈ R and a diagonal matrixD ∈ Rn×n

such thatL/β ≤ αD−1AD ≤ Uβ,

for the least possible β ∈ R. The entities in A are ‘pusheddown’ by the upper bound U and ‘pushed up’ thy thelower bound L. Hence squish scaling. Squish scalings arecomputed using max-plus algebra and have applications ingraph theory and eigenvalue computation.

James HookSchool of MathematicsUniversity of [email protected]

MS30

Using Matrix Scaling to Identify Block Structure

We can apply a two-sided diagonal scaling to a nonnegativematrix to render it into doubly stochastic form if and onlyif the matrix is fully indecomposable. The scaling oftenreveals key structural properties of the matrix as the effectsof element size and connectivity are balanced. Exploitingkey spectral properties of doubly stochastic matrices, wewill show how to use the scaling to reveal hidden blockstructure in matrices without any prior knowledge.

Philip KnightUniversity of Strathclyde, Glasgow, [email protected]

MS30

Hungarian Scaling of Polynomial Eigenproblems

We study the behaviour of the eigenvalues of a paramet-ric matrix polynomial P in a neighbourhood of zero. Ifwe suppose that the entries of P have Puiseux series ex-pansions we can build an auxiliary matrix polynomial Qwhose entries are the leading exponents of those of P. Weshow that preconditioning P via a diagonal scaling based

LA15 Abstracts 95

on the tropical eigenvalues of Q can improve conditioningand backward error of the eigenvalues.

Marianne Akian, Stephane GaubertINRIA and CMAP, Ecole [email protected], [email protected]

Andrea [email protected]

Francoise TisseurThe University of [email protected]

MS30

Max-Balancing Hungarian Scalings

A Hungarian scaling is a two-sided diagonal scaling of amatrix, which is typically applied along with a permuta-tion to sparse indefinite nonsymmetric linear system beforecalling a direct or iterative solver. A Hungarian scaled andreordered matrix has all its entries of modulus less thanor equal to one and entries of modulus one on the diago-nal. We use max-plus algebra to characterize the set of allHungarian scalings for a given matrix and show that max-balancing a Hungarian scaled matrix yields the most diago-nally dominant Hungarian scaled matrix possible. We alsopropose an approximate max-balancing Hungarian scalingwhose computation is embarrassingly parallel.

Francoise TisseurThe University of ManchesterSchool of [email protected]

James HookSchool of MathematicsUniversity of [email protected]

Jennifer PestanaSchool of MathematicsThe University of [email protected]

MS31

A High-Accuracy SR1 Trust-Region SubproblemSolver for Large-Scale Optimization

In this talk we present an SR1 trust-region subproblemsolver for large-scale optimization. This work makes useof the compact representation of an SR1 matrix and a QRfactorization to obtain the eigenvalues of the SR1 matrix.Optimality conditions are used to find a high accuracy solu-tion to each subproblem even in the so-called ”hard case”.Numerical results will be presented.

Jennifer ErwayWake Forest [email protected]

MS31

Compact Representation of Quasi-Newton Matri-ces

Very large systems of linear equations arising from quasi-Newton methods can be solved efficiently using the com-

pact representation of the quasi-Newton matrices. In thistalk, we present a compact formulation for the entire Broy-den convex class of updates for limited-memory quasi-Newton methods and how they can be used to solve large-scale trust-region subproblems with quasi-Newton Hessianapproximations.

Roummel F. MarciaUniversity of California, [email protected]

MS31

Obtaining Singular Value Decompositions in a Dis-tributed Environment

Matrix decompositions like the singular value decomposi-tions are important analysis tools for processing and an-alyzing large amounts data. It is thus important to de-velop stable robust approaches to efficiently processinglarge datasets in a distributed environment. We present re-sults for processing both sparse and dense data for big dataproblems. We further explore the efficiency of some promis-ing approximate optimization-based approaches that pro-vide more flexibility in customizing the canonical problemformulation.

Ben-Hao Wang, Joshua GriffinSAS Institute, [email protected], [email protected]

MS31

Linear Algebra Software in Nonlinear Optimization

We discuss some practical issues associated with sequen-tial quadratic programming (SQP) methods for large-scalenonlinear optimization. In particular, we focus on SQPmethods that can exploit advances in linear algebra soft-ware. Numerical results are presented for the latest versionof the software package SNOPT when utilizing third-partylinear algebra software packages.

Elizabeth WongUniversity of California, San DiegoDepartment of [email protected]

MS32

Low-Rank Cross Approximation Methods for Re-duction of Parametric Equations

The low-rank decomposition of matrices and tensors is aknown data compression technique. One of the problemsis to build a low-rank representation, given a procedure toevaluate a particular element of a tensor. The cross ap-proximation algorithms select a few entries to constructthe low-rank factorization. However, these methods as-sume that each fixed element of a tensor is a single number,and that different elements can be evaluated independentlyon each other. For stochastic PDEs this is not the case:each parameter point produces a vector of the discrete so-lution of the PDE. This would hinder the performance ofthe cross algorithms, since a large portion of data is likelyto be wasted. We extend the TT cross algorithm to ap-proximate many tensors simultaneously in the same low-rank format. Besides, for a linear SPDE we can use thecomputed low-rank factors as model reduction bases, andincrease the efficiency further.

Sergey Dolgov

96 LA15 Abstracts

MPI - [email protected]

MS32

Hierarchical Low Rank Approximation for ExtremeScale Uncertainty Quantification

We consider the problem of uncertainty quantification forextreme scale parameter dependent problems where an un-derlying low rank property of the parameter dependencyis assumed. For this type of dependency the hierarchicalTucker format offers a suitable framework to approximate agiven output function of the solutions of the parameter de-pendent problem from a number of samples that is linear inthe number of parameters. In particular we can a posterioricompute the mean, variance or other interesting statisticalquantities of interest. In the extreme scale setting it is al-ready assumed that the underlying fixed-parameter prob-lem is distributed and solved for in parallel. We providein addition a parallel evaluation scheme for the samplingphase that allows us on the one hand to combine severalsolves and on the other hand parallelise the sampling.

Lars GrasedyckRWTH [email protected]

Jonas BallaniEPF [email protected]

Christian LoebbertRWTH [email protected]

MS32

Interpolation of Inverse Operators for Precon-ditioning Parameter-Dependent Equations andProjection-Based Model Reduction Methods

We present a method for the construction of precondition-ers for large systems of parameter-dependent equations,where an interpolation of the matrix inverse is computedthrough a projection of the identity with respect to randomapproximations of the Frobenius norm. Adaptive interpo-lation strategies are then proposed for different objectivesin the context of projection-based model order reductionmethods: error estimation, projection on a given approxi-mation space, or recycling of computations.

Anthony NouyEcole Centrale Nantes, [email protected]

MS32

Stochastic Galerkin Method for the Steady StateNavier-Stokes Equations

We study steady state Navier-Stokes equations in the con-text of stochastic finite element discretizations. We assumethat the viscosity is given in terms of a generalized polyno-mial chaos (gPC) expansion. We formulate the model andlinearization schemes using Picard and Newton iterationsin the framework of the stochastic Galerkin method, andcompare the results with that of stochastic collocation andMonte Carlo methods. We also propose a preconditionerfor systems of equations solved in each step of the stochas-

tic (Galerkin) nonlinear iteration and we demonstrate itseffectiveness in a series of numerical experiments.

Bedrich SousedikUniversity of Maryland, Baltimore [email protected]


MS33

Global and Local Methods for Solving the Gravi-tational Lens Equation

The effect of gravitational microlensing can be modeledby the lens equation R(z) − z, where R(z) is a complexrational function. The zeros of this function correspond toimages of a distant light source, created by the action ofgravity on light. We discuss the numerical solution of thelens equation, including a reformulation as a rootfindingproblem of a polyanalytic polynomial, and discuss localoptimization methods in the complex variables z and z.

Robert LuceTU [email protected]

MS33

Fast and Stable Computation of the Roots of Poly-nomials

A stable algorithm to compute the roots of polynomials inthe monomial basis is presented. Implicit QR steps are ex-ecuted on a suitable representation of the iterates, whichremain unitary plus low rank. The computational com-plexity is reduced to a minimum by exploiting the redun-dancy present in the representation allowing to ignore thelow rank part. By ignoring the low rank part the remain-ing computations are solely based on rotations and thusunitary.

Thomas MachTU [email protected]

Jared AurentzUniversity of OxfordMathematical [email protected]


David S. WatkinsWashington State UniversityDepartment of [email protected]

MS33

Stable Polefinding-Rootfinding and Rational Least-Squares Fitting Via Eigenvalues

One way of finding the poles/roots of a meromorphic func-tion is rational interpolatation, followed by rootfinging for

LA15 Abstracts 97

the denominator. This is a two-step process and the type ofthe rational interpolant is required as input. Moreover, thenumerical stability has remained largely uninvestigated.This work introduces an algorithm with the features: (i)automatic detection of the numerical type (ii) pole/root-finding via a generalized eigenvalue problem in a one-stepfashion, and (iii) backward stability, defined appropriately.

Shinji ItoUniversity of Tokyoshinji [email protected]

Yuji NakatsukasaDepartment of MathematicsUniversity of [email protected]

MS33

Resultant Methods for Multidimensional Rootfind-ing Are Exponentially Unstable

Hidden-variable resultant methods are a class of algorithmsfor the solution of systems of polynomial equations. Whenthe number of variable is 2, when care is taken, they can becompetitive. Yet, in higher dimensions they are known tomiss solutions, calculate roots to low precision, and intro-duce spurious ghost solutions. We show that the popularhidden-variable resultant method based on the Cayley re-sultant is inherently and spectacularly numerically unsta-ble, by a factor that grows exponentially with the dimen-sion. We also show the ill-conditioning of the Sylvesterresultant for solving systems of two bivariate polynomialequations. This provides a rigorous explanation of the nu-merical difficulties that practitioners are frequently report-ing.


Alex TownsendDepartment of [email protected]

MS34

Singular Values and Vectors under Random Per-turbation

Computing the singular values and singular vectors of alarge matrix is a basic task in high dimensional data anal-ysis with many applications in computer science and statis-tics. In practice, however, data is often perturbed by noise.A natural question is the following. How much does a smallperturbation to the matrix change the singular values andvectors? Classical (deterministic) theorems, such as thoseby Davis-Kahan, Wedin, and Weyl, give tight estimatesfor the worst-case scenario. In this talk, I will considerthe case when the perturbation is random. In this setting,better estimates can be achieved when our matrix has lowrank. This talk is based on joint work with Van Vu andKe Wang.

Sean O’RourkeUniversity of Colorado [email protected]

Van Vu

Yale [email protected]

Ke WangInstitute for Mathematics and its ApplicationsUniversity of [email protected]

MS34

The Stochastic Block Model and Communities inSparse Random Graphs: Detection at OptimalRate

We consider the problem of communities detection insparse random graphs. Our model is the so-called Stochas-tic Block Model, which has been immensely popular in therecent statistics literature. Let X1, ...Xk be vertex sets ofsize n each (where k is fixed and n is large). One drawsrandom edges inside each Xi with probability p/n, and be-tween Xi and Xj with probability q/n, for some constantsp and q. Given one instance of this sparse random graph,our goal is to recover the sets Xi as correctly as possible.We are going to present a fast spectral algorithm whichdoes this job at the optimal rate (namely the relation be-tween p, q and the number of mistakes in the recovery isoptimal). Our algorithm is based on spectral properties ofrandom sparse matrices and is easy to implement. We willalso discuss several related works and algorithms and anopen question concerning perturbation of random matri-ces.

Anup [email protected]

MS34

Robust Tensor Decomposition and IndependentComponent Analysis

ICA is a classical model of unsupervised learning originat-ing and widely used in signal processing. In this model,observations in n-space are obtained by an unknown lineartransformation of a signal with independent components.The goal is to learn the the transformation and therebythe hidden basis of the independent components. In thistalk, we’ll begin with a polynomial-time algorithm calledFourier PCA, for underdetermined ICA, the setting wherethe observed signal is in a lower dimensional space than theoriginal signal, assuming a mild, verifiable condition on thetransformation. The algorithm is based on a robust tensordecomposition method applied to a higher-order derivativetensor of the Fourier transform and its analysis uses theanticoncentration of Gaussian polynomials. The runningtime and sample complexity are polynomial, but of highconstant degree, for any constant error; a MATLAB im-plementation of the algorithm appears to need 104 samplesfor 10-dimensional observations. We then present a moreefficient recursive variant for the fully determined case ofstandard ICA, which needs only a nearly linear number ofsamples and has polynomial time complexity. Its analysisis based on the gaps between the roots of polynomials ofGaussian random variables.

Santosh VempalaGeorgia Institute of [email protected]

Navin GoyalMicrosoft Research

98 LA15 Abstracts

[email protected]

Ying [email protected]

MS34

Applications of Matrix Perturbation Bounds withRandom Noise in Matrix Recovery Problems

The general matrix recovery problem is the following: A isa large unknown matrix. We can only observe its noisy ver-sion A+E, or in some cases just a small part of it. We wouldlike to reconstruct A or estimate an important parameteras accurately as possible from the observation. In a recentwork, we study how much the singular value or singularvector of data matrix will change under a small perturba-tion. We show that better estimates can be achieved if thedata matrix is low rank and the perturbation is random,improving the classical perturbation results. In this talk,I would like to discuss some applications of our results inthe matrix recovery problems. This is joint work with SeanO’Rourke and Van Vu.

