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# Preconditioned Inverse Iteration for Hierarchical Matrices

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### Preconditioned Inverse Iteration for Hierarchical Matrices

1. 1. 82nd GAMM Annual Scientiﬁc Conference Graz, April 19, 2011 Preconditioned Inverse Iteration for Hierarchical Matrices Peter Benner and Thomas Mach Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURGMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 1/28
2. 2. Hierarchical (H-)Matrices PINVIT Numerical Results Outline 1 Hierarchical (H-)Matrices 2 PINVIT 3 Numerical ResultsMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 2/28
3. 3. 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)Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 3/28
4. 4. 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 −1 ∗H , HLU(·), (·)H 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 7 11 12 11 23 10 15 8 9 11 9 7 10 13 51Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 4/28
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 −1 ∗H , HLU(·), (·)H 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 7 11 12 11 23 10 15 8 9 11 9 7 10 13 51 T B1 T B2 T B1 A1 + A2 = A1 A2 T B2Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 4/28
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 )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 5/28
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 )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 5/28
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.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 6/28
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) = T (Mx − xµ(x)) , x xMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 6/28
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) = T (Mx − xµ(x)) , x xMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 6/28
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) = T (Mx − xµ(x)) , x x + preconditioning ⇒ update equation: xi+1 := xi − B −1 (Mxi − xi µ(xi )) .Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 6/28
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 )) .Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 6/28
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 ∈ Rn×n 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 and the number of iterations is independent of n.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 7/28
14. 14. 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.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 8/28
15. 15. Hierarchical (H-)Matrices PINVIT Numerical Results Variants of PINVIT [Neymeyr 2001: A Hierarchy of Precond. Eigens. for Ellipt. Diff. Op.] 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(·).Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 9/28
16. 16. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm and Complexity The number of iterations is independent of matrix size n. H-PINVIT(1,d) Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R R := MXi − Xi µ, µ = XiT MXi endMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 10/28
17. 17. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm and Complexity The number of iterations is independent of matrix size n. H-PINVIT(1,d) Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H O(n (log n)2 k (c)2 ) T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R O(n (log n) k (c)2 ) R := MXi − Xi µ, µ = XiT MXi O(n (log n) k (c)) end The complexity of the algorithm is determined by the H-matrix inversion/Cholesky decomposition: ⇒ O(n (log n)2 k (c)2 ).Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 10/28
18. 18. 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 )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 11/28
19. 19. 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 )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 11/28
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 )? How to ﬁnd λi in O(n (log n)α k β )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 12/28
21. 21. 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 λ1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 13/28
22. 22. 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 λ1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 13/28
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 log n, shift with σ near λi . ... ... 0λn λi+1 λi λi−1 λ1 σMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 13/28
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, shift with σ near λi . ... ... 0λn λi+1 λi λi−1 λ1 σ But (M − σI) is not positive deﬁnite.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 13/28
25. 25. 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 vMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 14/28
26. 26. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ or LLT = 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.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 15/28
27. 27. Hierarchical (H-)Matrices PINVIT Numerical Results Folded Spectrum Method 1 Choose σ. 2 Compute 2 a) Mσ = (M − σI) and −1 b) Mσ or LLT = 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.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 15/28
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 or LLT = 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. If M is sparse, then shifting, squaring and inverting is prohibitive.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 15/28
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 or LLT = 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. If M is sparse, then shifting, squaring and inverting is prohibitive.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 15/28
