Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Upcoming SlideShare
Loading in …5
×

# Preconditioned Inverse Iteration for Hierarchical Matrices

902 views

Published on

Published in: Technology
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

### 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