Sean O’RourkeUniversity of Colorado [email protected]

Van VuYale [email protected]

Ke WangInstitute for Mathematics and its ApplicationsUniversity of [email protected]

MS35

A Hybrid LSMR Algorithm for Large-ScaleTikhonov Regularization

We develop a hybrid iterative approach for computing solu-tions to large-scale ill-posed inverse problems via Tikhonovregularization. We consider a hybrid LSMR algorithm,where Tikhonov regularization is applied to the LSMR sub-problem rather than the original problem. We show that,contrary to standard hybrid methods, hybrid LSMR iter-ates are not equivalent to LSMR iterates on the directlyregularized Tikhonov problem. Instead, hybrid LSMRleads to a different Krylov subspace problem. We showthat hybrid LSMR shares many of the benefits of standardhybrid methods such as avoiding semiconvergence behav-ior. In addition, since the regularization parameter can beestimated during the iterative process, it does not need tobe estimated a priori, making this approach attractive forlarge-scale problems. We consider various methods for se-lecting regularization parameters and discuss stopping cri-teria for hybrid LSMR, and we present results from imageprocessing.

Julianne ChungVirginia [email protected]

Katrina PalmerDepartment of Mathematical SciencesAppalachian State University

[email protected]

MS35

Lanczos Bidiagonalization with Subspace Auge-mentation for Discrete Inverse Problems

The regularizing properties of Lanczos bidiagonalizationwere studied and advocated by many researchers, includ-ing Dianne O’Leary. This approach is powerful when theunderlying Krylov subspace captures the dominating com-ponents of the solution. In some applications the regular-ized solution can be further improved by augmenting theKrylov subspace with a low-dimensional subspace that rep-resents specific prior information. Inspired by earlier workon GMRES we demonstrate how to carry these ideas overto the Lanczos bidiagonalization algorithm.

Per Christian HansenTechnical University of DenmarkInformatics and Mathematical [email protected]

MS35

A Flexible Regression Model for Count Data

Poisson regression is a popular tool for modeling countdata and is applied in a vast array of applications. Realdata, however, are often over- or under-dispersed and, thus,not conducive to Poisson regression. We instead proposea regression model based on the ConwayMaxwell-Poisson(COM-Poisson) distribution to address this problem. TheCOM-Poisson regression generalizes the well-known Pois-son and logistic regressions, and is suitable for fitting countdata with a wide range of dispersion levels.

Kimberly F. SellersMathematics and Statistics, Georgetown [email protected]

MS35

Solving Eigenvalue and Linear System Problems onGPU for Three-Dimentional Photonic Device Sim-ulations

Solutions of discretized Maxwell’s equations are essentialto three-dimensional photonic device simulations. For fullbandgap diagrams of photonic crystals, we demonstratehow the wanted interior eigenvalues can be computed ina ultrafast manner by the nullspace free algorithm on amultiple-GPU cluster. For linear systems in frequency-domain simulation, we explore the homogeneous and pe-riodic structures to develop a domain decomposition typealgorithm. The algorithm significantly reduces the memoryusage with satisfactory scalability and acceleration perfor-mance.

Weichung WangInstitute of Applied Mathematical SciencesNational Taiwan [email protected]

MS36

Rotated Block Two-by-Two Preconditioners Basedon Pmhss

Motivated by the HSS iteration method, to solve thedistributed control problem we have proposed modifiedHSS (MHSS) as well as preconditioned and modified HSS

LA15 Abstracts 99

(PMHSS) iteration methods. Theoretical analyses and nu-merical experiments have shown the effectiveness of theseiteration methods when used either as linear solvers oras matrix preconditioners for Krylov subspace methods.Moreover, we have developed the PMHSS iteration methodto block two-by-two matrices of non-Hermitian sub-blocks,resulting in the additive-type and the rotated block tri-angular preconditioners. Spectral analyses and numericalcomputations have shown that these preconditioners cansignificantly accelerate the convergence rates of the Krylovsubspace iteration methods when they are used to solvethe block two-by-two linear systems, and the inexact vari-ants these preconditioners are as effective and robust asthe corresponding exact ones.

Zhong-Zhi BaiInstitute of Computational MathematicsChinese Academy of Sciences, Beijing, [email protected]

MS36

Optimal Order Multigrid Preconditioners for Lin-ear Systems Arising in the Semismooth NewtonMethod Solution Process of a Class of Control-Constrained Problems

We introduce multigrid preconditioners for the semismoothNewton solution process of certain optimal control prob-lems with control-constraints. Each semismooth Newtoniteration essentially requires inverting a principal subma-trix of the matrix entering the normal equations of theassociated unconstrained optimal control problem. Usinga piecewise constant discretization of the control space andnon-conforming coarse spaces, we show that, under reason-able geometric assumptions, the preconditioner approxi-mates the inverse of the desired matrix to optimal order.

Andrei DraganescuDepartment of Mathematics and StatisticsUniversity of Maryland, Baltimore [email protected]

Jyoti SaraswatThomas More [email protected]

MS36

The Iterative Solution of Systems from Pde Con-strained Optimization: An Overview

In this, the introductory talk of the minisymposium ”Lin-ear Algebra for PDE-constrained optimization”, we set thescene by introducing the PDE-constrained optimizationproblem and describe a few of the ways such problems canbe discretized, give the resulting linear systems. Solvingthis linear system is typically the bottleneck in codes thatsolve such problems, and fast and efficient iterative meth-ods to do this are the topic of this session. We describe anumber of the common methods to solve such systems, andset the scene for the rest of the talks in the minisymposium.

Tyrone ReesRutherford Appleton Lab, [email protected]

MS36

Inexact Full-space Methods for PDE-constrained

Optimization

We discuss a hierarchy of inexact full-space methods forlarge-scale optimization known as matrix-free sequentialquadratic programming (SQP) methods. The hierarchy isbased on the increasing complexity of linear systems and,consequently, the increasing flexibility in the design of ef-ficient linear solvers and preconditioners. We examine thesolver performance on large-scale optimization problemswith linear and nonlinear PDE constraints, related to theoptimal design of acoustic fields and optimal control of ra-diative heat transfer.

Denis RidzalSandia National [email protected]

MS37

Eigenvalues and Beyond

Spectral graph partitioning and graph-based signal pro-cessing are mathematically based on the spectral decom-position of the graph Laplacian. Computing the full eigen-value decomposition is impractical, since the graph Lapla-cian can be of an enormous size. Thus, iterative approachesare of great importance such as, e.g., approximate low-pass filters resulted from Chebyshev and Krylov based it-erations. We present an overview of some of our recentlypublished results in this area.

Andrew KnyazevMitsubishi Electric Research [email protected]

MS37

Implementation of LOBPCG for Symmetric Eigen-value Problems in SLEPc

SLEPc is a parallel library that provides a collectionof eigenvalue solvers such as Krylov-Schur and Jacobi-Davidson. We have recently added an implementationof LOBPCG, the Locally Optimal Block PreconditionedConjugate Gradient method [Knyazev, SIAM J. Sci. Com-put. 23 (2001)]. In this talk we discuss several imple-mentation details, such as locking, block-orthogonalization,and multi-vector operations, and compare the performancewith the solver provided in BLOPEX (which is also inter-faced in SLEPc).

Jose E. RomanUniversidad Politecnica de [email protected]

MS37

Spectrum Slicing by Polynomial and RationalFunction Filtering

Filtering techniques are presented for solving eigenvaluelarge Hermitian problems by spectrum slicing. Polynomialfiltering techniques can be quite efficient in the situationwhere the matrix-vector product operation is inexpensiveand when a large number of eigenvalues is sought, as isthe case in electronic structure calculations for example.Rational filtering techniques will also be described with afocus on showing the advantages of such filters obtained bya least-squares approach.

Yousef Saad

100 LA15 Abstracts

Department of Computer ScienceUniversity of [email protected]

MS37

Techniques for Computing a Large Number ofEigenpairs of Sparse, Hermitian Matrices

Computing a large number, m, of extreme or interior eigen-values of sparse Hermitian matrices remains an outstandingproblem in many applications, including materials scienceand lattice QCD. Current state-of-the-art methods are lim-ited by the cost of the orthogonalization or the RayleighRitz step which grows cubicly with m. We present a win-dow approach that uses high degree polynomials to avoidboth steps, at the expense of more matrix-vector products.


Eloy Romero AlcaldeComputer Science DepartmentCollege of Williams & [email protected]

MS38

Data Sparse Techniques for Parallel Hybrid Solvers

In this talk we will describe how H-matrix data sparse tech-niques can be implemented in a parallel hybrid sparse linearsolver based on algebraic non overlapping domain decom-position approach. Strong-hierarchical matrix arithmeticand various clustering techniques to approximate the localSchur complements will be investigated, aiming at reduc-ing workload and memory consumption while complyingwith structures of the local interfaces of the sub-domains.Utilization of these techniques to form effective global pre-conditioner will be presented.

Yuval [email protected]

Eric F. DarveStanford UniversityMechanical Engineering [email protected]

Luc GiraudInriaJoint Inria-CERFACS lab on [email protected]

Emmanuel [email protected]

MS38



Gunnar MartinssonUniv. of Colorado at Boulder

[email protected]

MS38

Fast Solvers for H2 Matrices Using Sparse Algebra

We will present a new class of preconditioners for sparsematrices, based on hierarchical matrices. Similar to theincomplete LU algorithm, this preconditioner controls thesparsity of the matrix as the LU factorization is progress-ing. However, low-rank approximations instead of thresh-olding are used to remove the fill-in. This solver also sharessimilarities with algebraic multigrid in its use of multiplescales to represent the problem. Many variants of thesenew preconditioners exist. We will focus on variants thatuse either a domain decomposition approach (simpler toimplement), or multifrontal approach (optimal but moredifficult to program). These solvers were shown to guar-antee a bounded number of iterations independent of thematrix size N . The cost of the preconditioner is O(r2N)where r is the rank used in the approximation. r has a slowdependence on N (logarithmic or slower).

Mohammad Hadi Pour [email protected]


Zhengyu Huang, Pieter [email protected], [email protected]

MS38

A Tensor Train Decomposition Solver for IntegralEquations on Two and Three Dimensions

We present an efficient method based on TT (tensor train)decomposition for solving volume and boundary integralequations in 2D and 3D. This method proceeds by inter-preting the corresponding matrix as a high-dimensionaltensor, exploiting a matrix-block low-rank structure to pro-duce a highly compressed approximation of the inverse.For a variety of problems, computational cost and storageneeded to construct a preconditioner are very low, com-pared to existing methods, with experimentally observedcomplexity O(logN) or lower. Once computed, it allowsfor a fast O(N logN) solve. For volume and boundary inte-grals in simple geometries, our method yield a practical fastO(N) direct solver. For boundary integrals in complex ge-ometries, we show that a low accuracy approximation canbe reliably used as a preconditioner.

Denis ZorinComputer Science DepartmentCourant Institute, New York [email protected]

Eduardo CoronaThe Courant Institute of Mathematical SciencesNew York [email protected]

Abtin RahimianGeorgia Institute of Technology

LA15 Abstracts 101

[email protected]

MS39

Approximating Kernel Matrix with Recursive Low-Rank Structure

Given a positive-definite function φ and a set of n points{xi}, a kernel matrix Φ has the (i, j) element beingφ(xi, xj). Kernel matrices play an important role in Gaus-sian process modeling and are widely used in statistics andmachine learning. A challenge, however, is that many cal-culations with these matrices are O(n3). We present an ap-proximation with a recursive low-rank structure that pre-serves the positive definiteness of the matrix. A benefit ofthe structure is that it allows an O(n) cost for many ma-trix computations of interest. We also present bounds tocharacterize the asymptomatic convergence of the approx-imation.

Jie ChenArgonne National [email protected]

MS39

A Robust Algebraic Schur Complement Precondi-tioner Based on Low Rank Corrections

In this talk we discuss LORASC, a robust algebraic pre-conditioner for solving sparse linear systems of equationsinvolving symmetric and positive definite matrices. Thegraph of the matrix is partitioned by using k-way par-titioning with vertex separators into N disjoint domainsand a separator formed by the vertices connecting the Ndomains. The obtained permuted matrix has a block ar-row structure. The preconditioner relies on the Choleskyfactorization of the first N diagonal blocks and on approx-imating the Schur complement corresponding to the sepa-rator block. The approximation of the Schur complementinvolves the factorization of the last diagonal block anda low rank correction obtained by solving a generalizedeigenvalue problem or a randomized algorithm. The pre-conditioner can be build and applied in parallel. Numericalresults on a set of matrices arising from the discretizationby the finite element method of linear elasticity modelsillustrate the robusteness and the efficiency of our precon-ditioner.


Frederic NatafLaboratoire J.L. [email protected]

Soleiman YousefIFPENsoleiman [email protected]

MS39

A Comparison of Different Low-Rank Approxima-tion Techniques

Linear solvers and preconditioners based on structured ma-trices have gained popularity in the past few years. Inparticular, Hierarchically Semi-Separable (HSS) matrices,Hierarchically Off-Diagonal Low-Rank (HODLR) matrices

and Block Low-Rank (BLR) matrices have been used todesign both dense and direct solvers. In this presentationwe compare these three techniques from a theoretical andpractical standpoint, using large dense and sparse problemsfrom real applications.

