Hierarchical Matrices: Concept, Application and Eigenvalues

November 5, 2010 Hierarchical Matrices: Concept, Applications and Eigenvalues Thomas Mach Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURGHIERARCHICAL STRUCTURESMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 1/28

Concept Application Eigenvalues Householder Notation Scalar λ∈RMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 2/28

Concept Application Eigenvalues Householder Notation Vector   x1  x2    x =  x 3  ∈ Rn   . .. xnMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 3/28

Concept Application Eigenvalues Householder Notation Matrix   a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n    A = a31 a32 a33 . . . a3n  ∈ Rn×n   . . . .. .    .. . . . . . .  . an1 an2 an3 . . . annMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 4/28

Concept Application Eigenvalues Dense Matrix Format Store n vectors (Fortran):         a11 a12 a13 a1n a21  a22  a23  a2n          a31  a32  a33        · · · a3n     .   .   .   .   .   .   .  . . .  .  . an1 an2 an3 ann n2 entries in the storage Ax costs O(n2 ) flops AB costs O(nδ ) flops, δ ≥ 2.376 usually δ = 3 A−1 costs O(n3 ) flopsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 5/28

Concept Application Eigenvalues Dense Matrix Format Landau Symbol Store n vectors (Fortran):    that    more than cn2 flops There is a constant c, so  Ax costs not a11 a12 a13 a1n a21  a22  a23  a2n                · · · a3n  a31  a32  a33    Flops  .   .   .  .  .  .   .  .   . . .  . 1 flop = (αβ + γ → γ) an1 an2 an3 ann n2 entries in the storage Ax costs O(n2 ) flops AB costs O(nδ ) flops, δ ≥ 2.376 usually δ = 3 A−1 costs O(n3 ) flopsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 5/28

Concept Application Eigenvalues Dense Matrix Format Store n vectors (Fortran):         a11 a12 a13 a1n a21  a22  a23  a2n          a31  a32  a33        · · · a3n     .   .   .   .   .   .   .  . . .  .  . an1 an2 an3 ann n2 entries in the storage expensive! Ax costs O(n2 ) flops AB costs O(nδ ) flops, δ ≥ 2.376 usually δ = 3 A−1 costs O(n3 ) flopsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 5/28

Concept Application Eigenvalues Sparse Matrix Format 4 −1 0 −1 0 0 0 0 0   −1 4 −1 0 −1 0 0 0 0    0 −1 4 0 0 −1 0 0 0   −1 0 0 4 −1 0 −1 0 0    0 −1 0 −1 4 −1 0 −1 0    0  0 −1 0 −1 4 0 0 −1 0  0 0 −1 0 0 4 −1 0   0 0 0 0 −1 0 −1 4 −1 0 0 0 0 0 −1 0 −1 4Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 6/28

Concept Application Eigenvalues Sparse Matrix Format 4 −1 0 −1 0 0 0 0 0   −1 4 −1 0 −1 0 0 0 0    0 −1 4 0 0 −1 0 0 0   −1 0 0 4 −1 0 −1 0 0    0 −1 0 −1 4 −1 0 −1 0    0  0 −1 0 −1 4 0 0 −1 0  0 0 −1 0 0 4 −1 0   0 0 0 0 −1 0 −1 4 −1 0 0 0 0 0 −1 0 −1 4 non-zero entries per row/column = c ≤ 5Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 6/28

Concept Application Eigenvalues Sparse Matrix Format 4 −1 −1 −1   −1 −1 −1 4   −1 4 −1 −1   −1 −1 4 −1   4  −1 −1 −1  −1 −1 −1 4   −1 −1 4    −1 4 −1  4 −1 −1 non-zero entries per row/column = c ≤ 5Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 6/28

Concept Application Eigenvalues Sparse Matrix Format (Matlab) Store vector of tripel:   (1, 1, 4) (2, 1, −1)   (4, 1, −1)   (1, 2, −1) .   . . O(n) entries in the storage (if #non-zeros/column < c n) Ax costs O(n) ﬂops AB may be much denser ⇒ better use A(Bx) A−1 not possible, only solve Ax = bMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 7/28

