Wavelet based matrix compression for boundary integral equations on complex geometries

1 Wavelet based matrix compression for boundary integral equations on complex geometries Ulf Kähler Chemnitz University of Technology Workshop on Fast Boundary Element Methods in Industrial Applications (Hirschegg) 16.10.2003

2 Overview Motivation - presentation of problem

2 Overview Motivation - presentation of problem Wavelet basis - stiffness matrix

2 Overview Motivation - presentation of problem Wavelet basis - stiffness matrix Wavelet construction

2 Overview Motivation - presentation of problem Wavelet basis - stiffness matrix Wavelet construction Computation of the stiffness matrix

2 Overview Motivation - presentation of problem Wavelet basis - stiffness matrix Wavelet construction Computation of the stiffness matrix Numerical results

3 Preliminaries Aρ = f on Γ = Ω R 2 A : H q (Γ) H q (Γ) (Aρ)(x) = k(x, y)ρ(y) Γ y Γ

4 single layer potential: q = 1 2, A = K K = 1 log y x ρ(y) Γ y 2π double layer potential: q = 0, A = 1 2 + K K = 1 2π Γ Γ n(y), y x y x 2 ρ(y) Γ y

5 Galerkin scheme Variational formulation: find ρ H q (Γ) : (Aρ, v) L 2 (Γ) = (f, v) L 2 (Γ) v H q (Γ) V N = span{φ 1,..., φ N } H q (Γ) A Φ ρ Φ = f Φ

6 Geometry Γ N - polygonial approximations of the surface Γ finest level is fixed diam(ω) < 1

7 ansatzfunctions: φ i (x) = 1 Γ i Γ, for x Γ i, 0, else

7 ansatzfunctions: φ i (x) = 1 Γ i Γ, for x Γ i, 0, else φ i (x(s)) = 13 s Γ i 1 Γ+ 1 3 1 s 13 Γ Γ+ 1 i 1 3 Γ i Γ for x(s) Γ i 1, Γ i Γ for x(s) Γ i, 0 else

8 Motivation - presentation of problem classical single scale method

8 Motivation - presentation of problem classical single scale method advantages: reduction of the space dimension good convergence rates

8 Motivation - presentation of problem classical single scale method advantages: reduction of the space dimension good convergence rates disadvantages: densly populated stiffness matrix expensive quadrature rules

8 Motivation - presentation of problem classical single scale method advantages: reduction of the space dimension good convergence rates disadvantages: densly populated stiffness matrix expensive quadrature rules

Objectives 9

9 Objectives sparse stiffness matrix

9 Objectives sparse stiffness matrix low complexity of solving

Wavelet basis - stiffness matrix 10

10 Wavelet basis - stiffness matrix hierarchical wavelets:

10 Wavelet basis - stiffness matrix hierarchical wavelets: hierarchical structure

10 Wavelet basis - stiffness matrix hierarchical wavelets: hierarchical structure ψ i := N k=1 ω i,kφ k, ω i1 ω i2 = δ i1 i 2

10 Wavelet basis - stiffness matrix hierarchical wavelets: hierarchical structure ψ i := N k=1 ω i,kφ k, ω i1 ω i2 = δ i1 i 2 (ψ i, x α ) = 0, α < d ( x α = x α 1 1 xα 2 2 )

10 Wavelet basis - stiffness matrix hierarchical wavelets: hierarchical structure ψ i := N k=1 ω i,kφ k, ω i1 ω i2 = δ i1 i 2 (ψ i, x α ) = 0, α < d ( x α = x α 1 1 xα 2 2 ) Dahmen-Prößdorf-Schneider/ von Petersdorff-Schwab: A ψ is a quasi sparse matrix with O(N log(n)) entries.

11 idea: = Γ k(x, y)ψ k (x)ψ k (y) Γ x Γ y Γ (D α+β k)(x 0, y 0 ) (α,β) N 2 0 N 2 0 Γ ψ k(x)(x x 0 ) α Γ x α! Γ ψ k (y)(y y 0) β Γ y β!

11 idea: = Γ k(x, y)ψ k (x)ψ k (y) Γ x Γ y Γ (D α+β k)(x 0, y 0 ) (α,β) N 2 0 N 2 0 Γ ψ k(x)(x x 0 ) α Γ x α! D α+β k(x 0, y 0 ) ( 1 C x 0 y 0 Γ ψ k (y)(y y 0) β Γ y β! ) α+β+1 2q

Wavelet construction 12

12 Wavelet construction hierarchical structure coarsening cluster tree

12 Wavelet construction hierarchical structure coarsening cluster tree [ ] [ ] Φ ν,j 1 V ν,j 1 Ψ ν,j 1 0 = V ν,j 1 Φ ν,j 0

13 Let M ν,j 1 be the moment matrix of the cluster ν from level j 1 [ ] M ν,j 1 = x α Φ ν,j dx Γ α < d

13 Let M ν,j 1 be the moment matrix of the cluster ν from level j 1 [ ] M ν,j 1 = x α Φ ν,j dx Γ α < d SV D M ν,j 1 = UΣV = U [S, 0] [ V ν,j 1 0 V ν,j 1 0 ]

13 Let M ν,j 1 be the moment matrix of the cluster ν from level j 1 [ ] M ν,j 1 = x α Φ ν,j dx Γ α < d SV D M ν,j 1 = UΣV = U [S, 0] [ V ν,j 1 0 V ν,j 1 0 ] constant/linear ansatzfunctions Ψ is orthonormal/ Riesz-basis.