Francois-Henry RouetLawrence Berkeley National [email protected]

Patrick AmestoyENINPT-IRIT, Universite of Toulouse, [email protected] or [email protected]

Cleve AshcraftLivermore Software [email protected]

Alfredo ButtariCNRS-IRIT, [email protected]

Pieter GhyselsLawrence Berkeley National LaboratoryComputational Research [email protected]

Jean-Yves L’ExcellentINRIA-LIP-ENS [email protected]


Theo MaryUniversite de Toulouse, INPT [email protected]

Clement WeisbeckerINP(ENSEEIHT)[email protected]

MS39

An Algebraic Multilevel Preconditioner with Low-Rank Corrections for General Sparse SymmetricMatrices

In this talk we present a multilevel Schur low rank (MSLR)preconditioner for general sparse symmetric matrices. Thispreconditioner exploits a low-rank property that is satisfiedby the difference between the inverses of the local Schurcomplements and specific blocks of the original matrix andcomputes an approximate inverse of the original matrixfollowing a tree structure. We demonstrate its efficiencyand robustness through various numerical examples.

Yuanzhe XiPurdue [email protected]

Yousef SaadDepartment of Computer ScienceUniversity of [email protected]

102 LA15 Abstracts

Ruipeng LiDepartment of Computer Science & EngineeringUniversity of [email protected]

MS40

Spectral Clustering with Tensors

Two of the fundamental analyses of networks are clustering(partitioning, community detection) and the frequency ofnetwork motifs, or patterns of links between nodes. How-ever, these analyses are disjoint as community structure istypically defined by link relationships and ignore motifs.Here, we unify these two ideas through motif-based spec-tral clustering. Our framework represents motifs throughadjacency tensors and uses Markov chains derived fromthese tensors to find motif-based clusters.

Austin BensonStanford [email protected]


Jure LeskovecStanford [email protected]

MS40

Spacey Random Walks, Tensor Eigenvalues, andMultilinear PageRank

This presentation will present an overview of the mini-symposia. Then we’ll introduce the spacey-random walk,a new type of stationary distribution that gives rise to atensor eigenvalue problem. We’ll conclude by discussingthe multilinear PageRank vector and conditions when thePageRank eigenvector is unique and easy to compute.


Lek-Heng LimUniversity of [email protected]

Austin BensonStanford [email protected]

MS40

Eigenvalues of Tensors and a Second Order MarkovChain

It is possible to generalize the notion of eigenvalues of ma-trices to eigenvalues of tensors, from both algebraic and ge-ometric perspectives. Many interesting properties of eigen-values of matrices have analogues for eigenvalues of ten-sors. Especially, there are Perron-Frobenius type theoremsfor eigenvalues of nonnegative tensors. The theory can beapplied to study a particular second order Markov chainmodel. This talk will present some attempts along thisway.

Shenglong Hu

The Hong Kong Polytechnic [email protected]

MS40

The Anatomy of the Second Dominant Eigenvaluein a Transition Probability Tensor — The PowerIteration for Markov Chains with Memory

Though ineffective for general eigenvalue computation, thepower iteration has been of practical usage for comput-ing the stationary distribution of a stochastic matrix. Fora Markov chain with memory m, the transition ”matrix”becomes an order-m tensor. Under suitable tensor prod-ucts, the evolution of the chain with memory toward thelimiting probability distribution can be interpreted as twotypes of power-like iterations — one traditional and theother shorter alternative. What is not clear is the makeupof the “second dominant eigenvalue” that affects the con-vergence rate of the iteration, if the method converges atall. Casting the power method as a fixed-point iteration,this paper examines the local behavior of the nonlinearmap and identifies the cause of convergence or divergence.Equipped with this understanding, it is found that thereexists an open set of irreducible and aperiodic transitionprobability tensors where the short-cut type of power iter-ates fail to converge.

Moody T. ChuNorth Carolina State [email protected]

Sheng-Jhih WuSoochow [email protected]

MS41

Solving the Bethe-Salpeter Eigenvalue Problem us-ing Low-rank Tensor Factorization and a ReducedBasis Approach

The Bethe-Salpeter equation (BSE) is a reliable model forestimating the absorption spectra in molecules on the ba-sis of accurate calculation of the excited states from firstprinciples. Direct diagonalization of the BSE matrix is in-tractable due to the O(N6) complexity w.r.t. the atomicorbitals basis set. We present a new approach, yielding acomplexity of O(N3), based on a low-rank tensor approx-imation of the dense matrix blocks and a reduced basisapproach.

Peter BennerMax Planck Institute, Magdeburg, [email protected]

Venera KhoromskaiaMax Planck Institute for Mathematics in the [email protected]

Boris KhoromskijMax-Planck Institute, [email protected]

MS41

Fast Direct Solvers as Preconditioners for TimeEvolution Problems

For problems involving time evolution of geometries, thecost of a time step is often dominated by the cost of solv-

LA15 Abstracts 103

ing an integral equation. Typically, iterative solution tech-niques requiring many iterations are used as the computa-tion of a direct solver is too expensive to be done at eachtime step. This talk will illustrate the potential of cre-ating a high accuracy approximate inverse for a referencegeometry as preconditioner for the evolved geometries.

Adrianna GillmanRice UniversityDepartment of Computational and applied [email protected]

MS41

Hierarchical Matrix Factorization of the Inverse.

Given a sparse matrix, its LU-factors, inverse and inversefactors typically suffer from substantial fill-in, leading tonon-optimal complexities in their computation as well astheir storage. We will focus on the construction and rep-resentation of the (LU-) factors of the inverse of a matrix.We introduce a blockwise approach that permits to replace(exact) matrix arithmetic through (approximate but effi-cient) H-arithmetic. We conclude with numerical resultsto illustrate the performance of different approaches.

Sabine Le BorneTennessee Technological UniversityDepartment of [email protected]

MS41

H-Matrices for Elliptic Problems: Approximationof the Inverses of FEM and BEM Matrices.

Typically, the inversion or factorization of a rank-structured matrix cannot be performed exactly. A ba-sic question, therefore, is whether the inverse or the fac-tors can at least be approximated well in the chosen tar-get format. For the format of H-matrices (introduced byW. Hackbusch), which are blockwise low-rank matrices, weshow that indeed the inverses of Galerkin matrices arisingin FEM and in BEM can be approximated at an exponen-tial rate in the block rank.

Markus Melenk, Markus Faustmann, Dirk PraetoriusVienna University of [email protected], [email protected],[email protected]

MS42

Matrix Computations and Text Summarization

Automatic text summarization is an approach to address-ing information overload. Machine-generated summariesare applied to a range of genres from newswire to scien-tific reports. A user is presented with the key elements,extracted from one or more documents and arranged tobe both coherent and nonredundant. This talk will focuson matrix computations in text summarization, includingMarkov models, the QR decomposition, nonnegative ma-trix factorization and robust regression.

John M. ConroyIDA Center for Computing [email protected]

MS42

Towards Textbook Multigrid for the Helmholtz

Equation

The paper [H.C. Elman, O. G. Ernst and D. P. O’Leary:A multigrid method enhanced by Krylov subspace itera-tion for discrete Helmholtz equations] introduced a novelway of combining Krylov subspace iterations with a stan-dard multigrid V-cycle to obtain a robust solver for theHelmholtz equation which scales favorably with the wavenumber parameter. It also gave a simple model prob-lem eigenvalue analysis which explained why the standardmultigrid V-cycle will, in general, fail for the large wave-number case. In this talk we present some advances thathave been made since then, and show how a similar anal-ysis can be used to examine the behavior of the Wave-RayMethod of Brand and Livshits.

Oliver G. ErnstTU ChemnitzDepartment of [email protected]

MS42

Symmetric Tensor Analysis

A symmetric tensor is a multiway array that is invariantunder permutation of its indices. We consider the problemof decomposing a real-valued symmetric tensor as the sumof symmetric outer products of real-valued vectors, includ-ing the consideration of constraints such as nonnegativityand orthogonality. Application of standard numerical op-timization methods can be used to solve these problems.We also consider the data structures for efficiently storingsymmetric tensors and the related decompositions.

Tamara G. KoldaSandia National [email protected]

MS42

Blocks, Curves, and Splits: A Glimpse into theO’Leary Toolbox

Along with mathematical results, Dianne O’Leary’s bodyof work includes a wide variety of numerical algorithms,techniques, and strategies that can, in her words, be viewedas “tools in our virtual toolbox”. Among these are blocksthat capture relationships in form, structure, and sparsity;curves that adaptively trace an enlightening, improvingpath; and splits that partition unknowns and equationsto improve speed, accuracy, and flexibility. This talk willpresent an overview of her work from this perspective, be-ginning with her earliest days as a PhD student.

Margaret H. WrightCourant InstituteNew York [email protected]

MS43

Solving Large Scale NNLS Problems Arising InComputer Vision

Given a set of images of a scene, the problem of recon-structing the 3D structure of the scene and estimating thepose of the camera(s) when acquiring these images can beformulated as non-linear least squares problem. Relativelysmall instances (tens of images) of this problem lead to Ja-cobians with thousands of columns and tens of thousandsof rows. I will describe some new techniques for efficiently

104 LA15 Abstracts

solving problems with hundreds of thousands of images.These techniques combine ideas from Domain Decomposi-tion, data clustering and truncated Newton methods.

Sameer AgarwalGoogle [email protected]

MS43

Modulus Iterative Methods for Least SquaresProblems with Box Constraints

For the solution of large sparse box constrained leastsquares problems (BLS), a new iterative method is pro-posed using generalized SOR method for the inner itera-tions, and the accelerated modulus iterative method for theouter iterations for the solution of the linear complementar-ity problem resulting from the Karush-Kuhn-Tucker condi-tion of the BLS problem. Theoretical convergence analysisis presented. Numerical experiments show the efficiencyof the proposed method with less iteration steps and CPUtime compared to projection methods.

Ning ZhengSOKENDAI (The Graduate University for AdvancedStudies)[email protected]


Jun-Feng YinDepartment of MathematicsTongji University, Shanghai, P.R. [email protected]

MS43

LU Preconditioning for Full-Rank and SingularSparse Least Squares

Sparse QR factorization is often the most efficient approachto solving sparse least-squares problems min ‖Ax−b‖, espe-cially since the advent of Davis’s SuiteSparseQR software.For cases where QR factors are unacceptably dense, we con-sider sparse LU factors from LUSOL for preconditioning aniterative solver such as LSMR. LUSOL computes factorsof the form P1AP2 = LU , with permutations chosen topreserve sparsity while ensuring L is well-conditioned. Forfull-rank problems, one can right-precondition with eitherU or B, where B is a basis from the rows of A defined byP1 [Saunders, 1979]. More recently, Duff and Arioli haverecommended that B be chosen to have maximum volume.We experiment with LUSOL and LSMR on many realisticexamples. For singular problems we make use of LUSOL’sthreshold rook pivoting option, and investigate whetherthreshold complete pivoting increases the volume of B ina useful way.

Michael A. SaundersSystems Optimization Laboratory (SOL)Dept of Management Sci and Eng, [email protected]

Nick W. HendersonStanford UniversityInstitute for Computational and [email protected]

Ding MaStanford [email protected]

MS43

The State-of-the-Art of Preconditioners for SparseLinear Least Squares

In recent years, a number of different preconditioningstrategies have been proposed for use in solving m × noverdetermined large sparse linear least squares problems.However, little has been done in the way of comparing theirperformances in terms of efficiency, robustness and appli-cability. In this talk, we briefly review the different precon-ditioners and present a numerical evaluation of them usinga large set of problems arising from practical applications.

Jennifer ScottRutherford Appleton LaboratoryComputational Science and Engineering [email protected]

Nick GouldNumerical Analysis GroupRutherford Appleton [email protected]

MS44

A Structure-preserving Lanczos Algorithm for theComplex J-symmetric Eigenproblem

A structure-preserving Lanczos algorithm for the eigen-problem HCx = λx with

HC =

[A C

D −AT

]

and n × n complex matrices A,C = CT , D = DT will beconsidered. Matrices HC are called complex-J-symmetric

where J =

[0 In−In 0

]and In is the n × n identity.

The eigenvalues of HC display a symmetry: they appear inpairs (λ,−λ). The structure-preserving algorithm is basedon similarity transformations with 2n × 2n complex sym-plectic matrices SC , ST

CJSC = J. These matrices formthe automorphism group associated with the Lie algebraof complex-J-symmetric matrices. The similarity transfor-mation S−1

C HCSC yields a complex-J-symmetric matrix.

Heike FassbenderTU [email protected]

Peter BennerMax Planck Institute, Magdeburg, [email protected]

Chao YangLawrence Berkeley National [email protected]

MS44

Structure Preserving Algorithms for Solving theBethe–Salpeter Eigenvalue Problem

In the numerical computation of electron-hole excitations,

LA15 Abstracts 105

we need to solve a complex J-symmetric eigenvalue prob-lem arising from the discretized Bethe–Salpeter equation.In general all eigenpairs are of interest in this dense eigen-value problem. We develop new algorithms by investigat-ing the connection between this problem and the Hamilto-nian eigenvalue problem. Compared to several existing ap-proaches, our algorithm is fully structure preserving, and iseasily parallelizable. Computational experiments demon-strate the efficiency and accuracy of our algorithm.

