A Method for Estimating a Distribution of Eigenvalues using the AS Method Kenta SENZAKI 1) Hiroto TADANO 2) Tetsuya SAKURAI 1) Zhaojun BAI 3) 1) Graduate School of Systems and Information Engineering, University of Tsukuba 2) Graduate School of Informatics, Kyoto University 3) Department of Computer Science, University of California, Davis
Table of Contents Motivation and Background Earlier studies Technical issues What's new in this study Combination of AS and CIRR Flow of AS + CIRR Estimating a distribution of eigenvalue Speed up estimating a distribution Numerical examples Model of 8 DNA base pairs Epidermal Growth Factor Receptor (EGFR) Summary and future works RANMEP2008 Jan. 8, 2008 at NCTS − 1 −
Table of Contents Motivation and Background Earlier studies Technical issues What's new in this study Combination of AS and CIRR Flow of AS + CIRR Estimating a distribution of eigenvalue Speed up estimating a distribution Numerical examples Model of 8 DNA base pairs Epidermal Growth Factor Receptor (EGFR) Summary and future works RANMEP2008 Jan. 8, 2008 at NCTS − 2 −
Earlier studies We consider to solve the large-scale eigenvalue problem A x = λ B x , where A, B ∈ R n × n are symmetric, and B is positive definite. These eigenvalue problems appear in many applications such as Electronic structure calculation Vibration analysis Molecular orbital (MO) computation …etc. Our target problem RANMEP2008 Jan. 8, 2008 at NCTS − 3 −
Earlier studies Problem description The matrix size n = 2K ~ 200K. The number of nonzero elements nnz = 100K ~ 400M. • Sparse matrices, however, relatively dense. About 100 interior eigenpairs ( λ j , x j ) which is related to chemical reaction are needed. • MOs around Highest Occupied MO (HOMO) and Lowest Unoccupied MO (LUMO) are important. Which solver should we use? RANMEP2008 Jan. 8, 2008 at NCTS − 4 −
Earlier studies Dence type: LAPACK, ScaLAPACK Impractical in terms of computational time and memory. Sparse type: Iterative methods • Shift-invert Lanczos method • Jacobi-Davidson method … etc. Projection methods • Algebraic Sub-structuring method • Contour Integral Rayleigh-Ritz method …etc. RANMEP2008 Jan. 8, 2008 at NCTS − 5 −
Technical issues in iterative method Shift-invert Lanczos (SIL) method We need tens of eigenvalues, hence we may decompose the matrix A − σ B as A − σ B = LL T to reuse the matrix L in the iterative loop. However, in the near future, it will become unrealistic to decompose the matrix A − σ B because the matrix appears in MO computation tends to have many nonzero elements and a complicated sparsity pattern. Furthermore, it is difficult to set the appropriate shift value σ without the knowledge of eigenvalue distribution. RANMEP2008 Jan. 8, 2008 at NCTS − 6 −
Technical issues in projection method Algebraic Sub-structuring (AS) method Features AS method uses the structure of matrix which is reordered and partitioned by Nested Dissection (ND) algorithm. AS can compute hundreds of approximate eigenpairs fast. Issues The accuracy of AS tends to be lower than SIL. If there is a large substructure after ND ordering, it takes long time to complete AS. RANMEP2008 Jan. 8, 2008 at NCTS − 7 −
Technical issues in projection method Contour Integral Rayleigh-Ritz (CIRR) method Features CIRR method uses contour integration to construct the Rayleigh- Ritz subspace. CIRR is designed to be better adapted to parallel computational environment. Issue If the appropriate circular domain is put, the solutions are accurate. However, it is difficult to put such circle without the knowledge of eigenvalue distribution. RANMEP2008 Jan. 8, 2008 at NCTS − 8 −
△ △ ○ ○ ◎ ○ △ △ △ △ ○ △ ○ ○ △ △ △ ○ △ ○ ◎ ○ ○ △ Technical issues Feature of solvers for MO computation SIL SIL AS AS CIRR CIRR Complete Complete Yes Yes No No No No decomposition decomposition Accuracy Accuracy (Good shift) (Good shift) Accuracy Accuracy (Bad shift) (Bad shift) Time Time Parallel Parallel RANMEP2008 Jan. 8, 2008 at NCTS − 9 −
What’s new in this work We use the AS method to obtain the rough approximate eigenvalues. From the result of AS, we estimate the eigenvalue distribution and set the circular domains for CIRR method. The accurate solution is computed by the CIRR method. Goal Fully parallelized solver for generalized eigenvale problems Both AS and CIRR can be implemented to be better adapted to parallel environment. RANMEP2008 Jan. 8, 2008 at NCTS − 10 −
