Daubechies Wavelets and Interpolating Scaling Functions and Application on PDEs

Transcription

1 Daubechies Wavelets and Interpolating Scaling Functions and Application on PDEs R. Schneider F. Krüger TUB - Technical University of Berlin November 22, 2007 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 1 / 1

2 Outline Outline R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 2 / 1

3 Daubechies Wavelets R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 3 / 1

4 Daubechies Wavelets Daubechies Wavelets Conditions that fulfill the Daubechies wavelets: 1. They perform an orthonormal basis. 2. They have p vanishing moments for a given p N. 3. They have the minimal support of all functions that fulfill the first two conditions. 4. They are rather smooth. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 4 / 1

5 Fourier Space Daubechies Wavelets The Fourier transform of a function f is defined by ˆf(ω) = f(x)e iωx dx R Given a scaling function ϕ and its corresponding wavelet ψ with its refinement equations ϕ = k ψ = k h k ϕ 1 k g k ϕ 1 k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 5 / 1

6 Daubechies Wavelets In Fourier space it gets to 2 ˆϕ(ω) = 2 h(ω 2 ) ˆϕ(ω 2 ) (1) 2 ˆψ(ω) = 2 g(ω 2 ) ˆϕ(ω 2 ) with h(ω) = k Z h k e ikω g(ω) = k Z g k e ikω. By succesive application of (??) we get h(2 k ω) ˆϕ(ω) = ˆϕ(0) 2 if ˆϕ is continuos at 0. k=1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 6 / 1

7 Daubechies Wavelets Computation of the Daubechies Wavelets There is no analytical formula for the Daubechies scaling functions and wavelets. But you can compute exactly the filters h and g. They are tabulated in the internet, for example at wikipedia. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 7 / 1

8 Daubechies Wavelets Computation of the Daubechies Wavelets With the filters you can use the formulae in the Fourier space. ˆϕ(ω) = k=1 h(2 k ω) 2 ˆψ(ω) = 1 2 g( ω 2 ) ˆϕ(ω 2 ) with h(ω) = h k e ikω k Z g(ω) = g k e ikω. k Z R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 8 / 1

9 ϕ Daubechies Wavelets 1.4 p = p = p = 4 p = Figure: Daubechies scaling functions ϕ with p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling Functions and November Application 22, 2007 on PDEs 9 / 1

10 ψ Daubechies Wavelets 2 p = 2 2 p = p = 4 p = Figure: Daubechies wavelets ψ with p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs10 / 1

11 Vanishing Moments Daubechies Wavelets A very important property of the Daubechies wavelets are the vanishing moments x k ψ(x)dx = 0, k = 0,..., p 1. R That also means that every polynomial of degree at most p 1 is in V 0 in a compact subset K of R. For example is f = 1 K V 0 K. The series ( n k= n ϕ k(x)) n N is converging pointwise to the unity. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs11 / 1

12 Partition of Unity Daubechies Wavelets Partition of Unity In the picture is shown the function 10 f = k=0 with ϕ the Daubechies scaling function of order p = 2. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs12 / 1 ϕ k

13 Interpolating Scaling Functions R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs13 / 1

14 Interpolating Scaling Functions An interpolating scaling function φ fulfills φ(k) = δ k, k Z. Since the Daubechies scaling functions ϕ k are orthonormal we get an interpolating scaling function φ (Interpolet) by folding the Daubechies function with itself. φ(t) = ϕ(u)ϕ(u t)du = ϕ ϕ(t), where R ϕ(t) =ϕ( t). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs14 / 1

15 Interpolating Scaling Functions The interpolation property can be easily seen: φ(k) = ϕ(u)ϕ(u k)du = ϕ, ϕ k = δ k. R For the filter h I corresponding to φ we get h I k = (h h) k. To hold the interpolating property for j Z we omit the normalisation factor: φ j k (x) = φ(2j x k). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs15 / 1

