Adaptive Wavelet BEM

Wolfgang Dahmen, Helmut Harbrecht, Reinhold Schneider,

Rob Stevenson, and Manuela Utzinger

Helmut Harbrecht

Department of Mathematics and Computer Science

University of Basel (Switzerland)

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Overview

• Boundary integral equations and boundary element methods

• Some results from approximation theory

• An adaptive algorithm of optimal complexity

•Why trees?

• Numerical Results

Helmut Harbrecht

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MotivationLet L be a second order elliptic operator with constant coefficients

(LU)(x) =−∆U(x)−κ2U(x).

In a Lipschitz domain Ω⊂ R3 with boundary Γ := δΩ, consider the problems

Laplace equation (κ = 0):

∆U = 0 in Ω

U = f in Γ

Helmholtz equation (κ > 0):

∆U +κ2U = 0 in Ω

U = f in Γ

∂U∂r− iκU = O(r−2) as r = ‖x‖→ ∞

Problems arising:

• Discretization of the three dimensional domain Ω is expensive.• In case of an exterior problem, the infinite expansion of Ω has to be handled (e.g. artificial

boundaries have to be introduced).

What can be done?Rewriting our problem as a boundary integral equation gives us the new problem to solve:Au = f on Γ for some boundary integral operator A .

Helmut Harbrecht

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Boundary Integral Equations

We are given a boundary integral equation:

Au = f on Γ = ∂Ω⊂ R3

A : Ht(Γ)→ H−t(Γ), (Au)(x) =∫

Γ

k(x,y)u(y)dσy

Indirect formulations of the Laplace equation ∆U = 0 in Ω:

• single layer operator: f Dirichlet data

A = V : H−1/2(Γ)→ H1/2(Γ), (V u)(x) =1

4π

∫Γ

1‖x−y‖

u(y)dσy

the solution is given by U(x) = (V u)(x) for all x ∈Ω.

• double layer operator: f Dirichlet data

A = K ±1/2 : L2(Γ)→ L2(Γ), (K u)(x) =1

4π

∫Γ

〈x−y,ny〉‖x−y‖3 u(y)dσy

the solution is given by U(x) = (K u)(x) for all x ∈Ω.

• hypersingular operator: f Neumann data

A = W : H1/2(Γ)→ H−1/2(Γ), (W u)(x) =− 14π

∂

∂nx

∫Γ

〈x−y,ny〉‖x−y‖3 u(y)dσy

the solution is given by U(x) = (K u)(x) for all x ∈Ω.

Helmut Harbrecht

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Galerkin Scheme

Variational formulation:

seek u ∈ Ht(Γ) such that

(Au,v)L2(Γ) = ( f ,v)L2(Γ) for all v ∈ Ht(Γ)

Galerkin scheme:

• V0 ⊂V1 ⊂ ·· · ⊂ Ht(Γ),⋃j≥0

V j = Ht(Γ),⋂j≥0

V j =V0

• V j = span f γ−1i : f is polynomial of order d on

seek u ∈V j: (Au,v)L2(Γ) = ( f ,v)L2(Γ) ∀v ∈V j

Linear system: AΦuΦ = fΦ

Problems: The system matrix AΦ is

• densely populated

• ill conditioned: condAΦ ∼ h−|2t|100 200 300 400 500 600 700 800 900 1000

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Helmut Harbrecht

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Explicit Kernel ApproximationDecay property of the kernels of boundary integral operators in the space∣∣∣∂α

x ∂βyk(x,y)

∣∣∣. α!β![q‖x−y‖]2+2t+|α|+|β| , q > 0

• Fast Multipole Method [Greengard, Rokhlin,. . . ]• Panel Clustering [Hackbusch-Nowak,. . . ]• H -matrices [Hackbusch-Khoromskij,. . . ]• Adaptive Cross Approximation [Rjasanow-Bebendorf]

cluster methods

Expanding the kernel yields

[AΦ]i, j =∫

Γ

∫Γ

k(x,y)φi(x)φ j(y)dσy dσx

≈ ∑|α|,|β|<p

∂αx ∂

βyk(x0,y0)

α!β!︸ ︷︷ ︸=:kα,β

∫Γ

φi(x)(x−x0)α dσx︸ ︷︷ ︸

=:bi,α

∫Γ

φ j(y)(y−y0)β dσy︸ ︷︷ ︸

=:c j,β

.