14 complexity of computing cluster tree and wavelets: O(N)

15 Computation of the stiffness matrix transformation matrix Ω Ψ,Φ := (ω i,k ) N i,k=1

15 Computation of the stiffness matrix transformation matrix Ω Ψ,Φ := (ω i,k ) N i,k=1 A Ψ ρ Ψ = Ω Ψ,Φ A Φ Ω Ψ,Φ Ω Ψ,Φ ρ Φ = Ω Ψ,Φ (f, φ i ) N i=1 = f Ψ

15 Computation of the stiffness matrix transformation matrix Ω Ψ,Φ := (ω i,k ) N i,k=1 A Ψ ρ Ψ = Ω Ψ,Φ A Φ Ω Ψ,Φ Ω Ψ,Φ ρ Φ = Ω Ψ,Φ (f, φ i ) N i=1 = f Ψ calculation of f Ψ in O(N) possible

16 Ω Ψ,Φ = ψ ν2 1 0 0 0 ψ ν1 1 0 ψ ν0 1 0 ψ ν2 2 0 0. 0. 0 0 ψ ν2 3 0 0 ψ ν1 2. 0 0 0 ψ ν2 4 0..

16 Ω Ψ,Φ = ψ ν2 1 0 0 0 ψ ν1 1 0 ψ ν0 1 0 ψ ν2 2 0 0. 0. 0 0 ψ ν2 3 0 0 ψ ν1 2. 0 0 0 ψ ν2 4 0.. ψ ν2 1 ψ ν1 1 ψ ν0 1 ψ ν2 2.. ψ ν2 3 ψ ν1 2. ψ ν2 4..

16 Ω Ψ,Φ = ψ ν2 1 0 0 0 ψ ν1 1 0 ψ ν0 1 0 ψ ν2 2 0 0. 0. 0 0 ψ ν2 3 0 0 ψ ν1 2. 0 0 0 ψ ν2 4 0.. O(log(N)) columns ψ ν2 1 ψ ν1 1 ψ ν0 1 ψ ν2 2.. ψ ν2 3 ψ ν1 2. ψ ν2 4..

17 Using multipole method the multipole method

17 Using multipole method the multipole method iterative solving of A Φ ρ Φ = f Φ

17 Using multipole method the multipole method iterative solving of A Φ ρ Φ = f Φ fast matrix-vector product

17 Using multipole method the multipole method iterative solving of A Φ ρ Φ = f Φ fast matrix-vector product expansion of kernel

17 Using multipole method the multipole method iterative solving of A Φ ρ Φ = f Φ fast matrix-vector product expansion of kernel k(x, y) = (D α+β k)(x 0, y 0 ) (α,β) N 2 0 N 2 0 (x x 0 ) α (y y 0 ) β α! β!

low rank approximation of parts of A Φ 18

18 low rank approximation of parts of A Φ (A Φ i,j) i I,j J XkY

18 low rank approximation of parts of A Φ subdivision of A Φ (A Φ i,j) i I,j J XkY

18 low rank approximation of parts of A Φ subdivision of A Φ hierarchical matrix (A Φ i,j) i I,j J XkY

18 low rank approximation of parts of A Φ (A Φ i,j) i I,j J XkY subdivision of A Φ hierarchical matrix cluster-cluster interactions possible

18 low rank approximation of parts of A Φ (A Φ i,j) i I,j J XkY subdivision of A Φ hierarchical matrix cluster-cluster interactions possible complexity of a matrix-vector product: O(N log 2 (N))

18 low rank approximation of parts of A Φ (A Φ i,j) i I,j J XkY subdivision of A Φ hierarchical matrix cluster-cluster interactions possible complexity of a matrix-vector product: O(N log 2 (N))

application on wavelets 19

19 application on wavelets using fast matrix-vector products on ψ ν2 1 ψ ν1 1 ψ ν0 1 ψ ν2 2.. ψ ν2 3 ψ ν1 2. ψ ν2 4..

19 application on wavelets using fast matrix-vector products on ψ ν2 1 ψ ν1 1 ψ ν0 1 ψ ν2 2.. ψ ν2 3 ψ ν1 2. ψ ν2 4.. complexity of computing A Ψ : O(N log 3 (N))

19 application on wavelets using fast matrix-vector products on ψ ν2 1 ψ ν1 1 ψ ν0 1 ψ ν2 2.. ψ ν2 3 ψ ν1 2. ψ ν2 4.. complexity of computing A Ψ : O(N log 3 (N))

H.Harbecht: Error estimates for entries of A Ψ 20

Numerical results 21

Future research 22

22 Future research application to the 3D-case

22 Future research application to the 3D-case improved combination of multipole and wavelets