on the use of sparse direct solver in a projection method for generalized eigenvalue problems using...
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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)
LAPACK (GEPP) sparse solver (MD) sparse solver (ND)
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
Example of an interval containing 3 eigenvalues
Distribution of the eigenvalues and the specified interval
eigenvaluesspecified interval
The errors were of the same order for all three solvers.
Relative errors in the eigenvalues for each algorithm (N=64)
Accuracy of the eigenvalues (FMO)
LAPACK (GEPP) sparse solver (MD) sparse solver (ND)
- 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
Example of an interval containing 4 eigenvalues
Distribution of the eigenvalues and the specified interval
eigenvaluesspecified interval
The errors were of the same order for all three solvers.
Relative errors in the eigenvalues for each algorithm (N=64)
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.