Meiyue ShaoLawrence Berkeley National [email protected]

Felipe da JornadaUniversity of California, [email protected]


Jack DeslippeNational Energy Research Scientific Computing [email protected]

Steven LouieUniversity of California, [email protected]

MS44

Preconditioned Locally Harmonic Residual Meth-ods for Interior Eigenvalue Computations

This talk describes a Preconditioned Locally HarmonicResidual (PLHR) method for computing several interioreigenpairs of a generalized Hermitian eigenvalue problem,without traditional spectral transformations, matrix fac-torizations, or inversions. PLHR is based on a short-termrecurrence, easily extended to a block form, computingeigenpairs simultaneously. The method takes advantage ofHermitian positive definite preconditioning. We also dis-cuss generalizations of PLHR on non-Hermitian eigenprob-lems and cases where only an indefinite preconditioner isat hand. We demonstrate that PLHR is an efficient androbust option for interior eigenvalue calculation, especiallyif memory requirements are tight.

Eugene VecharynskiLawrence Berkeley National [email protected]

MS44

Preconditioned Solvers for Nonlinear HermitianEigenproblems with Variational Characterization

Large-scale nonlinear Hermitian eigenproblems of the formT (λ)v = 0 satisfying the standard variational characteri-zation arises in a variety of applications. We review somerecent development of numerical methods for solving eigen-problems of this type, including variants of preconditionedconjugate gradient (PCG) methods for extreme eigenval-ues, and preconditioned locally minimal residual (PLMR)methods for interior eigenvalues. We discuss the develop-ment of search subspaces, projection and extractions, pre-conditioning, deflation, and convergence analysis of thesemethods. Numerical experiments demonstrate the effi-

ciency and competitiveness of our algorithms.

Fei XueUniversity of Louisiana at [email protected]


Eugene VecharynskiLawrence Berkeley National [email protected]

MS45

Fast Hierarchical Randomized Methods for the Ap-proximation of Large Covariance Matrices

We propose a new efficient algorithm for performing hierar-chical kernel MVPs in O(N) operations called the UniformFMM (UFMM), an FFT accelerated variant of the blackbox FMM by Fong and Darve. The UFMM is used to speedup randomized low-rank methods thus reducing their com-putational cost to O(N) in time and memory. Numericalbenchmarks include low-rank approximations of covariancematrices for the simulation of stationary random fields onvery large distributions of points.

Pierre [email protected]

Olivier [email protected]


MS45

Algebraic Operations with H2-Matrices

We present a family of algorithms that perform approx-imate algebraic operations on H2-matrices. These algo-rithms allow us to compute approximations of factoriza-tions, inverses, and products of H2-matrices in almost lin-ear complexity with respect to the matrix dimension. Com-pared to previous methods, our approach is based on locallow-rank updates that can be carried out in linear complex-ity. All other matrix operations can be expressed by theseupdates and simple matrix-vector products, and we obtaina fast and flexible technique for, e.g., constructing robustpreconditioners for integral and elliptic partial differentialoperators.

Steffen BormUniversity of [email protected]

Knut ReimerUniversitaet Kiel

106 LA15 Abstracts

[email protected]

MS45

Fast Numerical Linear Algebra for Kohn-ShamDensity Functional Theory via Compression

In this talk we discuss algorithms for accelerating the solu-tion of the non-linear eigenvalue and eigenvector problemin Kohn-Sham density functional theory. Solving this prob-lem self consistently requires the repeated computation andapplication of various linear algebraic operators. As part ofthis talk we demonstrate how to construct a localized, andhence sparse, representation of basis functions that allowsfor reductions in the overall computational cost.

Anil DamleStanford [email protected]

Lexing YingStanford UniversityDepartment of [email protected]

MS45

Comparison of FMM and HSS at Large Scale

We perform a direct comparison between the fast multipolemethod (FMM) and hierarchically semi separable (HSS)matrix-vector multiplication for Laplace, Helmholtz, andStokes kernels. Both methods are implemented with dis-tributed memory, thread, and SIMD parallelism, and theruns are performed on large Cray systems such as Edison(XC30, 133,824 cores) at NERSC and Shaheen2 (XC40,197,568 cores) at KAUST.

Rio YokotaKing Abdullah University of Science and [email protected]



MS46

A Stable and Efficient Matrix Version of the FastMultipole Method

It is well known that Fast Multipole Method (FMM) isvery efficient for evaluating matrix-vector products involv-ing discretized kernel functions. In contrast, hierarchicallysemiseparable (HSS) representations provide a fast and sta-ble way for many matrix operations, especially direct fac-torizations. However, HSS forms have always been con-structed based on algebraic compression(e.g., SVD) of off-diagonal blocks. In this work, we show how to directlyobtain an HSS representation from the matrix derived inFMM.

Difeng CaiPurdue [email protected]


MS46

Some Recent Progress on Algorithms for FMMMatrices

There are simple time efficient O(N2) algorithms for con-verting a general dense matrix to FMM form using a priorpartition tree. We show that there is a surprise as far asmemory efficiency goes, and present a O(N1.5) workingmemory algorithm and a corresponding worst case familyof partition trees. It is an open question whether thereis a construction algorithm that only requires O(N) workspace. On another front, there is a simple O(N) flops al-gorithm to multiply two HSS forms exactly. However thecorresponding question for FMM forms has remained open.We show how to close it.

Shivkumar ChandrasekaranUniversity of California, Santa BarbaraDepartment of Electrical and Computer [email protected]

Kristen LesselUniversity of California, Santa [email protected]

MS46

Fast Linear Solvers for Weakly Hierarchical Matri-ces

In recent years, we have seen the development of a newclass of linear solvers and preconditioners based on hierar-chical low-rank approximations of matrices. Such solvershave been very successful and can solve problems that arecurrently intractable or very expensive using conventionalpreconditioners such as block diagonal, incomplete LU, al-gebraic multigrid, etc. However, the computational cost ofthese solvers currently fails to scale linearly with the size ofthe matrix. This is true in particular for 3D elliptic partialdifferential equations. In this talk, we will present a newversion of these fast solvers that scale like O(N) for generalsituations. We will see how these solvers are able to repre-sent dense matrices using (slightly larger) sparse matrices,and how the fill-in that appears during the LU factoriza-tion of these sparse matrices can be removed, so that thealgebra remains sparse throughout the process. As a resultof this new approach, all steps are O(N), even in 3D. Thisresults in very fast linear solvers that scale favorably withthe problem size.


Pieter Coulier, Mohammad Hadi Pour [email protected], [email protected]

MS46

MHS Structures for the Direct Solution of Multi-Dimensional Problems

We propose multi-layer hierarchically semiseparable

LA15 Abstracts 107

(MHS) structures for the efficient factorizations of densematrices arising from multi-dimensional discretized prob-lems. The problems include discretized integral equationsand dense Schur complements in the factorizations of dis-cretized PDEs. Unlike existing work on hierarchicallysemiseparable (HSS) structures which are essentially 1Dstructures, the MHS framework integrates multiple layersof rank and tree structures. We lay theoretical founda-tions for MHS structures and justify the feasibility of MHSapproximations for these dense matrices. Rigorous rankbounds for the low-rank structures are given. Represen-tative subsets of mesh points are used to illustrate themulti-layer structures as well as the structured factoriza-tion. Systematic fast and stable MHS algorithms are pro-posed, particularly convenient direct factorizations. Thenew structures and algorithms can yield direct solvers withnearly linear complexity and linear storage for solving somepractical 2D and 3D problems.

Jianlin [email protected]

MS47

A Semi-Tensor Product Approach for ProbabilisticBoolean Networks

Modeling genetic regulatory networks is an important is-sue in systems biology. Various models and mathematicalformalisms have been proposed in the literature to solvethe capture problem. The main purpose in this paper isto show that the transition matrix generated under semi-tensor product approach (Here we call it the probabilitystructure matrix for simplicity) and the traditional ap-proach (Transition probability matrix) are similar to eachother. And we shall discuss three important problems inProbabilistic Boolean Networks (PBNs): the dynamic of aPBN, the steady-state probability distribution and the in-verse problem. Numerical examples are given to show thevalidity of our theory. We shall give a brief introduction tosemi-tensor and its application. After that we shall focuson the main results: to show the similarity of these twomatrices. Since the semi-tensor approach gives a new wayfor interpreting a BN and therefore a PBN, we expect thatadvanced algorithms can be developed if one can describethe PBN through semi-tensor product approach.

Xiaoqing Cheng, Yushan Qiu, Wenpin Hou, Wai-Ki ChingThe University of Hong [email protected], [email protected],[email protected], [email protected]

MS47

Positive Diagonal Scaling of a Nonnegative Tensorto One with Prescribed Slice Sums

Given a nonnegative 3-tensor T = [ti,j,k] of dimensionsm,n, p, when one can rescale it by three positive vectorsx = (x1, . . . , xm), y = (y1, . . . , yn), z = (z1, . . . , zp), suchthat the tensor [xiyjzkti,j,k] has given row, column anddepth slice sums ri, cj , dk. Here

ri =

n,p∑j,k=1

xiyjzkti,j,k, cj =

m,p∑i,k=1

xiyjzkti,j,k, dk =m.n∑i,j=1

xiyjzkti,j,k

for i = 1, . . . ,m, j = 1, . . . ,m, k = 1, . . . , l. We give aneccesary and sufficient condition and state a correspond-ing minimizing problem of a convex function that gives asolution to the diagonal scaling. These results generalize

to any d-mode nonnegative tensors. References: S. Fried-land, Positive diagonal scaling of a nonnegative tensor toone with prescribed slice sums, Linear Algebra Appl., vol.434 (2011), 1615-1619.

Shmuel FriedlandDepartment of Mathematics ,University of Illinois,[email protected]

MS47

Open Problems and Concluding Discussions

In this final session of our mini-symposia, we will presenta summary of the results from all of the other presentersas well as invite them to submit slides for an open problemdiscussion. The point is to identify a few key problems thatcould be solved to move the field forward.


Lek-Heng LimUniversity of [email protected]

MS47

The Laplacian Tensor of a Multi-Hypergraph

Given a uniform multi-hypergraph H, we define a newhyper-adjacency tensor A(H) and use it to define the Lapla-cian L(H) and the signless Laplacian Q(H). We proveL(H) and Q(H) are positive semi-definite for connectedeven uniform multigraphs. Some analysis on the largestand smallest H and Z-eigenvalues of L(H) are given aswell as a comparison of other adjacency tensors.

Kelly PearsonMurray State [email protected]

MS48

Basker: A Scalable Sparse Direct Linear Solver forMany-Core Architectures

In this talk, we describe challenges related to the imple-mentation of a scalable direct linear solver on many-corearchitectures. Basker focuses on solving irregular systemssuch as those from circuit simulation. These systems mayrequire full partial pivoting, and cannot be reorder for ef-ficient use of BLAS3 operations. We attempt to mitigatethese issues by using a parallel implementation of Gilbert-Peierls method, while developing with Kokkos, which is aframework that allows for portability on emerging many-core architectures.

Joshua D. BoothCenter for Computing ResearchSandia National [email protected]

Siva RajamanickamSandia National [email protected]

Erik G. BomanCenter for Computing Research

108 LA15 Abstracts

Sandia National [email protected]

MS48

Parallel Iterative Incomplete Factorizations andTriangular Solves

We present asynchronous fine-grained parallel algorithmsfor computing incomplete LU (ILU) factorizations and per-forming the corresponding sparse triangular solves. TheILU algorithm computes all the non-zeros of the factors inparallel (asynchronously) using an iterative method. Thetriangular solves can be performed approximately usinganother iterative method. We present several modifica-tions from previous work that improve the robustness ofthe method. We show results from Intel Xeon Phi andGPU using the Kokkos library.

Aftab PatelSchool of Computational Science and EngineeringGeorgia Institute of [email protected]




MS48

A Sparse Direct Solver for Distributed MemoryGPU and Xeon Phi Accelerated Systems

In this talk, we presents the first sparse direct solver fordistributed memory systems comprising hybrid multicoreCPU and Intel Xeon Phi co-processors or GPUs. It buildson the algorithmic approach of SuperLUDist, which isright-looking and statically pivoted. Our contribution is anovel algorithm, called the HALO. The name is shorthandfor highly asynchronous lazy offload ; it refers to the waythe algorithm combines highly aggressive use of asynchronywith accelerated offload, lazy updates, and data shadowing(a la halo or ghost zones), all of which serve to hide andreduce communication, whether to local memory, acrossthe network, or over PCIe. We further augment HALOwith a model-driven autotuning heuristic that chooses theintra-node division of labor among CPU and acceleratorcomponents. When integrated into HALO and evaluatedon a variety of realistic test problems in both single-nodeand multi-node configurations, the resulting implementa-tion achieves speedups of up to 2.5× over an already effi-cient multicore CPU implementation, and achieves up to83% of a machine-specific upper-bound that we have esti-mated.

Piyush SaoGeorgia Institute of [email protected]

Xing Liu

School of Computational Science and EngineeringGeorgia Institute of [email protected]

Richard VuducGeorgia Institute of [email protected]. edu


MS48

Parallel Multilevel Incomplete LU FactorizationPreconditioner with Variable-Block Structure

In this talk, we describe a distributed-memory implementa-tion of a variable-block multilevel algebraic recursive iter-ative preconditioner for general non-symmetric linear sys-tems. The preconditioner construction detects automati-cally exact or approximate dense blocks in the coefficientmatrix and exploits them to achieve better reliability andcomputation density. To find such a dense-block struc-ture, a graph-compression algorithm has been developed,which requires only one easy-to-use parameter. We presenta study of the numerical performance and parallel scalabil-ity using additive Schwarz and Schur complement type pre-conditioners on a variety of general linear systems as wellas in the analysis of turbulent Navier-Stokes equations.

