An Implementation and Analysis of the Refined Projection Method For - PowerPoint PPT Presentation

An Implementation and Analysis of the Refined Projection Method For (Jacobi-)Davidson Type Methods Lingfei Wu Department of Computer Science College of William and Mary Advisor: Professor Andreas Stathopoulos March 18, 2015 SIAM CSE15 1 / 13

Standard eigenvalue problem Introduction Find k eigenvalues and associated eigenvectors of a large, • The problem sparse symmetric matrix A ∈ ℜ n × n : • Related work Analysis of the Refined Projection Method Ax i = λ i x i , λ 1 ≤ . . . ≤ λ k Efficient Computation of Interior Eigenvalues Numerical Evaluation Our problem: compute a few λ i closest to a shift τ or multiple Conclusion and future work shifts τ 1 , τ 2 , . . . , τ k Our focus: efficiency and accuracy in general subspace (not Krylov subspace) SIAM CSE15 2 / 13

Problem of standard Rayleigh-Ritz Introduction Standard Rayleigh-Ritz extracts Ritz pairs ( θ, u ) where u ∈ V • The problem by imposing Galerkin condition • Related work Analysis of the Refined Projection Method Au − θu ⊥ V Efficient Computation of Interior Eigenvalues Numerical Evaluation Best convergence for extreme eigenvalues, but not for interior Conclusion and future work eigenvalues due to spurious Ritz values Reason for spurious Ritz values: the associated Ritz vector is a combination of nearby eigenvectors → meaningless vector Trouble and effect causing by spurious Ritz values: difficult to select appropriate vectors and cause irregular convergence SIAM CSE15 3 / 13

Harmonic and refined Rayleigh-Ritz Introduction Harmonic Rayleigh-Ritz extracts harmonic Ritz pairs ( � u i ) θ i , � • The problem • Related work by imposing Petrov-Galerkin condition Analysis of the Refined Projection Method ( A − � θ i I ) � u i ⊥ ( A − τI ) V Efficient Computation of Interior Eigenvalues Numerical Evaluation Refined Rayleigh-Ritz replaces Ritz vector with a vector ˆ u ∈ V Conclusion and future work minimize � A ˆ u i − θ i ˆ u i � , i = 1 , 2 , . . . , k. Refined Rayleigh-Ritz achieves monotonic convergence while computational costs are much more expensive Our goal: develop an efficient approach with similar costs to Rayleigh-Ritz SIAM CSE15 4 / 13

Computation and accuracy of refined projection Introduction Approach I: Solve a set of skinny tall SVD problems Analysis of the Refined Projection Method compute ( A − θ i I ) V = Q i R i , i = 1 , 2 , . . . , k. 1. • Analysis solve a set of small SVD problems on each R i . 2. • Comparison Efficient Computation of Merits: numerically stable; Drawbacks: O ( knm 3 ) per restart Interior Eigenvalues Numerical Evaluation Conclusion and future work Approach II: Solve a set of small eigenvalue problems compute λ min ( V T A T AV − 2 θ i V T AV + θ 2 i I ) 1. Merits: O ( km 4 ) per restart; Drawbacks: numerically unstable Approach III: Solve one skinny tall SVD problem Compute a set of the smallest singular triplets of R 1 . 1. Merits: numerically stable; Drawbacks: O ( nm 3 ) per restart and effectiveness of ˆ u i , i = 2 , . . . , k may reduce SIAM CSE15 5 / 13

Computation and accuracy of refined projection Introduction Approach IV: Solve one small eigenvalue problem Analysis of the Refined Projection Method 1. Compute a set of the smallest eigenpairs of • Analysis λ min ( V T A T AV − 2 θ 1 V T AV + θ 2 1 I ) . • Comparison Efficient Computation of Merits: O ( m 4 ) per restart; Drawbacks: numerically unstable Interior Eigenvalues Numerical Evaluation Conclusion and future work Matrix nos3: shift = 99.5 Matrix nos3: shift = 99.95 2 2 10 10 Refined - App I Refined - App I Refined - App II Refined - App II Refined - App III Refined - App III 0 0 Refined - App IV Refined - App IV 10 10 ||A||*1e-8 ||A||*1e-8 Residual norm Residual norm -2 -2 10 10 -4 -4 10 10 -6 -6 10 10 0 1000 2000 3000 4000 5000 6000 0 2000 4000 6000 8000 10000 Number of MatVecs Number of MatVecs (a) Seeking one (b) Seeking a few Approaches III and IV converge faster than approaches I and II SIAM CSE15 6 / 13

An efficient and accurate hybrid method Introduction Hybrid Approach: combining Approach III and IV Analysis of the Refined Projection Method Efficient Computation of Matrix Nos3: shift = 10 Matrix Nos3: shift = 10 5 5 10 10 Interior Eigenvalues Refined - App III Refined - App III • Hybrid method Refined - App IV Refined - DynSwitch Refined - DynSwitch ||A||*1e-15 0 0 ||A||*1e-15 Numerical Evaluation 10 10 Residual norm Residual norm Conclusion and future work -5 -5 10 10 -10 -10 10 10 -15 -15 10 10 0 500 1000 1500 0 500 1000 1500 2000 2500 Number of MatVecs Number of MatVecs (a) Seeking one (b) Seeking a few Advantages of hybrid approach: 1) converges similarly with approach III 2) needs much less computation cost SIAM CSE15 7 / 13