Suppose clusters π = ∪i∈I suppφi and π′ = ∪ j∈J suppφ j such that dist(π,π′)> 0, then

[AΦ]i∈I, j∈J ≈ [bi,α]i∈I, |α|<p[kα,β]|α|,|β|<p[c j,β]Tj∈J, |β|<p

|ε|. maxdiamπ,diamπ′p diamπdiamπ′

[qdist(π,π′)]2+2t+p

≈

Helmut Harbrecht

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Wavelet Matrix Compression[Beylkin-Coifman-Rokhlin, Dahmen-Proßdorf-Schneider, von Petersdorff-Schwab,. . . ]

Decay property of the kernel on the boundary∣∣∣∂α

x ∂β

yk(x, y)∣∣∣≤ cα,β

∥∥x− y∥∥−(2+2t+|α|+|β|)

Suppose compactly supported ansatz functions satisfying a cancellation property∣∣( f ,ψi)L2(Γ)

∣∣. diam(suppψi)d+1| f |

W d,∞(suppψ j)

of order d. Then, it holds that∣∣(Aψi,ψ j)L2(Γ)

∣∣. diam(suppψi)d+1 diam(suppψ j)

d+1

dist(suppψi,suppψ j)2+2t+2d.

A wavelet basis does have these properties!

Linear system: AΨuΨ = fΨ

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Helmut Harbrecht

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Fast Boundary Element MethodsI We can compress AΦ directly in the single-scale basis by low-rank approximations of

appropriate chosen blocks. This results in the well known cluster methods like the Fast

Multipole Method (FMM) or the Adaptive Cross Approximation (ACA).

I Another approach is the choice of a more sophisticated basis Ψ for V j which yields

a sparse representation of the system matrix AΨ by a-priorily neglecting non-relevant

matrix entries. This results in the Wavelet Galerkin Scheme (WGS).

I For illustration purposes, ACA and FMM can be thought of as hp-approximations and

the WGS as adaptive sparse grid approximation of the kernel function k.

I The memory requirements of the WGS is superior over the cluster methods!

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Helmut Harbrecht

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Adaptive Boundary Element Methods

Why adaptivity?For many problems, e.g. geometries with edges, the solution admits singularities. It ishence desirable to have an adaptive approach. One wants to refine towards the interestingparts of the solution without spending all the computational cost and/or the memory foruniform mesh refinement.

Flow chart of (standard) adaptive algorithms:

solve −→ estimate −→ mark −→ refine

Remarks:

• there exist reliable and efficient error estimators for the residuum (Faermann [1996++])

• optimal convergence rates have been proven (Gantumur / Praetorius [2013])

• in contrast to finite element methods it is impossible to compute the residuum exactly!

• how to approximate the residuum and by which accuracy?

• how to combine modern fast methods and adaptivity?

• computational cost?

Helmut Harbrecht

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Excursion: Best (Tree) N-Term Approximations

• Ht(Γ)-normalized wavelet basis Ψ and u ∈ `2

u := Ψu ∈ Ht(Γ) and ‖u‖ ∼ ‖u‖Ht(Γ)

• if u ∈ Ht+ns(Γ) uniform refinement gives

‖u−uN‖. N−s, s≤ s := (d− t)/n

• if u ∈ Bt+nsτ (Γ) with τ = (s+1/2)−1 then

infnnz(uN)=N

‖u−uN‖. N−s, s≤ s

• Ht+ns(Γ) is a proper subset of Bt+nsτ (Γ)

• Dahlke-DeVore: Bt+nsτ (Γ) is much weaker than Ht+ns(Γ)

• if u ∈ Bt+nsτ (Γ), then

|u|`wτ

:= supN∈N

Ns infnnz(uN)=N

‖u−uN‖< ∞

The best tree N-term approximation is nearly as good as the pure best N-term

approximation. However, it retains the tree structure of wavelet bases.

Helmut Harbrecht

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Problem Formulation Cohen-Dahmen-DeVore [2001]

The variational formulation is equivalent to an infinite system of equations in `2:

seek u = Ψu such that Au = f

where A := (AΨ,Ψ)L2(Γ), f := ( f ,Ψ)L2(Γ).