16 Interpolating Scaling Functions p = 2 p = p = p = Figure: Interpolating scaling function φ for p = 2, 3, 4 and 5. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs16 / 1

17 Interpolation Projection Interpolating Scaling Functions With Interpolets we have a very simple projection P J onto V J : P J f = k Z f(x k )φ J k, x k = 2 J k This projection is called interpolation projection or collocation projection. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs17 / 1

18 Computing Integrals R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs18 / 1

19 Moments Computing Integrals We want to compute the moments M k = R x k ϕ(x)dx = 2p 1 0 x k ϕ(x)dx. By using the refinement equation we get M k = 2 h l x k ϕ(2x l)dx l R 2 = 2 k+1 h l (x + l) k ϕ(x)dx 2 = 2 k+1 l l R k h l j=0 ( ) k l j M k j. j R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs19 / 1

20 Computing Integrals Using the fact that M 0 = 1 we get the recursion formula M k = 2p 1 1 2(2 k 1) l=0 k h l j=1 ( ) k l j M k j for k > 0. j With the M k we can easily compute x k, ϕ j l = x k ϕ j l (x)dx R =2 jk j/2 (x + l) k ϕ(x)dx R =2 jk j/2 k m=0 ( ) k l m M k m. m R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs20 / 1

21 Computing Integrals Therefore for any polynomial π the integral π, ϕ j l = π(x)ϕ j l (x)dx is exactly computable. For computing R f, ϕ j l where you know the function values of some nodes of f, you can do a polynomial interpolation with a polynomial π f and integrate π f (x)ϕ j l (x)dx f(x)ϕ j l (x)dx. R R R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs21 / 1

22 Computing Dϕ j, Dϕ j k Computing Integrals ϕ satisfies a refinement equation, too: Dϕ = k h k Dϕ 1 k. We can use this to compute a k = Dϕ, Dϕ k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs22 / 1

23 Computing Integrals a k = l =4 l =4 l h l h m Dϕ 1 l, Dϕ1 2k+m m h l h m 2k Dϕ, Dϕ m l m (h l h l+m 2k )a l m If you set H k,l = 4h l m h l+m 2k you have the eigen equation Ha = a i.e. you have to compute an eigenvector for the eigenvalue 1. It can be shown that this eigenvector is unique up to normalisation. To find the correct normalisation of the a ks we use the equation k 2 a k = 2, which holds for p 3. k R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs23 / 1

24 Computing Integrals For a j k = Dϕj, Dϕ j k you have a j k = 22j Dϕ, Dϕ k. Remark: It holds a j k = φj k (0). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs24 / 1

25 Computing ϕ, φ k Computing Integrals With the same trick as before you can compute b k = ϕ, φ k = h l h I m ϕ 1 l, φ1 2k+m l m =2 1/2 h l h I m 2k ϕ, φ m l l m =2 1/2 (h l h I l+m 2k )b l. l m R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs25 / 1

26 Computing Integrals With H k,l = 2 1/2 m hi l+m 2k we have the equation Hb = b, i.e. we search an eigenvector of H for the eigenvalue 1. Here you also have to find a further information to achieve a unique solution. For b j k = ϕj, φ j k we get b j k = 2 j/2 b k. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs26 / 1

27 Multivariate MRAs R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs27 / 1

28 Multivariate MRAs Until now we only considered one dimensional functions. If ϕ is a scaling function we define a d-dimensional scaling function Φ as follows. Φ(x 1,..., x d ) = d ϕ(x i ) i=1 For k = (k 1,..., k d ) Z d and j Z we define Then with Φ j k (x 1,..., x d ) = d ϕ j k i (x i ). i=1 V j = span{φ j k k Zd } = d i=1 V j we get a MRA for L 2 (R d ). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs28 / 1