Concept Application Eigenvalues Is there something in between? Yes, e.g. Low rank matrices and Tucker tensor format Toeplitz/Hankel matrices Semiseparable matrices [Gantmacher, Krein 1937] Cauchy matrices Fast Multipole Method (FMM) [Greengard, Rokhlin ’87] Mosaic-skeleton matrices [Tyrtyshnikov ’96] ...Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 8/28

Concept Application Eigenvalues Is there something in between? Yes, e.g. Low rank matrices and Tucker tensor format Toeplitz/Hankel matrices Semiseparable matrices [Gantmacher, Krein 1937] Cauchy matrices Fast Multipole Method (FMM) [Greengard, Rokhlin ’87] Mosaic-skeleton matrices [Tyrtyshnikov ’96] Hierarchical (H-) Matrices [Hackbusch ’98]Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 8/28

Concept Application Eigenvalues Integral Equation Fredholm equation of ﬁrst kind: Ω g (x, y )u(y )dy = f (x)Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 9/28

Concept Application Eigenvalues Integral Equation unknown Fredholm equation of ﬁrst kind: Ω g (x, y )u(y )dy = f (x)Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 9/28

Concept Application Eigenvalues Integral Equation Fredholm equation of ﬁrst kind: Ω g (x, y )u(y )dy = f (x) electrostatic problem: g (x, y ) = 4π 1x−y u(y ) charge density f (x) electrostatic potential inverse of elliptic diﬀerential operators population dynamics [Koch, Hackbusch, Sundmacher]Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 9/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Ω g (x, y )u(y )dy = f (x) Ritz-Galerkin method: u(y ) = i ui ψi (y ) ⇒ g (x, y )ψi (y )dy ψj (x)dx ui = f (x)ψj (x)dx i Ω Ω Ω :=Mji =fjMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 10/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] discretization error ∼ Ω1g,(x, <)u(y )dy = f (x) nκ 0 y κ<1 Ritz-Galerkin method: u(y ) = i ui ψi (y ) ⇒ g (x, y )ψi (y )dy ψj (x)dx ui = f (x)ψj (x)dx i Ω Ω Ω :=Mji =fjMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 10/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Ω g (x, y )u(y )dy = f (x) Ritz-Galerkin method: u(y ) = i ui ψi (y ) ⇒ g (x, y )ψi (y )dy ψj (x)dx ui = f (x)ψj (x)dx i Ω Ω Ω :=Mji =fj Mu = fMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 10/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Ω g (x, y )u(y )dy = f (x) Ritz-Galerkin method: u(y ) = i ui ψi (y ) ⇒ g (x, y )ψi (y )dy ψj (x)dx ui = f (x)ψj (x)dx i Ω Ω Ω :=Mji =fj k x,s y ,t x ∈ Ωs , y ∈ Ωt : g (x, y ) ≈ p=1 gp (x)gp (y )Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 10/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Ω g (x, y )u(y )dy = f (x) Ritz-Galerkin method: u(y ) = i ui ψi (y ) ⇒ g (x, y )ψi (y )dy ψj (x)dx ui = f (x)ψj (x)dx i Ω Ω Ω :=Mji =fj k x,s y ,t x ∈ Ωs , y ∈ Ωt : g (x, y ) ≈ p=1 gp (x)gp (y ) Mji = g (x, y )ψj (x)ψi (y )dydx Ω Ω k x y T ≈ gp (x)ψj (x)dx gp (y )ψi (y )dy = Aj· Bi· p=1 Ω ΩMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 10/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 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 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 25 9 7 3 11 10 10 19 11 6 5 13 8 8 11 9 k 11 16 3 11 11 x,s y ,t 11 8 8 11 11 31 15 6 9 15 x ∈ Ωs , y ∈ Ωt : g (x, y ) ≈ gp (x)gp (y ) 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 10 7 p=1 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 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 Mji = g (x, y )ψj (x)ψi (y )dydx 11 8 3 7 8 15 8 39 10 10 10 6 5 10 3 15 10 11 9 15 3 10 3 25 7 10 3 6 3 13 6 Ω Ω 13 7 10 7 22 7 10 11 9 8 5 7 12 10 10 6 3 3 7 10 10 19 10 31 11 9 16 12 11 8 8 15 8 12 k x y T 20 13 gp (x)ψj (x)dx gp (y )ψi (y )dy = Aj· Bi· 8 10 8 9 11 8 15 15 11 34 10 10 25 13 7 10 11 6 13 5 6 11 ≈ Ω Ω 13 13 p=1 11 8 13 7 7 11 10 9 16 10 11 61 11 23 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 5 8 7 3 7 3 10 61 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, Hierarchical Matrices 11/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 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 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 25 9 7 3 11 10 10 19 11 6 5 13 8 8 11 9 k 11 16 3 11 11 x,s y ,t 11 8 8 11 11 31 15 6 9 15 x ∈ Ωs , y ∈ Ωt : g (x, y ) ≈ gp (x)gp (y ) 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 10 7 p=1 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 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 Mji = g (x, y )ψj (x)ψi (y )dydx 11 8 3 7 8 15 8 39 10 10 10 6 5 10 3 15 10 11 9 15 3 10 3 25 7 10 3 6 3 13 6 Ω Ω 13 7 10 7 22 7 10 11 9 8 5 7 12 10 10 6 3 3 7 10 10 19 10 