Goal: If u = Ψu ∈ Bt+nsτ (Γ) with τ := (s+1/2)−1 for some s≤ s,

find for N degrees of freedom the approximate solution uN satisfying

‖u−uN‖. N−s

within complexity O(N). you cannot do better!

Helmut Harbrecht

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Basic Ingredients

• computing the right hand side: g = rhs[J ,ε]% computes g with ‖(g− f)|J ‖ ≤ ε within complexity O(ε−1/s|u|1/s

`wτ)

% realization requires some a-priori information on f

• operator-vector multiplication: w = apply[v,J ,ε]% computes w with ‖(w−Av)|J ‖ ≤ ε within complexity O(ε−1/s|v|1/s

`wτ+nnzv)

• thresholding routine: J = coarse[v,ε]% computes index set J with ‖v−v|J ‖ ≤ ε

% if ‖u−v‖ ≤ γε then complexity O(ε−1/s|u|1/s`wτ)

% realization by bucket sort or according to Binev-DeVore

One may consider here both, best N-term approximations

and best N-term tree approximations.

Helmut Harbrecht

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Galerkin Algorithm Gantumur-Harbrecht-Stevenson [2005]

• growing routine: [J ,r] = grow[v,r,ε]% produces r ∼ ‖f−Av‖% global parameters ω,θ

η = rdo until (η < r) or (r+2ωη < ε)

η := η/2res = rhs[`2,ωη]−apply[`2,v,ωη], r := ‖res‖

if (η > ε) then J := coarse[res,θr] else stop• Galerkin solver: v = galerkin[J ,v,r]

% damps the error on the index set J by a certain constant ∼ λ

res = rhs[J ,r/λ]−apply[J ,v,r/λ]

compute A1/λ such that ‖A−A1/λ‖ ≤ 1/λ and solve A1/λd = resv := v+d• main program: v = solve[ε]

initialization : J := /0, r := ‖ f‖do until (r < ε)

[J ,r] = grow[v,r,ε]v = galerkin[J ,v,r]

optimal algorithm without coarsening of the solution

Helmut Harbrecht

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Surface Representation

0 1

1γi

Γi

I Let Γ⊂ R3 be a piecewise smooth Lipschitz boundary, i.e., Γ = ∪Mi=1Γi, where Γi∩Γ j

consists at most of a common vertex or a common edge for i 6= j.

I We assume Γi = γi(), where := [0,1]2 and γi : → Γi are smooth diffeomorphismsfor all i = 1, . . . ,M.

I This surface representation is in contrast to the approximation of surfaces by panels. Infact there is no approximation to be performed if the surface is given in this form.

Examples.

Helmut Harbrecht

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Computer Aided Design (CAD)

I Such parametric surface representations are easily accessible by tools for Computer

Aided Design (CAD). They are recently studied in Isogeometric Analysis, which aims at

the direct implementation of CAD and numerical analysis.

IGES Entity ID number IGES-codeLine 110 LINECircular arc 100 ARCPolynomial/rational B-spline curve 126 B SPLINEComposite curve 102 CCURVESurface of revolution 120 SREVTabulated cylinder 122 TCYLPolynomial/rational B-spline surface 128 SPLSURFTrimmed parametric surface 144 TRM SRFTransformation matrix 124 XFORM

Helmut Harbrecht

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Construction of Biorthogonal Wavelet Bases• Based on the biorthogonal wavelet bases of Cohen-Daubechies-Feauveau [1992],

wavelets on the interval are constructed according to Dahmen-Kunoth-Urban [1996]:

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

φ2

−3 −2 −1 0 1 2 3 4−1

−0.5

0

0.5

1

1.5

ψ24

−3 −2 −1 0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

~ψ24

• Define the biorthogonal multiscale analysis on the unit square via tensor products of thescaling functions on the interval and construct suitable wavelets in the complementaryspaces:

• Define the wavelets on the manifold via parametric liftings and use the technique ofCanuto-Tabacco-Urban [1999], Cohen-Masson [2000] or Dahmen-Schneider [1999] toconstruct globally continuous wavelet bases (see also Harbrecht-Stevenson [2006]).