Masha SosonkinaOld Dominion UniversityAmes Laboratory/[email protected]

Bruno CarpentieriInsitute for Mathematics and ComputationUniversity of [email protected]

MS49

Solving Sequential Strongly Underdetermined Sys-tems Using Sherman-Morrison Iterations

We are interested in repeatedly solving a regularized lin-ear least squares problem of the form where data becomesequentially available. We assume we are given a stronglyunderdetermined matrix and a Tikhonov type regulariza-tion term. Using the Sherman-Morrison-Woodbury for-mula we were able to develop a fast solver which can com-pete with methods base on Cholesky factorization and it-erative methods such as LSQR, CGLS, and LSMR.

Matthias ChungDepartment of MathematicsVirginia [email protected]

Julianne ChungVirginia [email protected]

Joseph SlagelDepartment of MathematicsVirginia Tech

LA15 Abstracts 109

[email protected]

MS49

A Tensor-Based Dictionary Approach to Tomo-graphic Image Reconstruction

The problem of low-dose tomographic image reconstruc-tion is investigated using dictionary priors learned fromtraining images. The reconstruction problem is consideredas a two phase process: in phase one, we construct a ten-sor dictionary prior from our training data (a process for-mulated as a non-negative tensor factorization problem),and in phase two, we pose the reconstruction problem interms of recovering expansion coefficients in that dictio-nary. Experiments demonstrate the potential utility of ourapproach.

Sara SoltaniTechnical University of DenmarkDepartment of Applied Mathematics and [email protected]


Per Christian HansenTechnical University of DenmarkInformatics and Mathematical [email protected]

MS49

Some Inverse Problems in Optical Imaging for Re-mote Sensing

Hyperspectral, Light Detection and Ranging (LiDAR), andPolarimetric imaging are the pervasive optical imagingmodalities in remote sensing. Here, we describe methodsfor deblurring and analyzing 3D images of these types, aswell as 4D compressive spectro-polarimetric snapshot im-ages.

Robert PlemmonsWake Forest [email protected]

MS49

Efficient Iterative Methods for Quantitative Sus-ceptibility Mapping

Quantitative Susceptibility Mapping (QSM) is a MRI tech-nique that allows to image magnetic properties of tissue.QSM is applied more and more widely, for example, in neu-roscience, where iron concentration serves as a biomarker ofcertain neurological diseases. While the tissues susceptibil-ity cannot be measured directly it can be reconstructed bysolving a highly ill-posed inverse problem. In this talk, wewill present iterative methods for QSM reconstruction andtheir efficient implementation using Julia. We formulateQSM as a PDE constrained optimization problem with anon-smooth regularizer based on total variation (TV). Spe-cial emphasis will be put on iterative all-at-once methodsfor handling the PDE-constraints.

Lars RuthottoDepartment of Mathematics and Computer ScienceEmory University

[email protected]

MS50

Domain Decomposition Algorithms for Large Her-mitian Eigenvalue Problems

In this talk we will discuss Domain-Decomposition (DD)type methods for large Hermitian eigenvalue problems.This class of thechniques rely on spectral Schur comple-ments combined with Newton’s iteration. The eigenvaluesof the spectral Schur complement appear in the form ofbranches of some functions, the roots of which are eigen-values of the original matrix. It is possible to extract theseroots by a number of methods which range from a formor approximate Rayleigh iteration to an approximate in-verse iteration, in which a Domain Decomposition frame-work is used. Numerical experiments in parallel environ-ments will illustrate the numerical properties and efficiencyof the method.


Vasilis KalantzisCEID, School of EngineeringUniversity of Patras, [email protected]

MS50

A Contour Integral-based Parallel Eigensolver withHigher Complex Moments

Large-scale eigenvalue problems arise in wide variety of sci-entific and engineering applications such as nano-scale ma-terials simulation, vibration analysis of automobiles, dataanalysis, etc. In such situation, high performance paral-lel eigensolvers are required to exploit distributed paral-lel computational environments. In this talk, we presenta parallel eigensolver (SS method) for large-scale interioreigenvalue problems. This method has a good parallel scal-ability according to a hierarchical structure of the method.We also present parallel software for computing eigenvaluesand corresponding eigenvectors in a given region or inter-val. We illustrate the performance of the software withsome numerical examples.

Tetsuya Sakurai, Yasunori Futamura, Akira ImakuraDepartment of Computer ScienceUniversity of [email protected], [email protected], [email protected]

MS50

On the Orthogonality of Eigenvectors Obtainedfrom Parallel Spectral Projection Methods

A number of recent eigensolvers have been devised with theaid of some approximate projectors to invariant subspaces.One of the major advantages on the use of these approx-imate spectral projectors is the natural parallelism theypresent. Different independent processing tasks can workon different parts of the spectrum in parallel. It is wellknown that exact eigenvectors from Hermitian problemsare mutually orthogonal. On the other hand, some initialstudies cast doubts on the numerical orthogonalities of thevectors computed independently when spectral projection

110 LA15 Abstracts

methods are applied in a parallel setting. In this talk wewill present several practical approaches to maintain or-thogonalities without explicit reorthogonalization betweenindependent tasks.

Peter TangIntel Corporation2200 Mission College Blvd, Santa Clara, [email protected]

MS50

A Contour-Integral Based Algorithm for Countingthe Eigenvalues Inside a Region in the ComplexPlane

In many applications, the information about the numberof eigenvalues inside a given region is required. In this pa-per, we propose a contour-integral based method for thispurpose. The new method is motivated by two findings.Recently, some contour-integral based methods were devel-oped for estimating the number of eigenvalues inside a re-gion in the complex plane. But our method is able to countthe eigenvalues inside the given region exactly. An appeal-ing feature of our method is that it can integrate with re-cently developed contour-integral based eigensolvers to de-termine whether all desired eigenvalues are found by thesemethods. Numerical experiments are reported to show theviability of our new method.

Guojian Yin, Guojian YinShenzhen Institutes of Advanced Technology,Chinese Academy of [email protected], [email protected]

MS51

Parallel Algorithms for Computing Functions ofMatrix Inverses

Functions of entries of inverses of matrices like all diagonalentries of a sparse symmetric indefinite matrix inverse or itstrace arise in several important computational applicationssuch as density functional theory, covariance matrix analy-sis or when evaluating Green’s functions in computationalnanolelectronics. We will utilize an algorithm for selectiveinversion by Lin et al. (TOMS 2011) for approximatelycomputing selective parts of a sparse symmetric matrix in-verse. Its overall performance will be demonstrated forselected numerical examples and we will sketch paralleliza-tion aspects for raising the computational efficiency.

Matthias BollhoeferTU [email protected]

MS51

A Parallel Multifrontal Solver and Precondi-tioner Using Hierarchically Semiseparable Struc-tured Matrices

We present a sparse solver or preconditioner using hier-archically semi-separable (HSS) matrices to approximatedense matrices that arise during sparse matrix factoriza-tion. Using HSS, rank structured matrices with low-rankoff-diagonal blocks, reduces fill-in and for many PDE prob-lems, it leads to a solver with lower computational com-plexity than the pure multifrontal method. For HSS con-struction, we use a novel randomized sampling technique.Hybrid shared and distributed memory parallel results are

presented.

Pieter GhyselsLawrence Berkeley National LaboratoryComputational Research [email protected]



MS51

Iterative Construction of Non-Symmetric FactoredSparse Approximate Preconditioners

Factored sparse approximate inverses (FSAI) play a key-role in the efficient algebraic preconditioning of sparse lin-ear systems of equations. For SPD problems remarkable re-sults are obtained by building the FSAI non-zero pattern it-eratively during its computation. Unfortunately, an equiv-alent algorithm is still missing in the non-symmetric case.In the present contribution we explore the possibility of it-eratively computing FSAI for non-symmetric matrices byusing a incomplete Krylov subspace bi-orthogonalizationprocedure.

Carlo Janna, Massimiliano FerronatoDept. ICEA - University of [email protected], [email protected]

Andrea FranceschiniUniversity of [email protected]

MS51

Monte Carlo Synthetic Acceleration Methods forSparse Linear Systems

Stochastic linear solvers based on Monte Carlo SyntheticAcceleration are being studied as a potentially resilient al-ternative to standard methods. Our work has shown thatfor certain classes of problems, these methods can alsodemonstrate performance competitive with modern tech-niques. In this talk we will review recent developmentsin Monte Carlo solvers for linear systems. Our work tar-gets large, distributed memory systems by adapting MonteCarlo techniques developed by the transport community tosolve sparse linear systems arising from the discretizationof partial differential equations.


Tom EvansOak Ridge National [email protected]

Steven [email protected]

LA15 Abstracts 111

Massimiliano Lupo PasiniDepartment of Mathematics and Computer ScienceEmory [email protected]

Stuart SlatteryComputer Science and Mathematics DivisionOak Ridge National [email protected]

MS52

A Matrix Equation Approach for Incremental Lin-ear Discriminant Analysis


Delin ChuDepartment of Mathematics, National University ofSingaporeBlock S17, 10 Lower Kent Ridge Road, Singapore [email protected]

MS52

Riccati Equations Arising in Stochastic Control

The numerical treatment of the infinite dimensionalstochastic linear quadratic control problem requires solvinglarge-scale Riccati equations. We investigate the conver-gence of these Riccati operators to the finite dimensionalones. Moreover, the coefficient matrices of the resultingRiccati equation have a given structure (e.g. sparse, sym-metric or low rank). We proposed an efficient methodbased on a low-rank approach which exploit the givenstructure. The performance of our method is illustratedby numerical results.

Hermann MenaDepartment of MathematicsUniversity of Innsbruck, [email protected]

MS52

Splitting Schemes for Differential Riccati Equa-tions

I will consider the application of exponential splittingschemes to large-scale differential Riccati equations of theform

P = ATP + PA+Q− PSP. (3)

These numerical methods consider the affine and nonlin-ear parts of the equation separately, thereby decreasingthe necessary computational effort. I will focus on imple-mentation issues and how to utilize low-rank structure. Iwill also consider several natural generalizations such astime-dependent or implicit problems.

Tony StillfjordDepartment of MathematicsUniversity of Lund, [email protected]

MS52

Conditioning Number of Solutions of Matrix Equa-tions in Stability Analysis


Mingqing Xiao

Southern Illinois University at [email protected]

MS53

Kolmogorov Widths and Low-Rank Approximationof Parametric Problems

This talk is concerned with error estimates for low-rankapproximation of diffusion equations with several parame-ters. In particular, we focus on estimates of Kolmogorov n-widths of such solution manifolds, which also allow conclu-sions to be drawn concerning the performance achievableby model reduction based on the reduced basis method.Moreover, we discuss numerical schemes for the computa-tion of low-rank approximations of such problems. Ourtheoretical bounds are illustrated by numerical experi-ments, and the results indicate in particular a surprisinglystrong dependence on particular structural features of theproblems.

Markus BachmayrUniversite Pierre et Marie [email protected]

Albert CohenUniversite Pierre et Marie Curie, Paris, [email protected]

MS53

A Greedy Method with Subspace Point of View forLow-Rank Tensor Approximation in HierarchicalTucker Format.

The goal is to approximate high-order tensors in subspace-based low-rank formats. A strategy is proposed for com-puting adapted nested tensor subspaces in a greedy fashion.The best approximation in these subspaces is then com-puted with a DMRG-like algorithm, again usign a greedystrategy, allowing an automatic rank adaptation with a re-duced complexity compared to standard DMRG. A heuris-tic error indicator based on singular values enables to cap-ture a possible anisotropy of the tensor of interest.

Loıc GiraldiGeM, Ecole Centrale de [email protected]


MS53

Tensor Krylov Methods

High dimensional models with parametric dependenciescan be challenging to simulate. We use tensor Krylov tech-niques to reduce the parametric model, combining bothmoment matching and interpolatory model reduction inthe parameter space. If a low rank tensor formulation ispossible, this approach is competitive with classic para-metric model reduction. Furthermore, we look at modelscontaining stochastic parameters and construct a reducedmodel that outputs the mean over these parameters withinthe domain of interest.

Pieter LietaertKU Leuven

112 LA15 Abstracts

[email protected]

MS53

An Adaptive Tensor Method for High-DimensionalParametric PDEs

We apply the recently developed Tensor Train (TT) formatfor tensor decomposition to high-dimensional paramet-ric PDEs [Oseledets, Tensor-Train Decomposition, 2011,SIAM J. Sci. Comput. 33, pp. 2295-2317]. Adaptivesolvers have been introduced independently [Eigel et al.,Adaptive Stochastic Galerkin FEM, Comput. Method.Appl. M. 270, pp. 247-269] but computational cost growsexponentially with the number of dimensions. In par-ticular, the proposed error estimators become unfeasablevery quickly for larger problems. The TT format breaksthis curse of dimensionality and allows for the treat-ment of these problems. We solve the discretized prob-lem iteratively using the well-known Alternating LeastSquares (ALS) algorithm [Holtz et al., The alternating lin-ear scheme for tensor optimisation in the TT format, SIAMJ. Sci. Comput. 34, pp. A683-A713] . Furthermore, weexploit the component structure of the tensor format. Theerror estimation can be computed far more efficiently thisway. Numerical results support this approach by showinghigher accuracy even for problems of far greater size thanpreviously discussed.

