Intro Analysis-Reduction Coupling Conclusions
Uncertainty Quantification in Multiscale Models
H.N. Najm
Sandia National LaboratoriesLivermore, CA
Workshop onStochastic Multiscale Methods: Mathematical Analysis and Algorithms
Univ. Southern California, Aug 10–11, 2009
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Intro Analysis-Reduction Coupling Conclusions
Uncertainty Quantification in Multiscale Models
Complex physical systems involve wide ranges of lengthand time scalesUncertainties abound in
specification of the full detailed modelidentification of reduced submodels over ranges of scalescoupling between submodelsdata assimilationcalibration and validation of coupled multiscale models
UQ is needed in the analysis, construction, and utilizationof multiscale models
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Intro Analysis-Reduction Coupling Conclusions
Outline
1 Introduction
2 Analysis and Reduction of Multiscale Models
3 Coarse-Fine Coupling in Multiscale Models
4 Conclusions
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Intro Analysis-Reduction Coupling Conclusions
Analysis and Reduction of Multiscale Models
There are many methods for analysis of multiscale physicalsystems, enabling reduction of their size, complexity, and/ordimensionality.
A deterministic reduction requirement:
Given specified observables, desired error thresholds, and thedetailed model, identify the smallest acceptable model
A potential reduction requirement under uncertainty:
Given specified observables, desired error thresholds, and anuncertain detailed model, identify the smallest acceptable(deterministic or uncertain) model
Homescu, Petzold, & Serban, SIAM Review, 2007
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Intro Analysis-Reduction Coupling Conclusions
Multiscale Coupling
There are manyavailable methods forcoupling multiscalemodels across a rangeof application domains
Ma Mb
Coupling techniques generally pursue accuracy incommunication of relevant physical quantities acrossheterogeneous numerical representations
Communication of uncertainty across model interfaces isimportant from a general UQ perspective, but also morespecifically to account for multiscale coupling errors
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Intro Analysis-Reduction Coupling Conclusions
Physical Models Spanning a Large Range of Scales
The degree of coupling among length/time scales affectsthe extent to which a model can be reduced, arriving at asimplified coupled system of submodelsThere are many analysis methods for examining thiscoupling and identifying low order reduced models
PDE systemsProper Orthogonal DecompositionPrincipal Component Analysis
Elliptic systemsFast Multipole Methods
Stochastic systemsKarhunen-Loeve Expansion
ODE systemsEigenvalue and SVD methodsComputational Singular Perturbation (CSP)
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Intro Analysis-Reduction Coupling Conclusions
Structure of Stiff ODE Models
Macroscale chemical models typically involve largestrongly-nonlinear stiff ODE systems
Stiffness is associated with the presence of a large rangeof time scalesStiff ODE system dynamics typically exhibit lowdimensional algebraic manifolds
The solution exhibits fast attraction towards the manifold– the fast subspace
The system evolves slowly along the manifold– the slow subspace
Identifying and decoupling the fast and slow subspacesprovides means of model analysis, and reduction
– CSP (Lam & Goussis, 1988)
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Intro Analysis-Reduction Coupling Conclusions
CSP Highlights
dydt
= g(y), y(t = 0) = y0
g = a1f 1 + · · · + aNf N, f i = bi · g, τi < τi+1, i = 1, . . . , N
g = a1f 1 + · · · + aMf M
︸ ︷︷ ︸
gfast≈0
+ aM+1f M+1 + · · · + aNf N
︸ ︷︷ ︸
gslow
Manifold:
f i(y) ≈ 0, i = 1, · · · , M
Slow Evolution:
gslow =
N∑
s=M+1
asfs =
(
I −M∑
r=1
arbr
)
g = Pg
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Intro Analysis-Reduction Coupling Conclusions
Model Reduction based on CSP Analysis
Employ CSP slow/fast Importance Indices I ik
– Measure importance of process k to the slow/fastevolution of species i
Use a given database of solutions of the detailed model,along with
– a specification of observables of interest u ⊂ y– thresholds on importance indices
arrive at reduced ODE modelValorani et al., 2006
Algorithm maintains controlled accuracy in representing boththe low dimensional manifold and time evolution along it
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Intro Analysis-Reduction Coupling Conclusions
CSP reduction of nHeptane-air model
Detailed model:560 sp, 2538 rn
A posteriori erroranalysis forhomogeneousignition of annHeptane-airmixture:
# species
Err
or%
τ ign
100 150 200 250 300 350 400
10-1
100
101
102
T0 = 700 KT0 = 850 KT0 = 1100 K
Error in ignition time versus the no. ofspecies in the reduced model
(Valorani et al., 2007)
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Intro Analysis-Reduction Coupling Conclusions
Uncertainty in ODE model
Consider uncertainty in g(y) (structure/parameters).