Helmut Harbrecht

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Why Trees?

The natural tree structure of the wavelets is exploited

to speed-up the code. However, matrix assembling

is still the most time consuming part in the (uniform)

wavelet Galerkin scheme. Finally, by the nature of

adaptivity, wavelet and element trees suggest them-

selves as data structures for our implementation.3 4 5 6 7 8

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level

cpu−

time

in p

erce

nt

compressionquadraturesolving

Algorithmical ingredients:

• exploit the tree structure of wavelets to compute the matrix pattern• reduce quadrature to element-element interactions element-based assembling• adaptive hp-quadrature scheme and recycling

Γi, j,k

Γi′, j′,k′−→

• use symmetry of the matrix pattern

Helmut Harbrecht

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Operator Compression

Define the operator A j by setting all coefficients (Aψλ,ψλ′)L2(Γ) to zero where

•∣∣|λ|− |λ′|∣∣> j/(n−1),

• dist(suppψλ,suppψλ′)> a2−min|λ|,|λ′|max

1,2b( j/n−||λ|−|λ′||)

,

• dist(suppψλ,singsuppψλ′)> amax

γ|λ|−|λ′|− j/n2 j−n|λ|+(n−1)|λ′|,2−|λ|

,

dist(suppψλ′,singsuppψλ)> amax

γ|λ|−|λ′|− j/n2 j−n|λ′|+(n−1)|λ|,2−|λ′|

,

and∣∣|λ|− |λ′|∣∣> j/n.

Then, there holds ∥∥A−A j∥∥. 2− js, nnz

(suppAi

j). 2 j.

first compression second compression

︸ ︷︷ ︸dist(suppψλ,suppψ

λ′)

ψλ ψλ′

︸︷︷︸dist(suppψλ,singsuppψ

λ′)

ψλψλ′

Helmut Harbrecht

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Visualization of the Compression

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nz = 79082

Helmut Harbrecht

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Tree Based Realization of Apply

Consider a finite vector v and a given target accuracy ε. Choose trees J j according to

‖v−v|J j‖ ≤ 2 jsε, j = J, . . . ,0,

and define the layers ∆ j := J j \ J j−1. Then, the approximation

[Av]approx =

AJ AJ−1 AJ−2 . . . A1

·

v|∆Jv|∆J−1v|∆J−2...v|∆1

← 2Jsε

← 2(J−1)sε

← 2(J−2)sε

← ε

satisfies

‖Av− [Av]approx‖.J

∑j=1‖A−AJ‖

∥∥v|∆ j

∥∥. Jε

and is of complexity O(ε−1/s|v|1/s`wτ+nnzv).

Helmut Harbrecht

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Symmetric Realization of Apply

The support of [Av]approx is called the prediction set J−1, which is again a tree.

With ∆0 := J0 \ J−1 we conclude

[Av]approx =

AJ AJ−1 AJ−2 . . . A1 A0

·

v|∆Jv|∆J−1...v|∆10∆0

.Symmetrization leads to

[Av]approx =

AJ,J AJ,J−1 . . . AJ,0

AJ−1,J AJ−1,J−1 . . . AJ−1,0... ... ...

A0,J A0,J−1 . . . A0,0

·

v|∆Jv|∆J−1...

0∆0

.satisfying

‖Av− [Av]approx‖. Jε.

Helmut Harbrecht

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Remarks

• quadratic matrices enable to consider the infinite diagonally scaled system

better condition numbers

• galerkin might be realized by solving the systemAJ+`,J+` AJ+`,J−1+` . . . AJ+`,`

AJ−1+`,J+` AJ−1+`,J−1+` . . . AJ−1+`,`... ... ...