Max PfefferTU [email protected]

MS54

Steady-state Analysis of a Multi-class MAP/PH/cQueue with Acyclic PH Retrials

A multi-class c-server retrial queueing system in which cus-tomers arrive according to a class-dependent Markovian ar-rival process (MAP) is considered. Service and retrial timesfollow class-dependent phase-type (PH) distributions. Anecessary and sufficient condition for ergodicity is given.A Lyapunov function is used to obtain a finite state spacewith a given steady-state probability mass. The truncatedsystem is described as a multi-dimensional Markov chainand a Kronecker representation of its generator matrix isanalyzed numerically.

Tugrul DayarBilkent UniversityDept of Computer [email protected]

M. Can OrhanBilkent [email protected]

MS54

The Computation of the Key Properties of MarkovChains using a Variety of Techniques

Techniques for finding the stationary distribution, themean first passage times and the group inverse of finiteirreducible Markov chains is presented. We extend thetechnique of [Kohlas J., Numerical computation of meanfirst passage times and absorption probabilities in Markovand semi-Markov models, Zeit fur Oper Res, 30, (1986),197-207], include some new results involving perturbationtechniques as well as some generalised matrix inverse proce-

dures. Accuracies are compared using some test problemsin the literature.

Jeffrey J. HunterAuckland University of [email protected]

MS54

General Solution of the Poisson Equation for QBDs

If T is the transition matrix of a finite irreducible Markovchain and b is a given vector, the Poisson equation (I −T )x = b has a unique solution, up to an additive constant,

given by x = (I − T )#b + α1, for any α scalar, where 1 isthe vector of all ones and H# denotes the group inverse ofH . This does not hold when the state space is infinite. Weconsider the Poisson equation (I − P )u = g, where P isthe transition matrix of a Quasi-Birth-and-Death (QBD)process with infinitely many levels, g is a given infinite–dimensional vector and u is the unknown vector. By usingthe block tridiagonal and block Toeplitz structure of P ,we recast the problem in terms of a set of matrix differenceequations. We provide an explicit expression of the generalsolution, relying on the properties of Jordan triples of ma-trix polynomials and on the solutions of suitable quadraticmatrix equations. The uniqueness and boundedness of thesolutions are discussed, according to the properties of theright–hand side g.

Dario BiniDept. of MathematicsUniversity of Pisa, [email protected]

Sarah Dendievel, Guy LatoucheUniversite libre de [email protected], [email protected]

Beatrice MeiniDept. of MathematicsUniversity of Pisa, [email protected]

MS54

Complex Nonsymmetric Algebraic Riccati Equa-tions Arising in Markov Modulated Fluid Flows

Motivated by the transient analysis of stochastic fluid flowmodels, we introduce a class of complex nonsymmetric al-gebraic Riccati equations. The existence and uniqueness ofthe extremal solutions to these equations are proved. Nu-merical methods for the extremal solutions are discussed.

Changli LiuSchool of Mathematical Science, Sichuan [email protected]

Jungong XueFundan [email protected]

MS55

Recovering Sparsity in a Krylov Subspace Frame-work

For many ill-posed inverse problems, such as image restora-tion, the regularized solution may benefit from the inclu-

LA15 Abstracts 113

sion of sparsity constraints, which involve terms evaluatedin the 1-norm. Our goal is to efficiently approximate the1-norm terms by iteratively reweighed 2-norm terms. Afterpresenting some new results on the regularizing propertiesof Krylov subspace methods, we describe a couple of newapproaches to enforce sparsity within a Krylov subspaceframework.

Silvia GazzolaUniversity of [email protected]

James G. NagyEmory UniversityDepartment of Math and Computer [email protected]

Paolo NovatiUniversity of [email protected]

MS55

Structure Preserving Reblurring Preconditionersfor Image Deblurring

In the context of image deblurring, we extend a precon-ditioning technique proposed in [Dell’Acqua et al., JCAM2013] to a more general method whose preconditioners in-herit the boundary conditions of the problem without los-ing in computational cost. Furthermore, following the ideadeveloped in [Donatelli, Hanke, IP 2013], we introduce anonstationary version of the proposed method providingfurther improvements in terms of iterations and quality ofreconstruction. Numerical comparisons with the aforemen-tioned techniques are given.

Pietro Dell’AcquaUniversity of [email protected]

Marco DonatelliUniversity of [email protected]

Claudio EstaticoDepartment of Mathematics, University of Genova, [email protected]

Mariarosa MazzaUniversity of [email protected]

MS55

SVD Approximations for Large Scale ImagingProblems

A fundamental tool for analyzing and solving ill-posed in-verse problems is the singular value decomposition (SVD).However, in imaging applications the matrices are oftentoo large to be able to efficiently compute the SVD. Inthis talk we present an efficient approach to compute anapproximate SVD, which exploits Kronecker product andtensor structures that arise in imaging applications.

James G. NagyEmory UniversityDepartment of Math and Computer Science

[email protected]

MS55

Arnoldi Methods for Image Deblurring with Anti-Reflective Boundary Conditions

We consider image deblurring with anti-reflective boundaryconditions and a nonsymmetric point spread function, andcompare several iterative Krylov subspace methods usedwith with reblurring right or left preconditioners. The pur-pose of the preconditioner is not to accelerate the rate ofconvergence, but to improve the quality of the computedsolution and to increase the robustness of the regulariza-tion method. Right preconditioned methods based on theArnoldi process are found to perform well.

Marco DonatelliUniversity of [email protected]

David Martin, Lothar ReichelKent State [email protected], [email protected]

MS56

The Use of Stochastic Collocation Methods to Un-derstand Pseudo-Spectra in Linear Stability Anal-ysis

Eigenvalue analysis is a well-established tool for stabilityanalysis of dynamical systems. However, it is also knownthat there are situations where eigenvalues miss some im-portant features of physical models. For example, in mod-els of incompressible fluid dynamics, there are exampleswhere eigenvalue analysis predicts stability but transientsimulations exhibit significant growth of infinitesmal per-turbations. This behavior can be predicted by pseudo-spectral analysis. In this work, we show that an approachsimilar to pseudo-spectral analysis can be performed in-expensively using stochastic collocation methods and theresults can be used to provide quantitive information aboutthe nature and probability of instability.


David SilvesterSchool of MathematicsUniversity of [email protected]

MS56

Large-Scale Eigenvalue Calculations in ScientificProblems

We will consider the problem of computing a relativelylarge number of eigenvalues of a large sparse and sym-metric matrix on distributed-memory parallel computers.The problem arises in several scientific applications, suchas electronic structure calculations. We will discuss theuse of spectrum slicing and multiple shift-invert Lanczosin computing the eigenvalues. This is joint work withHasan Metin Aktulga, Lin Lin, Christopher Haine, andChao Yang.

Esmond G. NgLawrence Berkeley National Laboratory

114 LA15 Abstracts

[email protected]

MS56

Accurate Computations of Eigenvalues of Diago-nally Dominant Matrices with Applications to Dif-ferential Operators

In this talk, we present an algorithm that computes alleigenvalues of a symmetric diagonally dominant matrix tohigh relative accuracy. We further consider using the al-gorithm in an iterative method for a large scale eigenvalueproblem and we show how smaller eigenvalues of finite dif-ference discretizations of differential operators can be com-puted accurately. Numerical examples are presented todemonstrate the high accuracy achieved by the new algo-rithm.

Qiang YeDept of MathUniv of [email protected]

MS56

Convergence of the Truncated Lanczos Approachfor the Trust-Region Subproblem

The Trust-Region Subproblem plays a vital role in vari-ous other applications. The truncated Lanczos approachproposed in [Gould, Lucidi, Roma and Toint, SIAM J. Op-tim., 9:504–525 (1999)] is a natural extension of the Lanc-zos method and mimics the Rayleigh-Ritz procedure. Inthis paper, we analyze the convergence of the truncatedLanczos approach and reveals its convergence behavior intheory. Numerical examples are reported to support ouranalysis.

Leihong ZhangShanghai University of Finance and [email protected]

Ren-Cang LiUniversity of Texas - [email protected]

MS57

Low-Rank Approximation Preconditioners forSymmetric and Nonsymmetric Systems

This talk will present a few preconditioning techniquesbased on low-rank approximations, designed for the gen-eral framework of the domain decomposition (DD) ap-proach and distributed sparse matrices. The DD decou-ples the matrix and yields two variants of preconditioners.The first is an approximate inverse of the original matrixbased on a low-rank correction that exploits the Sherman-Morrison-Woodbury formula. The second method is basedon the Schur complement technique and it is based on ap-proximating the inverse of the Schur complement. Low-rank expansions are computed by the Lanczos/Arnoldi pro-cedure with reorthogonalization. Numerical experimentswith Krylov subspace accelerators will be presented forsymmetric indefinite systems and nonsymmetric systems.These methods were found to be more efficient and robustthan ILU and multilevel ILU-type preconditioners.

Ruipeng LiDepartment of Computer Science & EngineeringUniversity of Minnesota

[email protected]


MS57

An Algebraic Multigrid Method for NonsymmetricLinear Systems

Linear systems with multiple physical unknowns pose achallenge for standard multigrid techniques, particularlywhen the coupling between the unknowns is strong. Wepresent our efforts to develop an AMG-preconditionedKrylov solver for nonsymmetric systems of linear equa-tions. The preconditioner is designed to represent the cou-pling between the physical variables and account for theunderlying physics of the system. We present performanceresults for the solver on challenging flow and transport ap-plications.

Daniel Osei-Kuffuor, Lu WangLawrence Livermore National [email protected], [email protected]

Robert FalgoutCenter for Applied Scientific ComputingLawrence Livermore National [email protected]

MS57

Aggressive Accelerator-Enabled Local SmoothingVia Incomplete Factorization, with Applications toPreconditioning of Stokes Problems with Hetero-geneous Viscosity Structure

Hybrid supercomputers offer attractive performance perWatt, yet present communication-related bottlenecks. Weinvestigate “heavy smoothing’ with multigrid precondi-tioners; aggressive coarsening with accelerator-enabledsmoothing can reduce communication, maintaining scal-ability and admitting a tradeoff between local work andnon-local communication. We examine local polynomialsmoothing as well as the Chow-Patel fine-grained ILU de-composition. Our work aims to provide portable softwarecontributions within ViennaCL, PETSc, and pTatin3D.We apply the smoothers to ill-conditioned systems fromlithospheric dynamics.

Patrick SananUniversita della Svizzera [email protected]

Karl RuppInstitute for Analysis and Scientific Computing, TU WienInstitute for Microelectronics, TU [email protected]

Olaf SchenkUSI - Universita della Svizzera italianaInstitute of Computational [email protected]

MS57

Preconditioned Iterative Methods for Solving In-

LA15 Abstracts 115

definite Linear Systems

Sparse symmetric indefinite linear systems of equationsarise in many practical applications. A preconditionediterative method is frequently the method of choice. Inthe talk we consider both general indefinite systems andsaddle-point problems and we summarize our work alongthese lines based on the recently adopted limited memoryapproach. A number of new ideas proposed with the goalof improving the stability, robustness and efficiency of theresulting preconditioner will be mentioned.

Miroslav TumaAcademy of Sciences of the Czech RepublicInstitute of Computer [email protected]

Jennifer ScottRutherford Appleton LaboratoryComputational Science and Engineering [email protected]

MS58

The s-Step Lanczos Method and its Behavior inFinite Precision

The s-step Lanczos method can reduce communication bya factor of O(s) versus the classical approach. By translat-ing results of the finite-precision block process back ontothe individual elements of the tridiagonal matrix, we showthat the bounds given by Paige [Lin. Alg. Appl., 34:235–258, 1980] apply to the s-step case under certain assump-tions. Our results confirm the importance of the s-stepbasis conditioning and motivate approaches for improvingfinite precision behavior while still avoiding communica-tion.

Erin C. CarsonUC [email protected]

James W. DemmelUniversity of CaliforniaDivision of Computer [email protected]

MS58

Sparse Approximate Inverse Preconditioners forCommunication-Avoiding Bicgstab Solvers

Communication-avoiding formulations of Krylov solverscan reduce communication costs by taking s steps of thesolver at the same time. In this work, we address thelack of preconditioners for communication-avoiding Krylovsolvers. We derive a preconditioned variant of the s-stepBiCGStab iterative solver and demonstrate the effective-ness of Sparse Approximate Inverse (SAI) preconditionersin improving the convergence rate of the preconditioned s-step BiCGStab solver while maintaining the reduction incommunication costs.

Maryam Mehri [email protected]

Erin CarsonU.C. [email protected]


James W. DemmelUniversity of CaliforniaDivision of Computer [email protected]

David FernandezMcGill [email protected]

MS58

Enlarged Krylov Subspace Methods for ReducingCommunication

We introduce a new approach for reducing communicationin Krylov subspace methods that consists of enlarging theKrylov subspace by a maximum of t vectors per iteration,based on the domain decomposition of the graph of A. Theenlarged Krylov projection subspace methods lead to fasterconvergence in terms of iterations and parallelizable algo-rithms with less communication, with respect to Krylovmethods. We present three enlarged CG versions, MSDO-CG, LRE-CG and SRE-CG.

Sophie M. MoufawadLRI - INRIA Saclay, [email protected]


Frederic NatafLaboratoire J.L. [email protected]

MS58

Preconditioning Communication-Avoiding KrylovMethods

Krylov subspace projection methods are the most widely-used iterative methods for solving large-scale linear systemsof equations. Communication-avoiding (CA) techniquescan improve Krylov methods’ performance on modern com-puters. However, in practice CA-Krylov methods sufferfrom lack of good preconditioners that can work seamlesslyin a communication-avoiding fashion. I will outline a do-main decomposition like framework for a family of precon-ditioners that are suitable for CA Krylov methods. Withinthis framework, we can envision multiple preconditionersthat can work seamlessly with a CA-Krylov method. I willpresent the first few preconditioners we have developed sofar and present results comparing these preconditioned CAtechniques to traditional methods.