dydt
= gω(y)
Depending on the uncertain elements of gω
, this can implyuncertainty in the fast and/or slow subspaces
– with consequences to model reduction
Stochastic eigenvalue problem
– Galerkin versus collocation formulation
... work in progress
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Intro Analysis-Reduction Coupling Conclusions
Coupled Multiscale Models
Accuracy of multiscale models based on coupled, possiblyheterogeneous, submodels is function of:
accuracy of individual submodelsaccuracy of coupling at model interfaces
Need to communicate uncertainty at model interfaces for:Forward propagation of model/parametric uncertainty inotherwise deterministic coupled multiscale constructionsAccounting for information mismatch at the interface
– Coarse-graining, upscaling, lifting, restrictionAccounting for errors due to finite sample size in stochasticsubmodels, e.g.
– SDE coupled with a deterministic model– MD/BD-continuum coupling
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Intro Analysis-Reduction Coupling Conclusions
Model/Parametric Uncertainty in Multiscale Models
Ma(ωa) Mb(ωb)
Uncertainties in {Ma, ωa,Mb, ωb} need to becommunicated in both directions between the two models
Sampling/collocation based UQ methods compute Ndeterministic instances of the coupled multiscale system
– no algorithmic consequences for the interfaceIntrusive Polynomial Chaos (PC) methods
Galerkin projection of the original governing equationsSpectral representations of random variable or fields
– Spectral representations of uncertain quantitiesneed to be communicated at the interface
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Intro Analysis-Reduction Coupling Conclusions
Coarse Graining
ExamplesBrownian dynamicscomputations employinglumped atom models andrelevant force-fieldsDSMCHomogenization-upscaling
VILLIN-1QQV
Fitting rich information from a complex model, or from data,using a lower fidelity model with a smaller number ofdegrees of freedom
Identify effective force fields, material properties, or otherparameters of the low fidelity coarse model, by calibratingwith respect to the complex model and/or data
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Intro Analysis-Reduction Coupling Conclusions
Upscaling with Uncertainty Quantificaton
Define stochastic uncertain effective upscaled properties
Forward UQ provides uncertain coarse-model prediction
Least-squares regression and optimization can be used tocalibrate the uncertain effective properties
Arnst & Ghanem, 2008
The density of the uncertain model outputs provides thebasis for the likelihood function in a Bayesian inferenceprocedure to similarly solve the calibration problem
Ghanem & Doostan, 2006; Liu et al., 2009
In the presence of measurement data, the associatednoise convolves with the uncertain model output density toform the likelihood function
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Intro Analysis-Reduction Coupling Conclusions
Coarse-Fine Model CouplingA Multiscale Combustion system Model
Detailed 2D reactingflow model in smallcombustor region
Non-reacting 1Dcompressible flow modelin long exhaust zone
l L >> l
Averaging of a 2D upstream flow/mixture structure at theinterface, to that of a 1D model, at each coarse time-stepAccounting for uncertainty:
Discrepancy between the two models can be representedusing a statistical modelThis provides the uncertain BC for the 1D domain
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Intro Analysis-Reduction Coupling Conclusions
Statistical Model Construction
A statistical model can be built based on:Probability Distribution.
– Parametric/non-parametric– Gaussian process model
e.g. Kennedy & O’Hagan, 2000– Gaussian mixture models– Kernel density estimation
Moments– Spectral expansion for RV/Random Process
Optimal choice depends onthe availability of datawhether or not the model is directly observable
– model costthe intended use
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Intro Analysis-Reduction Coupling Conclusions
Limited Information at Interface
Information availablefrom/to microscale modelor data assimilation atselect spatial locations inthe macroscale model
Ma
Mb/D Mb/D Mb/D
Boundary condition for Ma can be formulated as astatistical model, expressing uncertainty in the field alongthe interface away from the available points
Parameters of the statistical model can be determinedusing Bayesian inference or least-squares methods
Measurement uncertainty resulting from noise in the datacan be directly incorporated in the inference procedure
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Intro Analysis-Reduction Coupling Conclusions
Statistical Model Correlation Structure
With limited information, the random process or statisticalmodel representing the uncertain field at the interface isnot sufficiently specified
Uncorrelated fine-scale/data inputsNo information on spatial correlation structure along themacro dimension
Yet there are physical constraints on the range ofcorrelation lengths
The structure of material-property variability along theinterfaceThe length scales inherent in the macroscale forcing
These need to be used to constrain the correlationstructure
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Intro Analysis-Reduction Coupling Conclusions
Operator-Upscaling, Variational Multiscale Method
Using the known detailed material properties in afull-domain solution is prohibitively expensive.