Table of Contents Motivation and Background Earlier studies Technical issues What's new in this study Combination of AS and CIRR Flow of AS + CIRR Estimating a distribution of eigenvalue Speed up estimating a distribution Numerical examples Model of 8 DNA base pairs Epidermal Growth Factor Receptor (EGFR) Summary and future works RANMEP2008 Jan. 8, 2008 at NCTS − 11 −
Flow of AS + CIRR A , B , σ AS method approximate eigenvalues θ 1 , θ 2 , …, θ N Estimating a distribution of eigenvalue circular domain CIRR method ( λ j , x j ) RANMEP2008 Jan. 8, 2008 at NCTS − 12 −
Estimating a distribution of eigenvalue Let θ 1 , θ 2 , …, θ N be the approximate eigenvalues calculated by AS, which are included in the interval [ t 1 , t 2 ]. Apply the Gauss function where w is some weight. G ( t ) implies the distribution of eigenvalues, dense or sparse, at the point t . | : eigenvalue | : G ( t ) RANMEP2008 Jan. 8, 2008 at NCTS − 13 −
How to put the circle By the CIRR feature, we set the circular domain in the following two strategies. Large circles are put at the sparse area of eigenvalues, and small circles are put at the dense area. The bound of circle is set at the sparse point of eigenvalue. RANMEP2008 Jan. 8, 2008 at NCTS − 14 −
Speed up for estimating a distribution If the target problem has the many nonzero elements, it tends to take a long time to complete AS. In this case, the quite large substructures tend to appear after partitioning and reordering. We use AS to obtain the rough eigenvalue distribution; we don’t need highly accurate solution from AS. We apply “Cutoff” to the target problem in order to reduce the number of nonzero elements, and to execute the AS method faster. We define Cutoff as follows with small positive value δ . RANMEP2008 Jan. 8, 2008 at NCTS − 15 −
Speed up for estimating a distribution Perturbation of eigenvalue by Cutoff value δ . Let A δ , B δ be perturbation matrices which satisfy A δ = A − A c , B δ = B − B c and The perturbation error is estimated by following equation. † Kravanja, P., Sakurai, T., Sugiura, H., Van Barel, M., A Perturbation Result for Generalized Eigenvalue Problems and its Application to Error Estimation in a Quadrature Method for Computing Zeros of Analytic Functions , J. Comput. Appl. Math. 161(2):339-347, 2003. In practice, the perturbation error by Cutoff tends not to exceed the Cutoff value δ . Example Model of 8 DNA base pairs • 1,980 × 1,980 symmetric matrices. Compute all eigenvalues in each Cutoff value by LAPACK ( DSYGV ) RANMEP2008 Jan. 8, 2008 at NCTS − 16 −
Speed up for estimating a distribution Numerical example of the perturbation exact exact 1e-7 1e-7 1e-6 1e-6 1e-5 1e-5 1e-4 1e-4 1e-3 1e-3 1e-2 1e-2 1e-1 1e-1 − 80 − 70 − 60 − 50 − 40 − 30 − 20 − 10 0 10 exact exact 1e-7 1e-7 1e-6 1e-6 1e-5 1e-5 1e-4 1e-4 1e-3 1e-3 1e-2 1e-2 1e-1 1e-1 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0.0 0.1 0.2 0.3 0.4 0.5 RANMEP2008 Jan. 8, 2008 at NCTS − 17 −
Speed up for estimating a distribution Numerical example of the perturbation The number of eigenvalues which satisfy | λ j − λ j ’ | < δ . δ 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1886 1862 1800 1784 1851 1665 1441 95% 94% 91% 90% 93% 84% 72% The number of eigenvalues which satisfy | λ j − λ j ’ | < δ under HOMO ( λ 1320 ). δ 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1320 1320 1320 1318 1320 1320 1320 100% 100% 100% 99% 100% 100% 100% RANMEP2008 Jan. 8, 2008 at NCTS − 18 −
Flow of AS with Cutoff + CIRR A , B, σ Cutoff A c , B c, σ AS method approximate eigenvalues θ 1 , θ 2 , …, θ N Estimating a distribution of eigenvalue A , B , circular domain CIRR method ( λ j , x j ) RANMEP2008 Jan. 8, 2008 at NCTS − 19 −
Table of Contents Motivation and Background Earlier studies Technical issues What's new in this study Combination of AS and CIRR Flow of AS + CIRR Estimating a distribution of eigenvalue Speed up estimating a distribution Numerical examples Model of 8 DNA base pairs Epidermal Growth Factor Receptor (EGFR) Summary and future works RANMEP2008 Jan. 8, 2008 at NCTS − 20 −
Recommend
More recommend