A`,J+` A`,J−1+` . . . A`,`

v = g

where ` := dlog2(λ)/se

• only one matrix needs to be assembled to perform one step of solve

Helmut Harbrecht

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Numerical Results I

Interior Dirichlet problem solved by the single layer operator

on the Fichera vertex. Discretization via piecewise constant

wavelets with three vanishing moments.

nnzres N ‖res‖ % cpu-time ‖U−UN‖∞

48 42 1.87 100 0.05 1.84 ·10−1

175 156 1.53 91.7 0.1 1.85 ·10−1

445 341 3.72 ·10−1 52.8 0.34 5.90 ·10−2

1678 916 2.39 ·10−1 36.2 2.62 1.96 ·10−2

4270 2237 1.37 ·10−1 16.2 9.06 2.93 ·10−3

11001 6037 8.51 ·10−2 6.49 34.16 1.78 ·10−3

26960 13690 5.36 ·10−2 2.90 103.73 6.61 ·10−4

65066 31377 3.53 ·10−2 1.24 308.92 5.23 ·10−4

110541 53233 2.19 ·10−2 0.66 469.27 4.92 ·10−4

224019 103435 1.46 ·10−2 0.34 1211.68 1.65 ·10−4

493140 214176 1.07 ·10−2 0.16 3688.67 6.11 ·10−5

1082059 450806 6.98 ·10−3 0.07 11733.6 2.92 ·10−5

2404721 968714 4.58 ·10−3 0.03 41460.7 2.40 ·10−5

3934070 1636714 2.45 ·10−3 0.02 71050.1 2.42 ·10−5

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10!5

10!4

10!3

10!2

10!1

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Number of unknowns

Singlelayer on ficheravertex with polynomial rhs

rate N!0.75

rate N!0.5

ResNormPotErr

Helmut Harbrecht

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Numerical Results II

Interior Dirichlet problem solved by the single layer operator on the Fichera vertex with

special right hand side. Discretization via piecewise constant wavelets with three vanish-

ing moments.

-20

-10

0

Cell_Density

-26.4

5.01

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10−5

10−4

10−3

10−2

10−1

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101

Number of unknowns

Singlelayer on ficheravertex with harmonical rhs

rate N−0.75

rate N−0.5

ResNormPotErr

Helmut Harbrecht

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Numerical Results III

Interior Dirichlet problem solved by the single layer opera-

tor on a crankshaft. Discretization via piecewise constant

wavelets with three vanishing moments.

nnzres N ‖res‖ % cpu-time ‖U−UN‖∞

2272 2058 49.9 33.0 10.63 0.923454 2723 9.03 22.3 3.56 3.78

12708 8286 10.0 12.5 39.38 2.3131572 17885 4.86 5.18 105.11 2.75106560 48111 3.95 2.19 651.78 4.16 ·10−1

143712 83755 9.87 ·10−1 0.97 625.12 1.19460403 184305 1.51 0.61 4250.46 3.34 ·10−2

447758 260031 2.93 ·10−1 0.26 2181.14 9.12 ·10−1

2778615 672337 9.33 ·10−1 0.21 63616.6 1.11 ·10−2

1920665 857625 1.59 ·10−1 0.06 12380.2 3.21 ·10−2

5159313 1814776 2.30 ·10−1 0.05 140813 1.09 ·10−2

5601415 2562202 1.18 ·10−1 0.05 114303 9.27 ·10−3

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107

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10−4

10−3

10−2

10−1

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102

Number of unknowns

Singlelayer on crankshaft with polynomial rhs

rate N−0.75

rate N−0.5

ResNormPotErr

Helmut Harbrecht

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Future work

Joint project Adaptive Wavelet and Frame Techniques for Acoustic BEM with the Uni-

versity of Marburg and the Acoustics Research Institute Vienna: Solve the exterior Helmholtz

for the head and find the HRTFs (Head Related Transfer Functions) which are important

for the description of the three dimensional hearing.

Helmut Harbrecht

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References

P. Binev, R. DeVore.Fast computation in adaptive tree approximation.Numer. Math. 97:193–217 (2004)

A. Cohen, W. Dahmen, R. DeVore.Adaptive wavelet methods for elliptic operator equations: Convergence rates.Math. Comput. 70:27–75 (2001)

W. Dahmen, H. Harbrecht, R. Schneider.Adaptive methods for boundary integral equations: Complexity and convergence estimates.Math. Comput. 76:1243–1274 (2005)

T. Gantumur, H. Harbrecht, R. Stevenson.An optimal adaptive wavelet method for elliptic equations without coarsening.Math. Comput. 76:615–629 (2007)

H. Harbrecht, R. Stevenson.Wavelets with patchwise cancellation properties.Math. Comput. 75:1871–1889 (2006).

Thank you!Helmut Harbrecht

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