29 Wavelet Spaces Multivariate MRAs The wavelet spaces are a little bit more complicated. With W j 0 = V j and W j 1 = W j we have V J = = d i=1 V J d (V J 1 W J 1 ) i=1 = d e {0,1} d i=1 W J 1 e i. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs29 / 1

30 Multivariate MRAs That means that in d dimensions we have 2 d 1 different wavelets. For example, for d = 3, we have ψ 1 =ϕ ϕ ψ, ψ 2 =ϕ ψ ϕ, ψ 3 =ϕ ψ ψ, ψ 4 =ψ ϕ ϕ, ψ 5 =ψ ϕ ψ, ψ 6 =ψ ψ ϕ, ψ 7 =ψ ψ ψ. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs30 / 1

31 Multivariate MRAs Isotropic and Anisotropic Structures If we pass to more levels, we can do it in two ways. 1. isotrop: V J = d J 1 V 0 i=1 j=0 e {0,1} d e 0 d i=1 W j e i 2. anisotrop: With the notation W 1 = V 0 V J = J 1 J 1 j 1 = 1 j 2 = 1 J 1 d j d = 1 i=1 W j i R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs31 / 1

32 Isotropic Decomposition Multivariate MRAs V 3 V 3 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs32 / 1

33 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 2 V 2 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs33 / 1

34 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 1 W 1 W 1 W 1 W 2 V 2 V 1 V 1 W 1 V 1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs34 / 1

35 Isotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 1 W 1 W 1 W 1 V 0 W 0 W 0 W 0 V 0 V 0 W 0 V 0 W 1 V 1 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs35 / 1

36 Anisotropic Decomposition Multivariate MRAs V 3 V 3 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs36 / 1

37 Anisotropic Decomposition Multivariate MRAs V 2 W 2 W 2 W 2 V 2 V 2 W 2 V 2 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs37 / 1

38 Anisotropic Decomposition Multivariate MRAs V 1 W 2 W 1 W 2 W 2 W 2 V 1 W 1 W 1 W 1 W 2 W 1 V 1 V 1 W 1 V 1 W 2 V 1 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs38 / 1

39 Anisotropic Decomposition Multivariate MRAs V 0 W 2 W 0 W 2 W 1 W 2 W 2 W 2 V 0 W 1 W 0 W 1 W 1 W 1 W 2 W 1 V 0 W 0 W 0 W 0 V 0 V 0 W 0 V 0 W 1 W 0 W 1 V 0 W 2 W 0 W 2 V 0 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs39 / 1

40 Galerkin and Collocation Schemes R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs40 / 1

41 Galerkin Scheme Galerkin and Collocation Schemes Given an equation Au f = 0 (2) for a given function f and operator A. We are looking for the solution u in a Hilbert space H. (??) is equivalent to Au f, v = 0 for all v H. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs41 / 1

42 Galerkin Scheme Galerkin and Collocation Schemes We want to approximate the solution u in a N-dimensional subspace of H and denote the approximated solution by u N. Let {v 1,..., v N } be an orthonormal basis for V N. We are getting the equations Au N f, v i = 0, i = {1,..., n}. We make the ansatz u N = N k=1 λ kv k. We define the matrix A and the vector f by A k,l = Av k, v l, f k = f, v k. Then we have to solve the linear system Aλ = f. R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs42 / 1

43 Collocation Scheme Galerkin and Collocation Schemes We make the ansatz u N = k u(x k )v k. Here the v k dont have to be orthonormal. Here are the Interpolets φ j k senseful. This leads to the equations By defining A and f by we get the linear system (Au N f)(x k ) = 0, k = 1,..., N. A k,l = (Av l )(x k ), f k = f(x k ) Au = f, with u k u(k). R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs43 / 1

44 References References S. Mallat, A Wavelet Tour of Signal Processing, 2nd. ed., Academic Press, 1999 A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier Science B.V., 2003 R. Schneider F. Krüger Daubechies Wavelets and Interpolating Scaling FunctionsNovember and Application 22, 2007 on PDEs44 / 1