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Hierarchical (H-)Matrices                         PINVIT                                       Numerical Results          ...
Hierarchical (H-)Matrices                           PINVIT                                               Numerical Results...
Hierarchical (H-)Matrices                    PINVIT                                       Numerical Results Outline       ...
Hierarchical (H-)Matrices                                 PINVIT                                          Numerical Result...
Hierarchical (H-)Matrices                                                                                                 ...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Preconditione...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Preconditio...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Preconditio...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Preconditi...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Preconditi...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Preconditio...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Preconditio...
Hierarchical (H-)Matrices                      PINVIT                                         Numerical Results Preconditi...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Why is it ...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Variants o...
Hierarchical (H-)Matrices                 PINVIT                                       Numerical Results Algorithm       H...
Hierarchical (H-)Matrices                   PINVIT                                       Numerical Results Complexity     ...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Folded Spectr...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Folded Spec...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Folded Spec...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results Folded Spec...
Hierarchical (H-)Matrices                          PINVIT                                       Numerical Results Folded S...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Folded Spectr...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Folded Spectr...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Folded Spectr...
Hierarchical (H-)Matrices                     PINVIT                                       Numerical Results Folded Spectr...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                       PINVIT                                         Numerical Results Goal: Eig...
Hierarchical (H-)Matrices                  PINVIT                                       Numerical Results Numerical Result...
Hierarchical (H-)Matrices                 PINVIT                                       Numerical Results Hlib       Hlib  ...
Hierarchical (H-)Matrices                       PINVIT                                       Numerical Results 2D Laplace,...
Hierarchical (H-)Matrices                                    PINVIT                                        Numerical Resul...
Hierarchical (H-)Matrices                                    PINVIT                                        Numerical Resul...
Hierarchical (H-)Matrices                   PINVIT                                       Numerical Results Complexity     ...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Complexity...
Hierarchical (H-)Matrices                    PINVIT                                       Numerical Results Complexity    ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                          PINVIT                                       Numerical Results 2D Lapla...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                                    PINVIT                                       Numerical Result...
Hierarchical (H-)Matrices                                    PINVIT                                       Numerical Result...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results 2D Laplace, ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results BEM example,...
Hierarchical (H-)Matrices                                 PINVIT                                         Numerical Results...