31 11 9 16 12 11 8 8 15 8 12 k x y T 20 13 gp (x)ψj (x)dx gp (y )ψi (y )dy = Aj· Bi· 8 10 8 9 11 8 15 15 11 34 10 10 25 13 7 10 11 6 13 5 6 11 ≈ Ω Ω 13 13 p=1 11 8 13 7 7 11 10 9 16 10 11 61 11 23 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 8s 7 11 11 8 5 8 8 15 9 7 3 7 10 3 10 61 10 15 9 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 tMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 11/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 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 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 rank (Ms×t ) = 19 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 11 11 store: 10 7 8 11 11 31 15 6 9 15 11 8 8 8 15 10 10 15 9 5 8 11 61 3 6 14 9 11 10 13 6 15 10 6 25 10 6 13 8 |s|×19 3 A ∈11R12 , 10 19 10 11 8 12 16 5 6 10 14 6 10 31 10 9 8 15 = 20 B 13 R|t|×19 13 8 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 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 15 10 11 with Ms×t 12 AB T = 9 15 3 10 3 25 7 10 3 6 3 13 6 13 7 10 7 22 7 10 11 9 8 5 10 3 7 19 10 11 8 7 12 10 6 3 11 10 10 12 31 9 16 8 15 8 ⇒ only 19(|s| + |t|) instead of |s| |t| storage 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 61 13 11 8 13 7 7 11 9 13 8 11 8 9 8 15 6 5 10 13 6 ⇒ reduces the required storage to 15% 9 12 16 10 11 20 9 9 39 10 9 3 3 7 7 10 11 23 15 10 12 13 11 8s 7 11 11 8 5 8 8 15 9 7 3 7 10 3 10 61 10 15 9 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 tMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 11/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 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 10 11 31 11 31 11 9 16 11 7 6 15 5 6 13 8 8 15 9 15 12 13 11 8 3 3 8 7 5 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 8 11 11 31 11 15 6 9 15 11 11 8 10 15 1 if x ≈ y then x−y is large 8 8 15 10 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 10 19 10 10 16 11 128 11 small matrices on the diag- 5 6 10 14 6 10 31 9 8 15 20 13 8 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 onal have no low rank ap- 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 proximation 3 9 15 3 10 3 25 7 10 3 6 3 15 10 13 116 13 7 10 7 22 7 10 11 9 8 5 7 12 10 10 3 7 19 10 11 16 12 12 11 8 ⇒ use dense matrix format 6 3 10 10 31 9 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 7 11 10 9 16 10 11 61 11 23 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 5 8 7 3 7 3 10 61 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, Hierarchical Matrices 11/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 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 11 10 31 11 6 5 13 8 15 9 13 8 11 11 8 11 31 11 9 3 16 11 7 3 8 15 6 7 5 8 15 12 non-admissible block 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 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 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 7 12 10 10 6 3 3 7 10 10 19 10 31 11 9 16 12 11 8 8 15 8 12 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 7 11 10 9 16 10 11 61 11 23 13 9 6 13 20 9 9 7 11 8 3 7 9 8 8 15 5 6 12 9 39 10 10 15 10 admissible block 3 12 13 11 8 8 8 15 9 10 15 9 7 11 11 5 8 7 3 7 3 10 61 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, Hierarchical Matrices 11/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] rank-k-matrix: Ms×t = AB T , A ∈ Rn×k , B ∈ Rm×k (k n, m) 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-matricesMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 12/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Hierarchical matrices H(TI × I , k) = M ∈ RI × I rank (Ms×t ) ≤ k ∀s × t 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 11 5 13 13 adaptive rank k(ε) 14 10 31 11 6 8 15 9 8 11 11 31 11 16 11 15 6 8 15 12 11 8 9 7 8 7 5 3 3 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 9 3 25 10 13 8 11 11 6 5 9 storage NSt,H (T , k) = O(n log n k(ε)) 7 10 19 8 11 11 16 3 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 10 19 10 16 11 8 12 complexity of approximate arithmetic 5 6 10 14 6 10 31 10 9 8 15 11 20 13 8 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 O(n log n k(ε)) 39 10 5 MH v 8 15 8 10 10 6 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 7 12 10 10 6 3 7 10 10 19 10 31 11 9 16 12 11 8 8 15 8 12 O(n log n k(ε)2 ) 3 20 +H , −H 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 7 11 10 9 16 10 11 61 11 23 ∗H , HLU(·), (·)−1 O(n (log n)2 k(ε)2 ) 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 H 11 8 8 8 15 9 10 15 9 7 11 11 5 8 7 3 7 3 10 61 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, Hierarchical Matrices 13/28