Andrey Prokopenko

116 LA15 Abstracts

Sandia National [email protected]


Michael HerouxSandia National [email protected]


MS59

Low-rank Tensor Approximation of Singular Func-tions

We study the quantized tensor train approximation of sin-gular functions, which are given, in low-order discretiza-tions, by coefficient tensors. The functions of interest areanalytic but with singularities on the boundary of the do-main. Such functions arise, for example, as solutions of lin-ear second-order elliptic problems, where the singularitiesare inherited from the data or occur due to the nonsmooth-ness of the domain. We show that the QTT approxima-tion achieves, theoretically and algorithmically, the opti-mal approximation accuracy ε with polylogarithmic ranksO(logκ ε−1). That renders the low-rank QTT represen-tation, which is amply adaptive and non-specific to theparticular class of functions, comparable or even superiorto special techniques such as hp approximations. CS wassupported by the European Research Council (ERC) underAdG 247277.

Vladimir [email protected]

Christoph SchwabETH [email protected]

MS59

A Variant of Alternating Least Squares TensorCompletion in TT-Format

We fit a low rank tensor A ∈ RI , I = {1, . . . , n}d, toa given set {Mi ∈ R | i ∈ P}, P ⊂ I being unchange-able. Considered is the TT-format with target rank r (rankadaption is realizable), while M ∈ RI (e.g. certain multi-variate functions) is assumed rank structured. A SOR-typesolver ADF with O(r2d#P ) computational complexity (r2

lower than alternating least squares (ALS)) is presented,working with #P = cdnr2 ∼ log(#I) samples. As goodresults as for ALS, yet a significantly lower runtime areobserved rank r = 4, . . . , 20, given d ≤ 50, n ≤ 100 andc ≥ 2.

Sebastian Kraemer, Lars Grasedyck, Melanie KlugeRWTH [email protected], [email protected],

[email protected]

MS59

Low Rank Approximation of High DimensionalFunctions Using Sparsity Inducing Regularization

Approximation of high dimensional stochastic functions us-ing functional approaches is often limited by the so calledcurse of dimensionality. In literature, approximation meth-ods often rely on exploiting particular structures of high di-mensional functions. One such structure which is increas-ingly found to be applicable in these functions is sparsityon suitable basis. Also, these functions exhibit optimal lowrank representation and can be approximated in suitablelow rank tensor subsets. In this work, we exploit sparsitywithin low rank representation to approximate high dimen-sional stochastic functions in a non intrusive setting usingfew sample evaluations. We will illustrate this approachon suitable examples arising in computational sciences andengineering.

Prashant [email protected]

Mathilde ChevreuilLUNAM Universite, Universite de Nantes, CNRS, [email protected]

Loic GiraldiEcole Centrale de [email protected]


MS59

Riemannian Optimization for High-DimensionalTensor Completion

Tensor completion aims to reconstruct a high-dimensionaldata set with a large fraction of missing entries. Assuminglow-rank structure in the underlying data allows us to castthe problem into an optimization problem restricted to themanifold of fixed-rank TT tensors. We present a Rieman-nian nonlinear conjugate gradient scheme scaling linearlywith the number of dimensions. In numerical experiments,we show the effectiveness of our approach and compare toadaptive sampling techniques such as cross-approximation.

Michael SteinlechnerEPF [email protected]

MS60

Rank-Structured PDE Solvers

We describe a “superfast” Cholesky code based onCHOLMOD, organized around level 3 BLAS for speed.We combine sparsity and low-rank structure and directlyfactor low-rank blocks using randomized algorithms. Fora nearly-incompressible elasticity problem, CG with ourrank-structured Cholesky converges faster than with in-complete Cholesky and the ML multigrid preconditioner,both in iteration counts and in run time. At 106 degrees of

LA15 Abstracts 117

freedom, we use 3 GB of memory; exact factorization takes30 GB.

David BindelCornell [email protected]

MS60

Approximation of Real Eigenvalues of Some Struc-tured Nonsymmetric Matrices

Matrix Sign iteration is an efficient algorithm for generaleigenvalue problem, but we explore its application to ap-proximation of real eigenvalues of structured matrices withfurther application to real polynomial root-finding. Ourformal analysis and extensive tests show that the approachis quite promising.

Victor [email protected]

MS60

Linear Time Eigendecomposition and SVD Algo-rithms

We present a set of superfast (nearly O(n) complexity intime and storage) algorithms for finding the full singu-lar value decomposition or selected eigenpairs for a (non-symmetric) rank-structured matrix. These algorithms relyon tools from Hierarchically Semiseperable (HSS) matrixtheory, as well as the Fast Multipole Method (FMM), ran-domization, and additional tools. We further show howthese algorithms can be used in applications, such as PDEsolvers. This is joint work with Jianlin Xia.

James VogelPurdue [email protected]


MS60

Butterfly Factorizations

The butterfly factorization is a data-sparse representationof the butterfly algorithm. It can be constructed efficientlyif either the butterfly algorithm is available or the entriesof the matrix can be sampled individually. For an N ×Nmatrix, the resulting factorization is a product of O(logN)sparse matrices, each with O(N) non-zero entries. Theapplication complexity of this factorization is O(N logN)with a prefactor significantly smaller than the original but-terfly algorithm.

Yingzhou [email protected]

Haizhao YangStanford UniversityMath Dept, Stanford [email protected]

Eileen R. MartinStanford [email protected]

Kenneth L. HoDepartment of MathematicsStanford [email protected]

Lexing YingStanford UniversityDepartment of [email protected]

PP1

On the Global Convergence of the Cyclic JacobiMethods for the Symmetric Matrix of Order 4

We prove the global convergence of the general cyclic Ja-cobi method for the symmetric matrix of order four. Inparticular, we study the method under the strategies thatenable full parallelization of the method. These strategies,unlike the serial ones, can force the method to be veryslow/fast within one cycle, depending on the underlyingmatrix. This implies that for the global convergence proofone has to consider several adjacent cycles.

Erna Begovic, Vjeran HariUniversity of [email protected], [email protected]

PP1

BFGS-like Updates of Constraint Preconditionersfor Saddle Point Linear Systems Arising in ConvexQuadratic Programming

Constraint preconditioners (CP) have revealed very effec-tive in accelerating iterative methods to solve saddle-pointtype linear systems arising in unconstrained optimizationproblems. To avoid a large part of the costly applicationof CP we propose to selectively compute and update it viasuitable low-rank BFGS-like modifications. We show underwhich conditions the new updated preconditioner guaran-tees convergence of the PCG method and test it onto a classof large-size problems taken from the CUTE repository.

Luca BergamaschiUniversity of [email protected]

Angeles MartinezUniversita di [email protected]

PP1

Pivotality of Nodes in Reachability Problems UsingAvoidance and Transit Hitting Time Metrics

We propose a ”pivotality” measure of how easy it is to reacha node t from a node s. This measure attempts to capturehow pivotal a role that some intermediate node k plays inthe reachability from node s to node t in a given network.We propose two metrics, the avoidance and transit hittingtimes, and show how these can be computed from the fun-damental matrices associated with the appropriate randomwalk matrices.

Daniel L. Boley

118 LA15 Abstracts

University of MinnesotaDepartment of Computer [email protected]

Golshan GolnariUniversity of MinnesotaDepartment of Electrical and Computer [email protected]

Yanhua LiUniversity of MinnesotaDepartment of Computer [email protected]

Zhi-Li ZhangUniversity of MinnesotaComputer [email protected]

PP1

A Constructive Proof of Upper Hessenberg Formfor Matrix Polynomials

We present a constructive proof that every n × n matrixpolynomial can be reduced to an upper Hessenberg matrixrational. The proof relies on a pseudo inner product to bedefined over the vector space of n tuples whose elementsare rational functions. Armed with this pseudo inner prod-uct we proceed to apply Arnoldi and Krylov type subspacemethods to obtain an upper Hessenberg matrix rationalwhose eigenvalues are the same as the original matrix poly-nomial. We present a numerical technique for computingthe eigenvalues of an upper Hessenberg matrix rational andone numerical example.

Thomas R. CameronWashington State [email protected]

PP1

A New Parallel Sparse Linear Solver

We will present a new parallel sparse linear solver based onhierarchical matrices. The algorithm shares some elementswith the incomplete LU factorization, but uses low-rankapproximations to remove fill-ins. In addition, it uses mul-tiple levels, in a way similar to multigrid, to achieve fastglobal convergence. Empirically, the current implementa-tion has nearly linear complexity on equations arising fromthe discretization of elliptic PDE, making it very efficientto solve large problems. Moreover, the task dependencygraph of this algorithm has a tree structure that we exploitin our MPI parallelization of the algorithm. Lastly, becausefill-ins are strictly controlled by the algorithm, communi-cation is needed only when boundary nodes are eliminatedon every process. Preliminary performance and scalabilityresults of the solver will be demonstrated by our numericalexperiments.

Chao ChenStanford [email protected]

PP1

Towards Batched Linear Solvers on AcceleratedHardware Platforms

Many applications need computations on very large num-

ber of small matrices, especially for GPUs, which are cur-rently known to be about four to five times more energyefficient than multicore CPUs. We describe the develop-ment of the main one-sided factorizations that work fora set of small dense matrices in parallel. Our approachis based on representing the algorithms as a sequence ofbatched BLAS routines for GPU-only execution.

Tingxing DongUniversity of [email protected]



Piotr LuszczekUniversity of [email protected]


PP1

Improved Incremental 2-Norm Condition Estima-tion of Triangular Matrices

Basically two incremental techniques to estimate the 2-norm condition number of triangular matrices were intro-duced, one using left and one using right approximate sin-gular vectors. With incremental we mean that they com-pute estimates of a leading size k submatrix from a cheapupdate of the estimates for the leading size k − 1 subma-trix. We show that a clever combination of both techniquesand information on the inverse triangular factors lead to asignificantly more accurate estimator.

Jurjen Duintjer TebbensInstitute of Computer SciencesAcademy of Sciences of the Czech Republic, [email protected]

Miroslav TumaAcademy of Sciences of the Czech RepublicInstitute of Computer [email protected]

PP1

A New Regularization Method for ComputationalColor Constancy

Computational color constancy is an image processingproblem that consists of correcting for colors in digital im-ages distorted by noise and illuminants. The color con-stancy is an ill-posed problem, therefore regularizationmethods need to be applied. In this work, we proposea new regularization method for computational color con-stancy and show its performance in several natural images.

LA15 Abstracts 119

Malena I. Espanol, Michael WranskyDepartment of MathematicsUniversity of [email protected], [email protected]

PP1

Design and Analysis of a Low-Memory MultigridAlgorithm

We develop an efficient multigrid solver with low memorycomplexity, in the spirit of A. Brandt, M. Adams, and oth-ers. The algorithm avoids storing the entire finest grid.We introduce new ideas on how to apply τ -corrections ina full-approximation scheme to preserve the accuracy of astandard full multigrid solver while keeping the computa-tion costs low. We use Fourier analysis to study the errorcomponents at different stages of the algorithm.

Stefan HennekingGeorgia Institute of [email protected]

PP1

Anderson Acceleration of the Alternating Projec-tions Method for Computing the Nearest Correla-tion Matrix

In a wide range of applications an approximate correla-tion matrix arises that is symmetric but indefinite and itis required to compute the nearest correlation matrix inthe Frobenius norm. The alternating projections methodis widely used for this purpose, but has (at best) a linearrate of convergence and can require many iterations. Weshow that Anderson acceleration, a technique for acceler-ating the convergence of fixed-point iterations, can suc-cessfully be applied to the alternating projections method,bringing a significant reduction in both the number of it-erations and the computation time. We also show thatAnderson acceleration remains effective when certain ele-ments of the approximate correlation matrix are requiredto remain fixed.

Nicholas HighamSchool of MathematicsThe University of [email protected]

Natasa StrabicSchool of MathematicsSchool of Mathematics The University of [email protected]

PP1

Block Preconditioning for Time-Dependent Cou-pled Fluid Flow Problems

Many important engineering and scientific systems requirethe solution of incompressible flow models or extensions tothese models such as coupling to other processes or incor-porating additional nonlinear effects. Finite element meth-ods and other numerical techniques provide effective dis-cretizations of these systems, and the generation of theresulting algebraic systems may be automated by high-level software tools such those in the Sundance project,but the efficient solution of these algebraic equations re-mains an important challenge. In addition, many impor-

tant applications require time-dependent calculations. Weexamine block preconditioners for time-dependent incom-pressible Navier-Stokes problems and some related coupledproblems. In some time-dependent problems, explicit timestepping methods can require much smaller time steps forstability than are needed for reasonable accuracy. Thisleads to taking many more time steps than would other-wise be needed. With implicit time stepping methods, wecan take larger steps, but at the price of needing to solvelarge linear systems at each time step. We consider im-plicit Runge-Kutta (IRK) methods. Suppose our PDE hasbeen linearized and discretized with N degrees of freedom.Using an s-stage IRK method leads to an sN × sN linearsystem that must be solved at each time step. These linearsystems are block s×s systems, where each block is N×N .We investigate preconditioners for such systems, where wetake advantage of the fact that each subblock is related toa linear system from the (coupled) fluid flow equations.