Evaluation: Test matrices Introduction Analysis of the Refined Table 1: Properties of the test matrices Projection Method Efficient Computation of Interior Eigenvalues Matrix pde2961 dw2048 SiNa Kuu Numerical Evaluation • Test matrices order 2961 2048 5743 7102 • Experiment I nnz(A) 14585 10114 198787 340200 • Experiment II κ ( A ) Conclusion and future 9.5E+2 5.3E+3 5.0E+2 1.6E+4 work � A � 2 1.0E+1 1.0E+0 2.6E+1 5.4E+1 Application Model Dielectric Quantum Structural PDE waveguide chemistry problem Two types of problems: � 0 � A T 1) Seek smallest magnitude eigenvalue of B = 0 A 2) Seek interior eigenvalue of real symmetric A SIAM CSE15 8 / 13

Seek smallest magnitude eigenvalue of B Introduction Matrix pde2961: shift = 0.0001 Matrix pde2961: shift = 0.0001 0 0 Analysis of the Refined 10 10 Refined - App III Refined - App III Projection Method Refined - DynSwitch Refined - DynSwitch -2 10 ||A||*1e-13 ||A||*1e-15 Efficient Computation of -4 -5 10 10 Interior Eigenvalues Residual norm Residual norm -6 10 Numerical Evaluation • Test matrices -8 -10 10 10 • Experiment I -10 10 • Experiment II -15 -12 10 10 Conclusion and future 0 1000 2000 3000 4000 5000 6000 0 0.5 1 1.5 2 Number of MatVecs Number of MatVecs 4 work x 10 (a) Seeking one (b) Seeking a few Matrix dw2048: shift = 0.0001 Matrix dw2048: shift = 0.0001 0 0 10 10 Refined - App III Refined - App III Refined - DynSwitch Refined - DynSwitch ||A||*1e-13 ||A||*1e-15 -5 10 Residual norm Residual norm -5 10 -10 10 -10 10 -15 10 0 1000 2000 3000 4000 5000 6000 0 0.5 1 1.5 2 Number of MatVecs Number of MatVecs 4 x 10 (c) Seeking one (d) Seeking a few SIAM CSE15 9 / 13

Seek interior eigenvalue of real symmetric A Introduction Matrix SiNa: shift = 2 Matrix SiNa: shift = 2 0 0 Analysis of the Refined 10 10 Refined - App III Refined - App III Projection Method Refined - DynSwitch Refined - DynSwitch -2 -2 10 10 ||A||*1e-13 ||A||*1e-13 Efficient Computation of -4 -4 10 10 Interior Eigenvalues Residual norm Residual norm -6 -6 10 10 Numerical Evaluation • Test matrices -8 -8 10 10 • Experiment I -10 -10 10 10 • Experiment II -12 -12 10 10 Conclusion and future 0 1000 2000 3000 4000 5000 0 5000 10000 15000 Number of MatVecs Number of MatVecs work (a) Seeking one (b) Seeking a few Matrix Kuu: shift = 1 Matrix Kuu: shift = 1 0 0 10 10 Refined - App III Refined - App III Refined - DynSwitch Refined - DynSwitch -2 -2 10 10 ||A||*1e-13 ||A||*1e-13 -4 -4 10 10 Residual norm Residual norm -6 -6 10 10 -8 -8 10 10 -10 -10 10 10 -12 -12 10 10 0 2000 4000 6000 8000 10000 0 0.5 1 1.5 2 2.5 3 3.5 Number of MatVecs Number of MatVecs 4 x 10 (c) Seeking one (d) Seeking a few SIAM CSE15 10 / 13

Comparing Matvecs and Time Introduction Analysis of the Refined Table 2: Seeking one Projection Method Efficient Computation of Mat: pde2961 dw2048 SiNa Kuu Interior Eigenvalues App MV Sec MV Sec MV Sec MV Sec Numerical Evaluation • Test matrices 7014 62 7536 56 4833 49 12774 292 RR • Experiment I 6054 123 5215 85 4458 102 9215 412 • Experiment II III Conclusion and future 5892 78 5023 52 4771 65 9468 280 Hyd work Table 3: Seeking a few Mat: pde2961 dw2048 SiNa Kuu App MV Sec MV Sec MV Sec MV Sec 17572 180 17602 135 12668 137 45325 577 RR 17313 362 14399 227 15424 367 34451 888 III 17862 249 14069 161 15433 228 33149 556 Hyd SIAM CSE15 11 / 13

Conclusion and future work Introduction Refined and Harmonic Rayleigh-Ritz methods are useful tools Analysis of the Refined to tackle interior eigenvalue problems. Projection Method Efficient Computation of • Present a promising novel efficient approach for computing Interior Eigenvalues refined Ritz vectors Numerical Evaluation Conclusion and future • A robust metric to monitor the error of the desired Ritz vector work • • Conclusion Study similar issues in the harmonic projection method • • Study an efficient approach for refined harmonic projection method SIAM CSE15 12 / 13

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