Concept Application Eigenvalues H-Matrices [Hackbusch ’98] Hierarchical matrices H(TI × I , k) = M ∈ RI × I rank (Ms×t ) ≤ k ∀s × t 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 11 5 13 13 adaptive rank k(ε) 14 10 31 11 6 8 15 9 8 11 11 31 11 16 11 15 6 8 15 12 11 8 9 7 8 7 5 3 3 19 13 11 61 11 10 7 8 9 16 11 11 8 11 8 13 13 9 3 25 10 13 8 11 11 6 5 9 storage NSt,H (T , k) = O(n log n k(ε)) 7 10 19 8 11 11 16 3 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 10 19 10 16 11 8 12 complexity of approximate arithmetic 5 6 10 14 6 10 31 10 9 8 15 11 20 13 8 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 O(n log n k(ε)) 39 10 5 MH v 8 15 8 10 10 6 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 7 12 10 10 6 3 7 10 10 19 10 31 11 9 16 12 11 8 8 15 8 12 O(n log n k(ε)2 ) 3 20 +H , −H 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 7 11 10 9 16 10 11 61 11 23 ∗H , HLU(·), (·)−1 O(n (log n)2 k(ε)2 ) 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 H 11 8 8 8 15 9 10 15 9 7 11 11 5 8 7 3 7 3 10 61 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, Hierarchical Matrices 13/28

Concept Application Eigenvalues Special Case: H -Matrices F1 B2 AT 2 B4 AT 4 T F3 A2 B2 B8 AT 8 F5 B6 A T 6 T A4 B4 T F7 A6 B 6 F9 B10 AT 10 B12 AT 12 T A10 B10 F11 T A8 B8 F13 B14 AT 14 T A12 B12 T A14 B14 F15 Figure: Structure of an H3 (k)-matrixMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 14/28

Concept Application Eigenvalues ApplicationsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 15/28

Concept Application Eigenvalues Applications Finite-Element-Method (FEM) Inverse of FEM matrices Integral equations Solving matrix equations for model order reduction AX + XAT + BB T = 0, A ∈ H, B ∈ Rn×k Sign function iteration [Baur ’07, Grasedyck ’03, et al.] Boundary-Element-Method (BEM)Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 16/28

Concept Application Eigenvalues BEM for Electrostatic Computations ·103 4 2 0 −2 −4 2 −4 −2 0 ·103 0 2 −2 4 3 ·10Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 17/28

