on the use of sparse direct solver in a projection method for generalized eigenvalue problems using...

On the Use of Sparse Direct Solver in a Projection Method for Generalized

Eigenvalue Problems Using Numerical Integration

Takamitsu Watanabe and Yusaku Yamamoto

Dept. of Computational Science & Engineering

Nagoya University

Outline

Background

Objective of our study

Projection method for generalized eigenvalue problems using numerical integration

Application of the sparse direct solver

Numerical results

Conclusion

Background

Generalized eigenvalue problems in quantum chemistry and structural engineering

real axis

eigenvalues

specified interval

BxAx

Problem characteristics A and B are large and sparse. A is real symmetric and B is

s.p.d. Eigenvalues are real. Eigenvalues in a specified

interval are often needed.

RBA ,Given , find and such that .nn R 0x

HOMO LUMO

Background (cont’d)

A projection method using numerical integrationSakurai and Sugiura, A projection method for generalized eigenvalue problems,

J. Comput. Appl. Math. (2003)

Reduce the original problem to a small generalized eigenvalue problem within a specified region in the complex plane.

By solving the small problem, the eigenvalues lying in the region can be obtained.

The main part of computation is to solve multiple linear simultaneous equations.

Suited for parallel computation.

Original problem

Small generalized eigenvalue problem within the region

regionBxAx

Objective of our study

Previous approach Solve the linear simultaneous equations by an iterative

method. The number of iterations needed for convergence

differs from one simultaneous equations to another. This brings about load imbalance between processors,

decreasing parallel efficiency.

Our study Solve the linear simultaneous equations by a sparse

direct solver without pivoting. Load balance will be improved since the computational

times are the same for all linear simultaneous equations.

Projection method for generalized eigenvalue problems using numerical integration

×λm+2

×λm+1

Suppose that has distinct eigenvalues and that we need that lie in a closed curve .

BxAx d ,,, 21 m ,,, 21 )( dm

1

2

m

d

Using two arbitrary complex vectors , define a complex function

Then, f (z) can be expanded asfollows:

nCvu ,

.

C, g(z): polynomial in z.,c

c

Projection method for generalized eigenvalue problems using numerical integration (cont’d)

0 1 1

1 22 ,

1 2 2

1 2

2 3 11 ,

1 2 1

:

:

m

m mm i j i j

m m m

m

m mm i j i j

m m m

H

H

　　

　　

Further define the moments by and two Hankel matrices by

1

2

m

d

.

Th. are the m roots of . m ,,, 21

The original problem has been reduced to a small problem through contour integral.

BxAx

Projection method for generalized eigenvalue problems using numerical integration (cont’d)

Path of integration

Set the path of integration to a circle with center and radius .

Approximate the integral using the trapezoidal rule.

Computation of the moments :k

The function valueshave to be computed for each

.

Solution of N independent linearsimultaneous equations is necessary(N = 64 128).

1

2

1mm

j

Application of the sparse direct solver

Application of the sparse direct solver For a sparse s.p.d. matrix, the sparse direct solver

provides an efficient way for solving the linear simultaneous equations.

We adopt this approach by extending the sparse direct solver to deal with complex symmetric matrices.

The coefficient matrix is a sparse complex symmetric matrix.

A and B: sparse symmetric matrices, : a complex number

j

The sparse direct solver

Characteristics Reduce the computational work and memory

requirements of the Cholesky factorization by exploiting the sparsity of the matrix.

Stability is guaranteed when the matrix is s.p.d. Efficient parallelization techniques are available.

ordering

symbolic factorization

Cholesky factorization

triangular solution

Find a permutation of rows/columns that reduces computational work and memory requirements. Estimate the computational work and memory requirements. Prepare data structures to store the Cholesky factor.

Extension of the sparse direct solver to complex symmetric matrices

Algorithm Extension is straightforward by using the Cholesky

factorization for complex symmetric matrices. Advantages such as reduced computational work,

reduced memory requirements and parallelizability are carried over.

Accuracy and stability Theoretically, pivoting is necessary when factorizing

complex symmetric matrices. Since our algorithm does not incorporate pivoting,

accuracy and stability is not guaranteed.

We examine the accuracy and stability experimentally by comparing the results with those obtained using GEPP.

Numerical results

Matrices used in the experiments

BCSSTK12 BCSSTK13 FMO

matrix N NNZ explanation

BCSSTK12 1473 17,857 Ore car -- consistent mass

BCSSTK13 2003 42,943 Fluid flow generalized eigenvalues

FMO 1980 365,030 Fragment molecular orbital method

Harwell-BoeingLibrary

For each matrix, we solve the equations with the sparse direct solver (with MD and ND ordering) and GEPP. We compare the computational time and accuracy of the eigenvalues.

Computational time

Computational time (sec.) for one set of linear simultaneous equations and speedup(PowerPC G5, 2.0GHz)

matrix LAPACK (GEPP) sparse solver (MD)

sparse solver (ND)

BCSSTK12 2.44 (1x) 0.017 (144x) 0.021 (116x)

BCSSTK13 6.12 (1x) 0.36 (17x) 0.43 (14x)

FMO 5.86 (1x) 2.93 (2.0x) 3.51 (1.7x)

The sparse direct solver is two to over one hundred times faster than GEPP, depending on the nonzero structure.

BCSSTK12 BCSSTK13 FMO

Accuracy of the eigenvalues (BCSSTK12)

Example of an interval containing 4 eigenvalues

LAPACK (GEPP) sparse solver (MD) sparse solver (ND)

1.1E- 08 2.4E- 09 4.5E- 092.1E- 10 9.8E- 10 7.6E- 102.8E- 09 1.0E- 08 2.9E- 081.0E- 08 1.3E- 08 3.4E- 08

Relative errors in the eigenvalues for each algorithm (N=64)

Distribution of the eigenvalues and the specified interval

eigenvaluesspecified interval

The errors were of the same order for all three solvers. Also, the growth factor for the sparse solver was O(1).

Accuracy of the eigenvalues (BCSSTK13)


2.4E- 11 4.9E- 11 4.6E- 114.5E- 10 1.6E- 10 2.5E- 111.2E- 10 5.4E- 11 3.7E- 11




The errors were of the same order for all three solvers.


Accuracy of the eigenvalues (FMO)


- 5.0E- 13 - 5.0E- 13 - 5.0E- 13- 1.2E- 10 - 8.5E- 11 - 2.2E- 11- 1.7E- 10 - 3.0E- 10 - 1.4E- 11- 8.4E- 12 - 3.5E- 12 - 3.5E- 12




The errors were of the same order for all three solvers.


Conclusion

Summary of this study We applied a complex symmetric version of the sparse

direct solver to a projection method for generalized eigenvalue problems using numerical integration.

The sparse solver succeeded in solving the linear simultaneous equations stably and accurately, producing eigenvalues that are as accurate as those obtained by GEPP.

Future work Apply our algorithm to larger matrices arising from

quantum chemistry applications. Construct a hybrid method that uses an iterative solver

when the growth factor becomes too large. Parallelize the sparse solver to enable more than N

processors to be used.

on the use of sparse direct solver in a projection method for generalized eigenvalue problems using...

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