Solve the fine-scale problem independently within eachcoarse-scale element
Solve the coarse-scale problem, with the same physicalfine-scale material properties but an upscaled operator
Arbogast, 1998; Hughes, 1995
Stochastic version employing stochastic variationalmultiscale methods with Polynomial Chaos andKarhunen-Loeve Expansions
Accounting for inherent randomness of material properties
Asokan and Zabaras, 2006
Ganapathysubramanian and Zabaras, 2007, 2009
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Intro Analysis-Reduction Coupling Conclusions
Projective Time Integration
Detailed full-scale solution time integration over shortbursts of time, possibly at select macroscale locations
Use resulting statistics to inform large-∆t coarse-level timeintegration
Lifting and Restriction operationsKevrekidis, 2000-
Used for modeling diffusion with uncertain diffusivityXiu and Kevrekidis, 2005
MC Sampled fine-scale computationsEquation-free coarse-scale time integration of uncertainPolynomial Chaos representation
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Intro Analysis-Reduction Coupling Conclusions
Stochastic Coupling
Averaged informationfrom stochastic modelcommunicated todeterministic model
Finite sample sizeslead to uncertainty inunconverged averages
Ma
Deterministic
Mb
StochasticSDE/MD/BD
A wealth of deterministic MD/Continuum coupling methods. . . Kevrekidis 2003, Prudhomme 2009, . . .
What about coupling while accounting for averaginguncertainty?
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Intro Analysis-Reduction Coupling Conclusions
Multiscale Modeling of Desalination Membranes
Electrochemicalacqueous transportof Na+ and Cl− innanoporousmembranes
Adalsteinsson, Debusschere, Long, and Najm, 2008
Continuum: Poisson-Nernst-Planck – finite element
Particle: Brownian Dynamics
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Intro Analysis-Reduction Coupling Conclusions
Multiscale Coupling
Subdomains solved in an alternating fashion– Schwarz alternating method
Hadjiconstantinou, JCP 1999; Schwarz, 1896
Electrostatic field computation– continuum: ignore particles inside the nanopore– nanopore: superpose continuum field and local
particle Coulomb interactions
Handshake zone coupling region– Continuum solution provides concentration BC for
the particle domain– BD solution provides flux BC for the continuum
domain
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Intro Analysis-Reduction Coupling Conclusions
Residual Averaging Noise in BD Fluxes
Multiple replica simulationsat each iteration step
Standard deviation ofreplica simulation fluxes iscomparable with theiteration-to-iteration noisein the mean
0 2 4 6 8 10iteration step
0.0e+00
5.0e-17
1.0e-16
1.5e-16
flux
[m
ol/s
]
JNa
, 150-50 mMJ
Cl, 150-50 mM
JNa
, 150-100 mMJ
Cl, 150-100 mM
Residual variability could be communicated to thecontinuum model as an uncertainty in the flux BC
– work in progress
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Intro Analysis-Reduction Coupling Conclusions
Summary
Numerous methods for analyzing, reducing, and couplingmultiscale models
Uncertainties abound in the inherent formulation andpractical implementation of these systems
Methods have been developed to handle and communicatethese uncertaintiesMany opportunities remain
Coupling multiple/all-relevant sources of uncertaintyBalancing error budgets and model reduction underuncertaintyHandling systems with strong coupling in length/time scales
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Intro Analysis-Reduction Coupling Conclusions
Acknowledgements
B.J. Debusschere,H. Adalsteinsson . . . Sandia National Labs, Livermore, CA
O.M. Knio . . . Johns Hopkins Univ., Baltimore, MDR.G. Ghanem . . . Univ. Southern California, Los Angeles, CAY.M. Marzouk . . . Massachusetts Inst. Tech., Cambridge, MAM. Valorani . . . La Sapienza Univ. Rome, ItalyD. Goussis . . . National Tech. Univ. Athens, Greece
This work was supported by the US Department of Energy (DOE), Office of Science (SC), Office of Basic EnergySciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences, the SciDAC ComputationalChemistry Program, and the Office of Advanced Scientific Computing Research (ASCR). Sandia NationalLaboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for theUnited States Department of Energy under contract DE-AC04-94-AL85000.