Victoria Howle, Ashley MeekTexas [email protected], [email protected]

PP1

Structured Computations of Block Matrices withApplication in Quantum Monte Carlo Simulation

A block matrix can be regarded as a reshaped fourth-ordertensor. A properly structured computation of the blockmatrix provides insight into the tensor and vice versa. Wewill present a fast sketching algorithm for computing theinversion of a block p-cyclic matrix, and our synergistic ef-fort in developing stable and high-performance implemen-tation of the algorithm for the applications of the fourth-order tensor computations arising from quantum Monte-Carlo simulation.

Chengming JiangUniversity of California, Davis, [email protected]

Zhaojun BaiDepartments of Computer Science and MathematicsUniversity of California, [email protected]

Richard ScalettarDepartment of Physics,University of California, Davis, [email protected]

PP1

MHD Stagnation Point over a Stretching Cylinderwith Variable Thermal Conductivity

This article addresses the behavior of viscous fluid nearstagnation point over a stretching cylinder with vari-able thermal conductivity. The effects of heat genera-tion/absorption are also encountered. Thermal conductiv-ity is considered as a linear function of temperature. Com-parison between solutions through HAM and Numerical ispresented. The effect of different parameters on velocityand temperature fields are shown graphically..

Farzana Khanstudent

120 LA15 Abstracts

[email protected]

PP1

On Simple Algorithm Approximating ArbitraryReal Powers Aα of a Matrix from Number Rep-resentation System

In this paper we present new algorithm for approximat-ing arbitrary real powers Aα of a matrix A ∈ Cn×n onlyusing matrix standard operation and matrix square rootalgorithms. Furthermore, although a real matrix with non-positive eigenvalues is given, we get real solution withoutcomplex arithmetic. Its accuracy is only depend on a con-dition number of A and that of matrix roots. Since the newalgorithm works for arbitrary real α and is independent onsingularity and defectiveness against Pade-approximationand Schur-Parllet algorithm respectively, it seems the onlyway to get a good approximation of arbitrary real powerwithout any handling when a given matrix has such con-ditions. To addition this algorithm is also well suited inparallel computation because it can be implemented onlyby standard matrix operations.

Yeonji KimDepartment of mathematicsPusan Nation [email protected]

Jong-Hyeon SeoBasic [email protected]

Hyun-Min KimDepartment of mathematicsPusan National [email protected]

PP1

Cycles of Linear and Semilinear Mappings

A mapping A from a complex vector space U to a complexvector space V is semilinear if

A(u+ u′) = Au+Au′, A(αu) = αAufor all u, u′ ∈ U and α ∈ C. We give a canonical form ofmatrices of a cycle of linear or semilinear mapping

V1 · · · Vt−1V2 ��Vtin which all Vi are complex vector spaces, each line is

an arrow −→ or ←−, and each arrow denotes a linear orsemilinear mapping. The talk is based on the article [LinearAlgebra Appl. 438 (2013) 3442–3453].

Tetiana KlymchukTaras Shevchenko National University of Kyiv, UkraineMechanics and Mathematics [email protected]

PP1

Regularization of the Kernel Matrix via CovarianceMatrix Shrinkage Estimation

The kernel trick concept, formulated as an inner productin a feature space, facilitates powerful extensions to manywell-known algorithms. While the kernel matrix involvesinner products in the feature space, the sample covariance

matrix of the data requires outer products. Therefore, theirspectral properties are tightly connected. This allows usto examine the kernel matrix through the sample covari-ance matrix in the feature space and vice versa. The useof kernels often involve a large number of features, com-pared to the number of observations. In this scenario,the sample covariance matrix is not well-conditioned noris it necessarily invertible, mandating a solution to theproblem of estimating high-dimensional covariance matri-ces under small sample size conditions. We tackle thisproblem through the use of a shrinkage estimator that of-fers a compromise between the sample covariance matrixand a well-conditioned matrix (also known as the ”tar-get”) with the aim of minimizing the mean-squared error(MSE). We propose a distribution-free kernel matrix regu-larization approach that is tuned directly from the kernelmatrix, avoiding the need to explicitly address the featurespace. Numerical simulations demonstrate that the pro-posed regularization is effective in classification tasks.

Tomer LancewickiUniversity of [email protected]

Itamar ArelUniversity of TennesseeDepartment of Electrical Engineering and [email protected]

PP1

Computationally Enhanced Projection Methodsfor Symmetric Lyapunov Matrix Equations

In the numerical treatment of large-scale Lyapunov equa-tions, projection methods require solving a reduced Lya-punov problem to check convergence. As the approxima-tion space expands, this solution takes an increasing por-tion of the overall computational efforts. When data aresymmetric, we show that convergence can be monitored atsignificantly lower cost, with no reduced problem solution.Numerical experiments illustrate the effectiveness of thisstrategy for standard and extended Krylov methods.

Davide Palitta, Valeria SimonciniUniversita’ di [email protected], [email protected]

PP1

Fast Interior Point Solvers for PDE-ConstrainedOptimization

We present a fast interior point method for solvingquadratic programming problems arising from a numberof PDE-constrained optimization models with bound con-straints on the state and control variables. Particular em-phasis is given to the development of preconditioned iter-ative methods for the large and sparse saddle point sys-tems arising at each Newton step. Having motivated andderived our recommended preconditioners, we present nu-merical results demonstrating the potency of our solvers.

John PearsonUniversity of [email protected]

PP1

An Optimal Solver for Linear Systems Arising from

LA15 Abstracts 121

Stochastic FEM Approximation of Diffusion Equa-tions with Random Coefficients

Stochastic Galerkin approximation of elliptic PDE prob-lems with correlated random data often results in hugesymmetric positive definite linear systems. The novel fea-ture of our preconditioned MINRES solver involves incor-poration of error control in the natural “energy’ norm to-gether with an effective a posteriori estimator for the PDEapproximation error leading to a robust and optimally ef-ficient stopping criterion: the iteration terminates whenthe algebraic error becomes insignificant compared to theapproximation error.

Pranjal Prasad, David SilvesterUniversity of [email protected],[email protected]

PP1

Approximation of the Scattering Amplitude UsingNonsymmetric Saddle Point Matrices

This poster presents iterative methods for solving the for-ward and adjoint systems where the matrix is large, sparse,and nonysmmetric, such as those used to approximate thescattering amplitude. We use a conjugate gradient-like it-eration for a nonsymmetric saddle point matrix that hasa real positive spectrum and a full set of eigenvectors thatare, in some sense, orthogonal. Numerical experimentsdemonstrate the effectiveness of this approach comparedto other iterative methods for nonsymmetric systems.

Amber S. RobertsonThe University of Southern [email protected]

PP1

Generalizing Block Lu Factorization:A Lower-Upper-Lower Block Triangular Decompositionwith Minimal Off-Diagonal Ranks

We propose a factorization that decomposes a 2×2-blockednon-singular matrix into a product of three matrices thatare respectively lower block-unitriangular, upper block-triangular, and lower block-unitriangular. In addition, wemake this factorization “as block-diagonal as possible’ byminimizing the ranks of the off-diagonal blocks. We presentsharp lower bounds for these ranks and an algorithm tocompute an optimal solution. One application of this fac-torization is the design of optimal logic circuits for stream-ing permutations.

Francois SerreETH [email protected]

Markus PuschelDepartment of Computer ScienceETH [email protected]

PP1

Existence and Uniqueness for the Inverse Prob-lem for Linear and Linear-in-parameters Dynam-ical Systems

Certain experiments are non-repeatable, because they re-sult in the destruction or alteration of the system under

study, and thus provide data consisting of at most a singletrajectory in state space. Before proceeding with parame-ter estimation for models of such systems, it is important toknow if the model parameters can be uniquely determined,or identified, from idealized (error free) single trajectorydata. For linear models and a class of nonlinear systemsthat are linear in parameters, I will present several charac-terizations of identifiability. I will establish necessary andsufficient conditions, solely based on the geometric struc-ture of an observed trajectory, for these forms of identi-fiability to arise. I will extend the analysis to considercollections of discrete data points, where, due to proper-ties of the fundamental matrix solution, we can recover theparameters explicitly in certain cases. I will then examinethe role of measurement error in the data and determineregions in the data space which correspond to parametersets with desired properties, such as identifiability, modelequilibrium stability, and sign structure.

Shelby StanhopeUniversity of [email protected]

PP1

Is Numerical Stability an Important Issue in Iter-ative Computations?

Although the answer seems obvious, the practice might benot. Full numerical stability analysis might be intriguing orour of reach. This contribution examines several situationswhere the presentation and analysis of iterative numericalalgorithms assumes exact arithmetic. Such approach canbe well justified in some cases and it can severely limitapplication of the obtained results in others.

Zdenek StrakosFaculty of Mathematics and Physics,Charles University in [email protected]

PP1

Computing Geodesic Rotations

An element of a flag manifold is a frame—a decomposi-tion of Euclidean space into orthogonal components—anda geodesic path on a flag manifold provides a “direct” ro-tation from one frame to another. Furthermore, a geodesicpath corresponds to a decomposition of unitary matricesthat generalizes the CS decomposition. We develop effi-cient methods for numerical computation on a flag mani-fold, including the location of geodesic paths, and considera question of uniqueness.

Brian D. SuttonRandolph-Macon [email protected]

PP1

On the Numerical Behavior of Quadrature-basedBounds for the A-Norm of the Error in CG

The A-norm of the error in the conjugate gradient (CG)method can be estimated using quadrature-based bounds.Our numerical experiments predict that some numericaldifficulties may arise when computing upper bounds basedon modified quadrature rules like Gauss-Radau, regardlesswhether we use the CGQL algorithm or our new CGQalgorithm for their computation. In this presentation we

122 LA15 Abstracts

analyze this phenomenon and try to explain when possiblenumerical difficulties can appear.

Petr TichyCzech Academy of [email protected]

Gerard MeurantRetired from Commissariat a l’Energie Atomique (CEA)[email protected]

PP1

Tensor Notation: What Would Hamlet Say?

It is hard to spread the word about tensor computations.Summations, transpositions, and symmetries are describedthrough vectors of subscripts and vectors of superscripts. Isthere a magic notation for handling that stuff? Einstein no-tation and Dirac notation are noble efforts. Other worthynotations flow from traditional ”SIAM style” matrix com-putations. I will suggest with examples how the researchcommunity might navigate among these schemes. Shake-speare points the way: ”There is nothing either good orbad [about any given tensor notation] but thinking makesit so”. Hamlet II,II,250.

Charles Van LoanCornell UniversityDepartment of Computer [email protected]

PP1

Best Rank-1 Approximations Without OrthogonalInvariance for the 1-norm and Frobenius 1-norm

We consider finding the best rank-1 approximations to amatrix A where the error is measured as the operator in-duced 1-norm (maximum column 1-norm) or Frobenius 1-norm (sum of absolute values over the matrix). Neither ofthese norms are unitarily invariant and the typical SVDprocedure is insufficient. For the case of a 2 × 2 matrixA, there is a simple solution procedure and the result-ing approximation bound for the Frobenius 1-norm is the| det(A)|maxij

|Aij | . We present some thoughts on how to extend the

arguments to larger matrices.

Varun VasudevanPurdue [email protected]

Tim PostonForus [email protected]


PP1

Pseudospectra Computation Via Rank-RevealingQR Factorization

The pseudospectrum of a matrix is a way to visualize thenon-normality of a matrix and the sensitivity of its eigen-values. In this paper, we propose an equivalent definitionof pseudospectra of matrices using Rank-Revealing QR(RRQR) factorization. We give the proof of the equiv-

alency between this definition and classical ones. And,we propose some generalizations of eigenvalue theoremsto pseudospectra theorems based on the equivalent defi-nition. Also, an algorithm of pesudospectra computationvia RRQR is given. Numerical experiments and compar-isons are given to illustrate the efficiency of the results inthis paper.

Zhengsheng WangNanjing University of Aeronautics and [email protected]

Lothar ReichelKent State [email protected]

Min Zhang, Xudong LiuDepartment of Mathematics, Nanjing University [email protected], [email protected]

PP1

Linear Response and the Estimation of AbsorptionSpectrum in Time-Dependent Density FunctionalTheory

The absorption spectrum of a molecular system can beestimated from the dynamic dipole polarizability associ-ated with the linear response of a molecular system (at itsground state) to an external perturbation. Although an ac-curate description of the absorption spectrum requires thediagonalization of the so-called Casida Hamiltonian, thereare more efficient ways to obtain a good approximation tothe general profile of the absorption spectrum without com-puting eigenvalues and eigenvectors. We will describe thesemethods that only require multipling the Casida Hamil-tonian with a number of vectors. When highly accurateoscillator strength is required for a few selected excitationenergies, we can use a special iterative method to obtainthe eigenvalues and eigenvectors associated with these en-ergies efficiently.


PP1

A Set of Fortran 90 and Python Routines for Solv-ing Linear Equations with IDR(s)

We present Fortran 90 and Python implementations of[Van Gijzen and Sonneveld, Algorithm 913: An ElegantIDR(s) Variant that Efficiently Exploits Bi-orthogonalityProperties. ACM TOMS, Vol. 38, No. 1, pp. 5:1-5:19,2011]. Features of the new software include: (1) the capa-bility of solving general linear matrix equations, includingsystems with multiple right-hand-sides, (2) the possibilityto extract spectral information, and (3) subspace recycling.

Martin B. van GijzenDelft Inst. of Appl. MathematicsDelft University of [email protected]

Reinaldo AstudilloDelft University of [email protected]

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