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 or LLT = 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. If M is sparse, then shifting, squaring and inverting is prohibitive.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 15/28
31. 31. 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 )? How to ﬁnd λi in O(n (log n)α k β )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 16/28
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 λn in O(n (log n)α k β )? How to ﬁnd λi in O(n (log n)α k β )?Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 16/28
33. 33. Hierarchical (H-)Matrices PINVIT Numerical Results Numerical Results Numerical ResultsMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 17/28
34. 34. Hierarchical (H-)Matrices PINVIT Numerical Results Hlib Hlib ¨ [Borm, Grasedyck, et al.] 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.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 18/28
35. 35. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 2D Laplace over [−1, 1] × [−1, 1], precond.: H-inversion ti N(ni ) Name ni error ti ti−1 N(ni−1 ) FEM8 64 5.6146E-010 0.01 FEM16 256 4.5918E-010 0.02 2.00 106.67 FEM32 1 024 3.7550E-010 0.12 6.17 27.08 FEM64 4 096 3.8009E-010 0.82 6.68 8.06 FEM128 16 384 4.4099E-010 5.84 7.09 5.44 FEM256 65 536 3.9651E-010 34.47 5.91 5.22 FEM512 262 144 3.7877E-010 194.00 5.63 6.51 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only H-PINVIT (without H-inversion)Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 19/28
36. 36. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 2D Laplace over [−1, 1] × [−1, 1], precond.: H-inversion ti N(ni ) Name ni error ti ti−1 N(ni−1 ) FEM8 64 5.6146E-010 0.01 FEM16 256 4.5918E-010 0.02 2.00 106.67 FEM32 1 024 3.7550E-010 0.12 6.17   27.08 FEM64 4 096 3.8009E-010 ˆ λ1 − λ1 0.82 6.68 8.06 FEM128 16 384 4.4099E-010 λ 7.09  ˆ 5.842 − λ2  5.44 FEM256 65 536 3.9651E-010  ˆ λ3 − λ3  34.47 5.91 5.22 FEM512 262 144 3.7877E-010 ˆ 194.004 − λ4 2 6.51 λ 5.63 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only H-PINVIT (without H-inversion)Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 19/28
37. 37. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 104 H-PINVIT(1,4) H-inversion 103 H-PINVIT(3,4) O(N(ni ) log ni ) O(N(ni )) 102 CPU time in s 101 100 10−1 10−2 10−3 16 32 64 128 EM256 EM512 FEM FEM FEM FEM F FMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 20/28
38. 38. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 104 H-PINVIT(1,4) H-inversion 103 H-PINVIT(3,4) O(N(ni ) log ni ) O(N(ni )) MATLAB eigs 102 CPU time in s 101 100 10−1 10−2 10−3 16 32 64 128 EM256 EM512 FEM FEM FEM FEM F FMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 20/28
39. 39. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm and Complexity H-PINVIT(1,d) Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H O(n (log n)2 k (c)2 ) T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R O(n (log n) k (c)2 ) R := MXi − Xi µ, µ = XiT MXi O(n (log n) k (c)) endMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 21/28
40. 40. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm and Complexity H-PINVIT(1,d) Input: MCompetitive to Rn×d (X0 X0 = I , e.g. randomly chosen) ∈ Rn×n , X0 ∈ MATLAB eigs. T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H O(n (log n)2 k (c)2 ) T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R O(n (log n) k (c)2 ) R := MXi − Xi µ, µ = XiT MXi O(n (log n) k (c)) endMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 21/28
41. 41. Hierarchical (H-)Matrices PINVIT Numerical Results Algorithm and Complexity H-PINVIT(1,d) Input: M ∈ Rn×n , X0 ∈ Rn×d (X0 X0 = I , e.g. randomly chosen) T Output: Xp ∈ R n×d , µ ∈ Rd×d , with MX − X µ ≤ p p B −1 = (M)−1 or B −1 = L−T L−1 H H H O(n (log n)2 k (c)2 ) T R := MX0 − X0 µ, µ = X0 MX0 for (i := 1; R F > ; i + +) do Xi := Orthogonalize Xi−1 − B −1 R O(n (log n) k (c)2 ) R := MXi − Xi µ, µ = XiT MXi O(n (log n) k (c)) end Expensive.Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 21/28
42. 42. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 2D Laplace over [−1, 1] × [−1, 1], precond.: H-Cholesky decomp. ti N(ni ) Name ni error ti ti−1 N(ni−1 ) FEM8 64 4.6920E-010 0.01 FEM16 256 4.7963E-010 0.02 2.00 106.67 FEM32 1 024 3.4696E-010 0.08 4.00 27.08 FEM64 4 096 4.6414E-010 0.48 6.00 8.06 FEM128 16 384 3.3206E-010 3.20 6.67 5.44 FEM256 65 536 3.8468E-010 13.90 4.34 5.22 FEM512 262 144 3.1353E-010 62.40 4.49 6.51 d = 4, c = 0.2, eig = 10−4 , N(ni ) = ni (log2 ni ) Csp Cid Time t only H-PINVIT (without H-Cholesky decomposition)Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 22/28
43. 43. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 104 H-PINVIT(1,4) H-Chol. decomp. 103 H-PINVIT(3,4) O(N(ni ) log ni ) O(N(ni )) 102 CPU time in s 101 100 10−1 10−2 10−3 16 32 64 128 EM256 EM512 FEM FEM FEM FEM F FMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 23/28
44. 44. Hierarchical (H-)Matrices PINVIT Numerical Results 2D Laplace, smallest eigenvalues 104 H-PINVIT(1,4) H-Chol. decomp. 103 H-PINVIT(3,4) O(N(ni ) log ni ) O(N(ni )) MATLAB eigs 102 CPU time in s 101 100 10−1 10−2 10−3 16 32 64 128 EM256 EM512 FEM FEM FEM FEM F FMax Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 23/28
45. 45. 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 −1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 24/28
46. 46. 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 −1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 24/28
47. 47. 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 −1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 24/28
48. 48. 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 −1Max Planck Institute Magdeburg Thomas Mach, Preconditioned Inverse Iteration for Hierarchical Matrices 24/28