Hierarchical (H-)Matrices                                 PINVIT                                         Numerical Results...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results Conclusions ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results Conclusions ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results Conclusions ...
Hierarchical (H-)Matrices                      PINVIT                                       Numerical Results Conclusions ...
Hierarchical (H-)Matrices                        PINVIT                                       Numerical Results Conclusion...
Appendix Adaptive H-Inversion [Grasedyck 2001]      Adaptive H-Inversion                                    c      Input: ...
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Preconditioned Inverse Iteration for Hierarchical Matrices

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Preconditioned Inverse Iteration for Hierarchical Matrices

  1. 1. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration for H-Matrices Peter Benner and Thomas Mach Mathematics in Industry and Technology Department of Mathematics TU Chemnitz 23rd Chemnitz FEM Symposium 2010 1/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  2. 2. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration for H-Matrices Peter Benner and Thomas Mach Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg 23rd Chemnitz FEM Symposium 2010 MAX−PLANCK−INSTITUT FÜR DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG 1/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  3. 3. Hierarchical (H-)Matrices PINVIT Numerical Results Outline 1 Hierarchical (H-)Matrices 2 PINVIT 3 Numerical Results 2/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  4. 4. Hierarchical (H-)Matrices PINVIT Numerical Results H-Matrices [Hackbusch 1998] Some dense matrices, e.g. BEM or FEM, can be approximated by H-matrices in a data-sparse manner. hierarchical tree TI block H-tree TI × I I = {1, 2, 3, 4, 5, 6, 7, 8} 12345678 12345678 12345678 12345678 1 1 1 1 2 2 2 2 {1, 2, 3, 4} {5, 6, 7, 8} 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 {1, 2} {3, 4} {5, 6} {7, 8} 7 7 7 7 8 8 8 8 {1}{2}{3}{4}{5}{6}{7}{8} dense matrices, rank-k-matrices rank-k-matrix: Ms×t = AB T , A ∈ Rn×k , B ∈ Rm×k (k n, m) 3/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  5. 5. Hierarchical (H-)Matrices PINVIT Numerical Results H-Matrices [Hackbusch 1998] Hierarchical matrices H(TI × I , k) = M ∈ RI × I rank (Ma×b ) ≤ k ∀a × b admissible 3 3 22 7 10 14 8 11 9 8 5 3 7 19 10 11 8 3 10 10 31 11 11 9 16 12 8 15 8 12 11 8 19 10 11 8 14 8 11 11 11 8 10 11 31 11 31 11 9 3 16 11 7 3 8 6 15 5 6 7 13 5 8 8 15 9 15 12 13 adaptive rank k(ε) 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 9 3 25 10 13 8 7 11 10 19 11 6 5 8 11 9 11 16 3 storage NSt,H (T , k) = O(n log n k(ε)) 8 11 11 31 11 15 6 9 15 11 11 8 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 10 7 13 6 15 10 3 6 25 10 6 13 8 12 5 6 10 14 6 10 19 10 10 31 10 16 9 11 8 8 15 11 12 20 13 8 complexity of approximate arithmetic 7 8 11 15 9 10 10 5 8 11 8 11 10 51 7 9 3 7 3 9 7 6 15 10 25 11 8 13 13 7 13 11 5 6 10 16 10 9 7 3 3 11 25 10 10 19 9 13 8 8 11 11 11 8 3 7 8 15 8 39 10 10 10 6 5 10 3 9 15 3 10 3 25 7 10 3 6 3 15 10 13 6 11 13 7 10 7 22 7 10 11 9 8 5 12 O(n log n k(ε)) 10 7 19 10 11 8 MH v 3 7 12 10 6 3 10 10 31 11 9 16 12 8 15 8 20 13 8 8 11 8 34 10 13 10 6 5 10 9 15 15 11 10 25 7 11 13 6 11 13 13 11 8 13 7 O(n log n k(ε)2 ) 7 11 10 16 61 11 23 +H , −H 9 10 11 13 9 6 13 20 9 9 7 11 8 3 7 9 8 8 15 5 6 12 9 39 10 3 10 15 10 12 13 11 8 8 8 15 9 10 15 9 7 11 11 3 3 61 10 ∗H , HLU(·), (·)−1 O(n (log n)2 k(ε)2 ) 5 8 7 7 10 9 11 19 13 9 6 13 20 9 9 7 9 8 13 8 8 11 5 6 10 15 10 9 34 10 13 12 12 15 10 H 9 9 7 11 11 23 10 8 11 7 13 51 4/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  6. 6. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λi in O(n (log n)α k β )? 5/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  7. 7. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λn in O(n (log n)α k β )? 5/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  8. 8. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Definition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient. 6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  9. 9. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Definition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient. Minimize the Rayleigh quotient by a gradient method: 2 xi+1 := xi − α µ(xi ), µ(x) = (Mx − xµ(x)) , xT x 6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  10. 10. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Residual r (x) = Mx − xµ(x). Minimize the Rayleigh quotient by a gradient method: 2 xi+1 := xi − α µ(xi ), µ(x) = (Mx − xµ(x)) , xT x 6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  11. 11. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Definition The function x T Mx µ(x) = µ(x, M) = xT x is called the Rayleigh quotient. Minimize the Rayleigh quotient by a gradient method: 2 xi+1 := xi − α µ(xi ), µ(x) = (Mx − xµ(x)) , xT x + preconditioning ⇒ update equation: xi+1 := xi − B −1 (Mxi − xi µ(xi )) . 6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  12. 12. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] Preconditioned residual B −1 r (x) = B −1 (Mx − xµ(x)). + preconditioning ⇒ update equation: xi+1 := xi − B −1 (Mxi − xi µ(xi )) . 6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  13. 13. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr 2009] xi+1 := xi − B −1 (Mxi − xi µ(xi )) If M symmetric positive definite and B −1 approximates the inverse of M, so that I − B −1 M M ≤ c < 1, 7/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  14. 14. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr 2009] xi+1 := xi − B −1 (Mxi − xi µ(xi )) If M symmetric positive definite and B −1 approximates the inverse of M, so that I − B −1 M M ≤ c < 1, then Preconditioned INVerse ITeration (PINVIT) converges. 7/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  15. 15. Hierarchical (H-)Matrices PINVIT Numerical Results Preconditioned Inverse Iteration [Knyazev, Neymeyr, et al.] The residual ri = Mxi − xi µ(xi ) converges to 0, so that ri 2 < is a useful termination criterion. 8/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  16. 16. Hierarchical (H-)Matrices PINVIT Numerical Results Why is it called Preconditioned Inverse Iteration? Inverse Iteration: xi+1 = µ(xi )A−1 xi , xi+1 = xi − A−1 Axi +µ(xi )A−1 xi , =0 −1 xi+1 = xi − A (Axi + µ(xi )xi ) . Replace A−1 with the inexact solution by the preconditioner B: xi+1 = xi − B −1 (Axi + µ(xi )xi ) . 9/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  17. 17. Hierarchical (H-)Matrices PINVIT Numerical Results Variants of PINVIT [Neymeyr 2001: A Hierarchy of Preconditioned Eigensolvers for Elliptic Diff. Operators] Classification by Neymeyr: PINVIT(1): xi+1 := xi − B −1 ri . PINVIT(2): xi+1 := arg minv ∈span{xi ,B −1 ri } µ(v ). PINVIT(3): xi+1 := arg minv ∈span{xi−1 ,xi ,B −1 ri } µ(v ). PINVIT(n): Analogously. PINVIT(·,d): Replacing x by a rectangular full rank matrix X ∈ Rn×d one gets the subspace version of PINVIT(·). 10/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  18. 18. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm H-PINVIT(1,d) Input: M ∈ Rn×n , X0 ∈ Rn×d (e.g. randomly chosen) Output: Xp ∈ Rn×d , µ ∈ Rd×d , with MXp − Xp µ ≤ B −1 = (M)−1H Orthogonalize X0 T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Xi−1 − B −1 R Orthogonalize Xi R := MXi − Xi µ, µ = XiT MXi end 11/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  19. 19. Hierarchical (H-)Matrices PINVIT Numerical Results Complexity The number of iterations is independent of matrix size n. Once: adaptive H-matrix inversion: O(n (log n)2 k (c)2 ). Per iteration: H-matrix-vector products with M and B −1 : O(n (log n) k (c)) and O(n (log n) k (c)2 ), and some dense arithmetic handling vectors of length n. The complexity of the algorithm is determined by the H-matrix inversion: ⇒ O(n (log n)2 k (c)2 ). 12/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  20. 20. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λn in O(n (log n)α k β )? 13/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  21. 21. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λn in O(n (log n)α k β )? 13/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  22. 22. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λi in O(n (log n)α k β )? 14/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  23. 23. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method How to find λi in O(n (log n)α k β )? If i = n − d with d < O(log n), use subspace version PINVIT(·,d ). ... 0λn λn−1 λn−2 λ1 15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  24. 24. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method How to find λi in O(n (log n)α k β )? If i = n − d with d log n? ... ... 0λn λi+1 λi λi−1 λ1 15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  25. 25. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method How to find λi in O(n (log n)α k β )? If i = n − d with d log n, shift with σ near λi . ... ... 0λn λi+1 λi λi−1 λ1 σ 15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  26. 26. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method How to find λi in O(n (log n)α k β )? If i = n − d with d log n, shift with σ near λi . ... ... 0λn λi+1 λi λi−1 λ1 σ But (M − σI) is not positive definite! 15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  27. 27. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method Folded Spectrum Method [Wang, Zunger 1994] Mσ = (M − σI)2 Mσ is s.p.d., if M is s.p.d. and σ = λi . Assume all eigenvalues of Mσ are simple. Mv = λv ⇔ Mσ v = (M − σI)2 v = M 2 v − 2σMv + σ 2 v = λ2 v − 2σλv + σ 2 v = (λ − σ)2 v 16/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  28. 28. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ . 3 Use PINVIT to find the smallest eigenpair (µσ , v ) of Mσ . 4 Compute µ = v T Mv /v T v . (µ, v ) is the nearest eigenpair to σ. 17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  29. 29. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ . 3 Use PINVIT to find the smallest eigenpair (µσ , v ) of Mσ . 4 Compute µ = v T Mv /v T v . (µ, v ) is the nearest eigenpair to σ. H-arithmetic enables us to shift, square and invert M resp. Mσ in linear-polylogarithmic complexity. 17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  30. 30. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ . 3 Use PINVIT to find the smallest eigenpair (µσ , v ) of Mσ . 4 Compute µ = v T Mv /v T v . (µ, v ) is the nearest eigenpair to σ. H-arithmetic enables us to shift, square and invert M resp. Mσ in linear-polylogarithmic complexity. 17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  31. 31. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ . 3 Use PINVIT to find the smallest eigenpair (µσ , v ) of Mσ . 4 Compute µ = v T Mv /v T v . (µ, v ) is the nearest eigenpair to σ. H-arithmetic enables us to shift, square and invert M resp. Mσ in linear-polylogarithmic complexity. 17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  32. 32. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λi in O(n (log n)α k β )? 18/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  33. 33. Hierarchical (H-)Matrices PINVIT Numerical Results Goal: Eigenvalues of symmetric H-Matrices M = M T ∈ H(T , k), Λ(M) = {λ1, λ2, . . . , λn }, λ1 ≥ λ2 ≥ · · · ≥ λn > 0 ⇓ How to find λi in O(n (log n)α k β )? 18/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  34. 34. Hierarchical (H-)Matrices PINVIT Numerical Results Numerical Results Numerical Results 19/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  35. 35. Hierarchical (H-)Matrices PINVIT Numerical Results Hlib Hlib [B¨rm, Grasedyck, et al.] o We use the Hlib1.3 (www.hlib.org) for the H-arithmetic operations and some examples out of the library for testing the eigenvalue algorithm. 20/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  36. 36. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 2D Laplace over [−1, 1] × [−1, 1] ti N(ni ) Name ni error ti ti−1 N(ni−1 ) FEM8 64 4.3466E-08 0.006 FEM16 256 1.2389E-07 0.014 2.33 80.00 FEM32 1 024 4.4308E-07 0.036 2.57 21.67 FEM64 4 096 1.8003E-06 0.228 6.33 6.72 FEM128 16 384 6.7069E-06 0.992 4.35 4.67 FEM256 65 536 1.6631E-05 2.397 2.42 4.57 FEM512 262 144 4.4015E-05 8.120 3.39 5.79 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only PINVIT (without H-inversion) 21/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  37. 37. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 103 PINVIT(1,4) 102 PINVIT(3,4) O(N(ni )) CPU time in s 101 H-inversion O(N(ni ) log ni ) 100 10−1 10−2 10−3 8 16 32 64 8 6 2 M 12 25 51 M M M FE M M M FE FE FE FE FE FE 22/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  38. 38. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 103 PINVIT(1,4) 102 PINVIT(3,4) O(N(ni )) CPU time in s 101 H-inversion O(N(ni ) log ni ) 100 eigs 10−1 10−2 10−3 8 16 32 64 8 6 2 M 12 25 51 M M M FE M M M FE FE FE FE FE FE 22/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  39. 39. Hierarchical (H-)Matrices PINVIT Numerical Results Complexity The number of iterations is independent of matrix size n. Once: adaptive H-matrix inversion: O(n (log n)2 k (c)2 ). Per iteration: H-matrix-vector products with M and B −1 : O(n (log n) k (c)) and O(n (log n) k (c)2 ), and some dense arithmetic handling vectors of length n. The complexity of the algorithm is determined by the H-matrix inversion: ⇒ O(n (log n)2 k (c)2 ). 23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  40. 40. Hierarchical (H-)Matrices PINVIT Numerical Results Complexity Competitive to MATLAB eigs. Once: adaptive H-matrix inversion: O(n (log n)2 k (c)2 ). Per iteration: H-matrix-vector products with M and B −1 : O(n (log n) k (c)) and O(n (log n) k (c)2 ), and some dense arithmetic handling vectors of length n. 23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  41. 41. Hierarchical (H-)Matrices PINVIT Numerical Results Complexity Once: adaptive H-matrix inversion: O(n (log n)2 k (c)2 ). Per iteration: H-matrix-vector products with M and B −1 : O(n (log n) k (c)) and O(n (log n) k (c)2 ), and some dense arithmetic handling vectors of length n. Expensive. 23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  42. 42. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, vn , FEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  43. 43. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, vn−1 , FEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  44. 44. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, vn−2 , FEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  45. 45. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, vn−3 , FEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  46. 46. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, inner eigenvalues 2D Laplace over [−1, 1] × [−1, 1] ti N(ni ) Name ni error ti ti−1 N(ni−1 ) FEM8 64 2.2560E-08 <0.01 FEM16 256 3.4681E-06 0.01 — 80.00 FEM32 1 024 1.1875E-04 0.04 4.00 21.67 FEM64 4 096 7.2115E-04 0.20 5.00 6.72 FEM128 16 384 2.0747E-05 0.93 4.65 4.67 FEM256 65 536 9.2370E-05 3.59 3.86 4.57 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only PINVIT (without H-inversion) 25/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  47. 47. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, smallest eigenvalues 2D Laplace over [−1, 1] × [−1, 1] [0, 1] × [0, 1] ti N(ni ) Name n error t ti−1 N(ni−1 ) LFEM8 48 6.0999E-07 0.023 LFEM16 192 5.7426E-07 0.037 1.57 10.86 LFEM32 768 1.4644E-06 0.180 4.91 10.11 LFEM64 3 072 4.4753E-06 0.497 2.76 108.78 LFEM128 12 288 1.8770E-04 0.770 1.55 9.49 LFEM256 49 152 — 2.133 2.77 4.59 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only PINVIT (without H-inversion) 26/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  48. 48. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, smallest eigenvalues 103 PINVIT(1,4) 102 PINVIT(3,4) O(N(ni )) CPU time in s 101 H-inversion O(N(ni ) log ni ) 100 10−1 10−2 10−3 8 16 32 64 8 6 M 12 25 EM EM EM E EM EM LF LF LF LF LF LF 27/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  49. 49. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, smallest eigenvalues 103 PINVIT(1,4) 102 PINVIT(3,4) O(N(ni )) CPU time in s 101 H-inversion O(N(ni ) log ni ) 100 eigs 10−1 10−2 10−3 8 16 32 64 8 6 M 12 25 EM EM EM E EM EM LF LF LF LF LF LF 27/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  50. 50. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, vn , LFEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  51. 51. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, vn−1 , LFEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  52. 52. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, vn−2 , LFEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  53. 53. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, L-shape, vn−3 , LFEM64 ·10−2 6 4 ·10−2 5 2 0 0 −2 1 −5 −4 −1 0 −0.5 0 0.5 −6 1 −1 28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  54. 