Concept Application Eigenvalues BEM for Electrostatic Computations ·103 Electrostatic problem: 4 Ω g (x, y )u(y )dy = f (x) 1 2 g (x, y ) = 4π x−y unknown: u(y ) charge density 0 given: f (x) electrostatic potential −2 −4 2 −4 −2 0 ·103 0 2 −2 4 3 ·10Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 17/28

Concept Application Eigenvalues BEM for Electrostatic ComputationsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 17/28

Concept Application Eigenvalues BEM for Electrostatic ComputationsMax Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 18/28

Concept Application Eigenvalues BEM for Electrostatic Computations 26 14 14 20 19 23 16 19 25 14 14 20 17 23 13 20 16 17 12 22 23 13 10 11 22 20 23 17 27 19 25 25 9 22 15 16 23 20 22 22 36 17 18 19 25 8 19 26 20 14 11 22 9 12 19 15 22 7 13 15 11 18 13 10 11 23 9 16 10 13 18 10 8 19 8 17 10 9 7 8 6 7 7 6 7 7 19 10 15 13 9 17 18 15 21 8 23 6 9 25 14 14 20 16 22 15 20 19 11 16 26 13 15 16 10 13 22 30 43 27 19 11 17 22 13 36 19 26 16 21 16 25 18 15 15 19 28 8 16 18 17 16 20 13 18 17 12 22 24 20 8 8 8 8 8 8 8 8 6 22 20 27 19 13 20 22 22 28 20 18 15 17 19 26 23 22 36 20 28 36 22 19 12 34 17 11 10 18 11 21 11 23 48 12 25 11 14 11 12 20 21 6 7 30 12 20 10 11 28 20 20 28 29 20 22 29 28 18 18 18 19 32 13 28 17 22 18 50 15 20 10 13 16 25 20 19 16 17 12 11 8 9 6 22 21 29 19 29 25 10 35 21 13 15 23 20 18 25 23 9 30 18 36 22 32 21 28 22 17 18 23 25 23 13 13 15 20 12 23 25 24 20 12 15 23 20 12 12 15 17 25 20 14 13 20 19 6 6 16 11 10 9 17 24 20 19 20 17 18 20 27 18 13 12 18 27 12 13 10 12 12 10 29 10 8 25 10 32 22 18 10 9 12 8 10 7 9 8 9 6 9 7 7 7 7 10 22 19 13 26 36 27 19 36 26 14 16 19 16 16 16 18 24 17 19 28 40 20 14 10 16 19 23 14 20 24 33 15 11 25 50 26 18 18 18 23 44 29 14 22 10 13 10 17 16 20 8 8 19 12 19 15 20 20 25 40 20 14 13 20 20 14 15 23 17 20 21 13 13 14 17 20 20 12 15 12 20 22 6 17 18 19 15 8 19 24 20 14 15 13 20 46 20 17 15 23 16 20 30 24 11 8 6 32 13 33 42 16 18 18 14 21 46 21 11 19 13 19 12 17 11 18 12 6 13 13 6 6 10 6 13 18 9 17 13 17 14 10 6 6 11 17 15 12 17 15 9 31 19 18 17 18 12 17 16 27 16 9 19 12 27 12 12 27 9 11 35 25 8 11 8 12 7 18 18 12 13 16 13 22 23 24 18 18 13 10 10 19 20 15 33 14 9 14 9 13 9 13 9 21 17 13 18 12 10 22 21 22 8 10 11 19 19 19 17 17 27 18 18 17 24 19 19 19 16 16 15 24 17 12 27 17 19 15 40 15 15 25 12 19 13 28 14 18 12 24 13 18 12 17 9 16 14 17 14 8 9 6 6 12 16 15 13 17 15 9 10 11 8 24 6 10 9 15 8 14 11 12 18 13 16 15 14 18 12 12 15 8 11 19 9 14 13 18 12 14 15 13 18 12 14 15 10 20 17 20 17 19 15 34 13 25 18 24 20 23 19 20 14 17 24 19 29 12 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Hierarchical Matrices: Concept, Application and Eigenvalues

by Thomas Mach on Mar 08, 2011

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Hierarchical Matrices: Concept, Application and Eigenvalues Presentation Transcript