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Intro Analysis-Reduction Coupling Conclusions
Relevant Papers – 1
H. Adalsteinsson, B.J. Debusschere, K.R. Long, and H.N. Najm, “Components for atomistic-to-continuummultiscale modeling of flow in micro- and nanofluidic systems”, Scientific Programming, 16(4):297-313,2008.
T. Arbogast, S.E. Minkoff, and P.T. Keenan, “An operator-based approach to upscaling the pressureequation”, Computational Methods in Water Resources XII, Vol. 1: Computational Methods inContamination and Remediation of Water Resources, V.N. Burganos et al. Eds., Computational MechanicsPublications, Southampton, U.K., pp. 405-412, 1998.
M. Arnst and R. Ghanem, “Probabilistic equivalence and stochastic model reduction in multiscale analysis”,Comput. Methods Appl. Mech. Engrg., 197:3584-35923, 2008.
B.V. Asokan and N. Zabaras, “A stochastic variational multiscale method for diffusion in heterogeneousrandom media”, J. Comp. Phys., 218:654-676, 2006.
P.T. Bauman, J.T. Oden, and S. Prudhomme, “Adaptive multiscale modeling of polymeric materials withArlequin coupling and Goals algorithms”, Comput. Methods Appl. Mech. Engrg., 198:799-818, 2009.
B. Ganapathysubramanian and N. Zabaras, “Modeling diffusion in random heterogeneous media:Data-driven models, stochastic collocation and the variational multiscale method”, J. Comp. Phys.,226:326-353, 2007.
B. Ganapathysubramanian and N. Zabaras, “A stochastic multiscale framework for modeling flow throughrandom heterogeneous porous media”, J. Comp. Phys., 228:591-618, 2009.
R. Ghanem and A. Doostan, “On the construction and analysis of stochastic models: Characterization andpropagation of the errors associated with limited data”, J. Comp. Phys., 217:63-81, 2006.
T.J.R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgridscale models, bubbles and the origins of stabilized methods”, Comput. Methods Appl. Mech. Engrg.,127:387-501, 1995.
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Intro Analysis-Reduction Coupling Conclusions
Relevant Papers – 2
V. Kolobov, R. Arslanbekov, ad A. Vasenkov, “Coupling Atomistic and Continuum Models for Multi-scaleSimulations of Gas Flows”, Y.Shi et al. (Eds.): ICCS 2007, Part I, LNCS 4487, pp. 858-865, Springer-VerlagBerlin Heidelberg 2007.
W.K. Liu, L. Siad, R. Tian, S. Lee, D. Lee, X. Yin, W. Chen, S. Chan, G.B. Olson, L.-E. Lingden, M.F.Horstmeyer, Y.-S. Chang, J.-B. Choi and Y.J. Kim, “Complexity science of multiscale materials via stochasticcomputations”, Int. J. Numer. Meth. Engng., submitted, 2009.
S. Prudhomme, H.B. Dhia, P.T. Bauman, N. Elkhodja, and J.T. Oden, “Computational analysis of modelingerror for the coupling of particle and continuum models by the Arlequin method”, Comput. Methods Appl.Mech. Engrg., 197:3399-3409, 2008.
S. Prudhomme, L. Chamoin, H.B. Dhia, and B.T. Bauman, “An adaptive strategy for the control of modelingerror in two-dimensional atomic-to-continuum coupling simulations”, Comput. Methods Appl. Mech. Engrg.,198:1887-1901, 2009.
M. Valorani, F. Creta, D.A. Goussis, J.C. Lee, and H.N. Najm, “Chemical Kinetics Mechanism Simplificationvia CSP”, Combustion and Flame, 146:29-51, 2006.
M. Valorani, F. Creta, F. Donato, H.N. Najm, and D.A. Goussis, “Skeletal Mechanism Generation andAnalysis for n-heptane with CSP”, Proc. Comb. Inst., 31:483-490, 2007.
D. Xiu, I.G. Kevrekidis, and R. Ghanem, “An Equation-Free Multiscale Approach to UncertaintyQuantification”, Computing in Science & Engineering, pp.18-23, May/June 2005.
D. Xiu and I.G. Kevrekidis, “Equation-free, multiscale computation for unsteady random diffusion”, MultiscaleModel. Simul., 4(3):915-935, 2005.
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