54. Hierarchical (H-)Matrices PINVIT Numerical Results BEM example, smallest eigenvalues dense matrix, but approximable by an H-matrix ⇒ MATLAB eigs is expensive ti N(ni ) Name ni error ti ti−1 N(ni−1 ) BEM8 258 2.0454E-05 0.01 BEM16 1 026 3.5696E-05 0.03 3.00 26.81 BEM32 4 098 7.1833E-05 0.14 4.67 14.42 BEM64 16 386 — 0.46 3.29 16.51 BEM128 65 538 — 2.00 4.36 26.35 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only PINVIT (without H-inversion) 29/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  55. 55. Hierarchical (H-)Matrices PINVIT Numerical Results BEM example, smallest eigenvalues PINVIT(1,4) 104 PINVIT(3,4) O(N(ni )) CPU time in s 102 H-inversion O(N(ni ) log ni ) 100 10−2 8 16 32 64 8 M 12 M M M BE M BE BE BE BE 30/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  56. 56. Hierarchical (H-)Matrices PINVIT Numerical Results BEM example, smallest eigenvalues PINVIT(1,4) 104 PINVIT(3,4) O(N(ni )) CPU time in s 102 H-inversion O(N(ni ) log ni ) eigs 100 10−2 8 16 32 64 8 M 12 M M M BE M BE BE BE BE 30/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  57. 57. Hierarchical (H-)Matrices PINVIT Numerical Results Conclusions Once we have inverted M ∈ H(T , k) finding the eigenvalues by PINVIT is cheap and storage efficient. For dense, data-sparse matrices, H-PINVIT can solve problems where eigs (implicitly, restarted Arnoldi/Lanczos iteration) is not applicable. The folded spectrum method enables us to compute inner eigenvalues, too. We will investigate the use of H-Cholesky factorizations instead of the H-inversion. 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  58. 58. Hierarchical (H-)Matrices PINVIT Numerical Results Conclusions Once we have inverted M ∈ H(T , k) finding the eigenvalues by PINVIT is cheap and storage efficient. For dense, data-sparse matrices, H-PINVIT can solve problems where eigs (implicitly, restarted Arnoldi/Lanczos iteration) is not applicable. The folded spectrum method enables us to compute inner eigenvalues, too. We will investigate the use of H-Cholesky factorizations instead of the H-inversion. 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  59. 59. Hierarchical (H-)Matrices PINVIT Numerical Results Conclusions Once we have inverted M ∈ H(T , k) finding the eigenvalues by PINVIT is cheap and storage efficient. For dense, data-sparse matrices, H-PINVIT can solve problems where eigs (implicitly, restarted Arnoldi/Lanczos iteration) is not applicable. The folded spectrum method enables us to compute inner eigenvalues, too. We will investigate the use of H-Cholesky factorizations instead of the H-inversion. 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  60. 60. Hierarchical (H-)Matrices PINVIT Numerical Results Conclusions Once we have inverted M ∈ H(T , k) finding the eigenvalues by PINVIT is cheap and storage efficient. For dense, data-sparse matrices, H-PINVIT can solve problems where eigs (implicitly, restarted Arnoldi/Lanczos iteration) is not applicable. The folded spectrum method enables us to compute inner eigenvalues, too. We will investigate the use of H-Cholesky factorizations instead of the H-inversion. 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  61. 61. Hierarchical (H-)Matrices PINVIT Numerical Results Conclusions Once we have inverted M ∈ H(T , k) finding the eigenvalues by PINVIT is cheap and storage efficient. For dense, data-sparse matrices, H-PINVIT can solve problems where eigs (implicitly, restarted Arnoldi/Lanczos iteration) is not applicable. The folded spectrum method enables us to compute inner eigenvalues, too. We will investigate the use of H-Cholesky factorizations instead of the H-inversion. Thank you for your attention. 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
  62. 62. Appendix Adaptive H-Inversion [Grasedyck 2001] Adaptive H-Inversion c Input: M ∈ H (TI ×I ), c = √ ˜ ∈R M 2 −1 −1 Output: MH , with I − MH M 2 < c ˜ −1 H Compute MH with local = c /(Csp (M) M ˜ 2 ) 0 −1 H δM := I − MH M 2 /˜c i > 1 do while δM −1 i Compute MH with local = local /δM i −1 H δM := I − MH M 2 /˜c end −1 c −1 I − MH M 2 <c = ˜ ⇔ I − MH M M ≤c M 2 31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices

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