Introduction to Wavelet Based Numerical Homogenization

Transcription

1 Introduction to Wavelet Based Numerical Homogenization Olof Runborg Numerical Analysis, School of Computer Science and Communication, KTH RTG Summer School on Multiscale Modeling and Analysis University of Texas at Austin Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 / 5

2 Wavelet based numerical homogenization [Beylkin,Brewster,Engquist,Dorobantu,Levy,Gilbert,O.R.,... ] Suppose L j+ u = f, L j+ L(V j+, V j+ ) u, f V j+, is a discretization (e.g. FD, FEM) of a differential equation on scale-level j + where L j+ contains small scales. Want to find an effective discrete operator L j, with j j that computes the coarse part of u. C.f. classical homogenization. Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 2 / 5

3 Wavelet based numerical homogenization [Beylkin,Brewster,Engquist,Dorobantu,Levy,Gilbert,O.R.,... ] Suppose L j+ u = f, L j+ L(V j+, V j+ ) u, f V j+, is a discretization (e.g. FD, FEM) of a differential equation on scale-level j + where L j+ contains small scales. Want to find an effective discrete operator L j, with j j that computes the coarse part of u. C.f. classical homogenization. Example (Elliptic eq, Haar) x r(x/ε) x u ε = f, L j+ = h 2 +R ε. where R ε is diagonal matrix sampling r(x/ε), and 2 j /ε. Here one could use L j = h 2 + R. Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 2 / 5

4 Wavelet transforms Simple to extract the coarse and fine part of u = {u k }: ( ) Uf Wu =, u V j+ U f W j, U c V j. U c For compactly supported wavelets, W is sparse. It is also orthonormal, W T W = I. In Haar basis on [, ], W = R 2j+ 2 j+. Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 3 / 5

5 Wavelet based numerical homogenization Wavelet decomposition of operator Start from equation L j+ u = f, L j+ L(V j+, V j+ ) u, f V j+. Decompose equation in coarse and fine part (use W T W = I) WL j+ W T Wu = Wf ( )( ) ( Aj B j U f F f = C j L j U c F c ). Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 4 / 5

6 Wavelet based numerical homogenization Wavelet decomposition of operator Start from equation L j+ u = f, L j+ L(V j+, V j+ ) u, f V j+. Decompose equation in coarse and fine part (use W T W = I) Eliminate U f, WL j+ W T Wu = Wf ( )( ) ( Aj B j U f F f = C j L j U c F c ). (L j C j A j B j )U c = F c C j A j F f. Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 4 / 5

7 Wavelet based numerical homogenization Wavelet decomposition of operator Start from equation L j+ u = f, L j+ L(V j+, V j+ ) u, f V j+. Decompose equation in coarse and fine part (use W T W = I) Eliminate U f, WL j+ W T Wu = Wf ( )( ) ( Aj B j U f F f = C j L j U c F c ). (L j C j A j B j )U c = F c C j A j F f. Supposing f smooth so F f = and F c = f. (L j C j A j B j )U c = f. Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 4 / 5

8 Wavelet based numerical homogenization Numerically homogenized operator We call the matrix L j = L j C j A j B j, Lj L(V j, V j ), the (numerically) homogenized operator. Since Half the size of original L j+. Given L j, f we can solve for coarse part of solution, U c. Takes influence of fine scales into account. Compare with classical homogenization: L = R(x/ε) L = R(x)dx where χ solves the (elliptic) cell problem. R(x) χ x dx Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 5 / 5

9 Wavelet based numerical homogenization Reduction can be repeated, L j L j L j 2..., Lj L(V j, V j ), to discard suitably many small scales / to get a suitably coarse grid. Also, condition number improves κ( L j ) < κ(l j+ ). Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 6 / 5

10 Wavelet based numerical homogenization Problem: L sparse (banded) L sparse (banded). (Must invert A j.) However: Approximation properties of wavelets imply elements of A j decay rapidly away from diagonal. Therefore: L diagonally dominant in many important cases and can be well approximated by a banded matrix. (Cf. a (local) differential operator.) Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 7 / 5

11 Different Approximation Strategies Crude truncation to ν diagonals, 2 Band projection to ν diagonals, defined by Mx = band(m, ν)x, x span{v, v 2,..., v ν }. v j = { j, 2 j,..., N j } T, j =,..., ν. C.f. probing, [Chan, Mathew], [Axelsson, Pohlman, Wittum]. 3 The above methods used on the matrix H instead, where e.g. L j+ = h 2 +R Lj = (2h) 2 +H. H can be seen as the effective material coefficient. 4 The above methods used on the matrix A j instead, where L j = L j C j A j B j. [Levy, Chertock] Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 8 / 5

12 Elliptic D case Consider the elliptic one-dimensional problem x a ε (x) x u =, u() = u () =, with standard second order discretization. Try two cases: a ε (x) = noise a ε (x) = narrow slit Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 9 / 5

13 Elliptic D case noise Different approximation strategies trunc(l, ν) trunc(h, ν) Exact ν=3 ν=5 ν= Exact ν=3 ν=5 ν=7.5 band(l, ν) band(h, ν) Exact ν=3 ν=5 ν= Exact ν= ν=3 ν=5.5 Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 / 5

14 Elliptic D case narrow slit Different approximation strategies trunc(l, ν) trunc(h, ν) Exact ν=3 ν=5 ν= Exact ν=3 ν=5 ν=7.5 band(l, ν) band(h, ν) Exact ν=7 ν=9 ν= Exact ν= ν=3 ν=5.5 Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 / 5

15 Elliptic D case narrow slit Matrix element size L 7 H Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 2 / 5

16 a Examples Helmholtz 2D case Simulate a wave hitting a wall with a small opening modeld by Helmholtz Wall with slit.2 a(x, y) u + ω 2 u =, x.2.4 y.6.8 Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 3 / 5

17 Examples Helmholtz 2D case Original homogenization homogenizations 3 homogenizations Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 4 / 5

18 Examples Helmholtz 2D case Untruncated operator ν= x.5 y.5 x.5 y ν=7 ν= x.5 y 2.5 x.5 y Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 4 / 5

19 Helmholtz 2D case Matrix element size Olof Runborg (KTH) Wavelet Based Homogenization Austin, August 28 5 / 5