Wavelets in Scattering Calculations

Transcription

1 Wavelets in Scattering Calculations W. P., Brian M. Kessler, Gerald L. Payne The University of Iowa Wavelets in Scattering Calculations p.1/43

2 What are Wavelets? Orthonormal basis functions. Compact support. Local pointwise representation of low-degree polynomials. Generated from a single function by translations and scale transforms. Wavelets in Scattering Calculations p.2/43

3 Why are Wavelets Interesting? Efficient representation of information. Used in the FBI s fingerprint archive. Used in the JPEG2000 image compression algorithm. Fast reconstruction of information. Natural basis for functions with structure on multiple scales. Wavelets in Scattering Calculations p.3/43

4 Physics Applications Scattering with large energy/momentum transfers requires a relativistic quantum treatment. Relativistic quantum models are naturally formulated in momentum space Wavelets in Scattering Calculations p.4/43

5 Physics Applications Momentum-space few-body equations for realistic systems have large dense matrices. Wavelet bases lead to equivalent linear equations with sparse matrices Wavelets in Scattering Calculations p.5/43

6 Useful Properties Wavelets are a local basis. Wavelets are an orthonormal basis. Basis functions never have to be computed. The fast wavelet transform automatically eliminates unimportant basis functions. Wavelets can locally pointwise represent polynomials. Wavelets lead to efficient treatment of scattering singularities. Wavelets in Scattering Calculations p.6/43

7 Interesting Properties Wavelets are fractals. Basis functions are generated from a single Mother function by translations and dyadic scale changes. Related to solution ( Father function = scaling function ) of a linear renormalization group equation. Wavelets in Scattering Calculations p.7/43

8 Main Elements of Wavelet Theory Dyadic scale changes: Integer translations: Dξ(x) := 1 2 ξ( x 2 ) T ξ(x) := ξ(x 1) These unitary transformations are used to generate bases from a single function. Wavelets in Scattering Calculations p.8/43

9 The Scaling Equation The scaling equation is the most important equation for numerical applications. The solution, φ(x), is called the scaling function: Dφ(x) = l h l T l φ(x) The quantities h l are numerical coefficients that determine the type of wavelet. Wavelets in Scattering Calculations p.9/43

10 Scaling Bases For each resolution, m, scaling basis functions are defined as φ mn (x) := D m T n φ(x) (φ mn, φ mn ) = δ nn The approximation space V m with resolution m is defined by V m := Span(φ mn ) L 2 (R) Wavelets in Scattering Calculations p.10/43

11 Multiresolution Analysis The scaling equation implies the relations among the approximation spaces: V 0 V 1 L 2 (R) V m 1 V m V m+1 { } Wavelet spaces W m are defined by V m 1 = V m W m. V m = V m+k W m+k W m+k 1 W m+1. Wavelets in Scattering Calculations p.11/43

12 The Mother Wavelet ψ(x) The Mother wavelet is a function in V 1 that generates a basis in W 0 by integer translations. It is defined by Dψ = l g l T l φ; g l := ( ) l h k l k odd. Wavelet basis functions ψ mn are defined by ψ mn (x) := D m T n ψ(x); (ψ mn, ψ m n ) = δ mm δ nn. Wavelets in Scattering Calculations p.12/43

13 The Scaling Coefficients The Daubechies scaling coefficients, h l, of order K are fixed by the requirements: ψ(x)x l = 0; l = 0, 1,, K. It follows that m, n: ψ mn (x)x l = 0; l = 0, 1,, K. Everything can be expressed in terms of the scaling coefficients. Wavelets in Scattering Calculations p.13/43

14 The Scaling Coefficients The scaling coefficients, h l, are the solution to the equations: 2K 1 h l = 2 2K 1 h l h l 2n = δ n0 l=0 l=0 2K 1 l=0 l m ( ) l h k l = 0 The solutions for K = 1, 2, 3 are: m = 0,, K Wavelets in Scattering Calculations p.14/43

15 Daubechies Scaling Coefficients h l K=1 K=2 K=3 h 0 1/ 2 (1 + 3)/4 2 ( )/16 2 h 1 1/ 2 (3 + 3)/4 2 ( )/16 2 h 2 0 (3 3)/4 2 ( )/16 2 h 3 0 (1 3)/4 2 ( )/16 2 h ( )/16 2 h ( )/16 2 Wavelets in Scattering Calculations p.15/43

16 Computation of φ(x) and ψ(x) The scaling function at integer points can be obtained by solving the linear equations: φ(n) = 2K 1 2hl φ(2n l) φ(n) = 1 l=0 n The support of φ(x) [0, 2K 1]. φ(n) = 0 n 0 or n 2K 1. Wavelets in Scattering Calculations p.16/43

17 Computation of φ(x) and ψ(x) The φ(n) can be used as input to recursively calculate φ(x) and ψ(x) at all dyadic rationals using φ( n 2K 1 2 ) = k l=0 ψ( n 2K 1 2 ) = k l=0 2hl φ( n l) 2k 1 2gl φ( n l) 2k 1 Wavelets in Scattering Calculations p.17/43

18 K = 3 Wavelet and Scaling Function Scaling Function Wavelet Function Wavelets in Scattering Calculations p.18/43

19 Approximation Spaces There are two approximation spaces related by a fast orthogonal transformation H = V m W m+1 W m+2 W m+k V m+k with orthonormal bases {φ mn (x)} n= {φ m+k n (x)} n= {ψ ln (x)},m+k n=,l=m+1 Wavelets in Scattering Calculations p.19/43

20 Local Polynomials L 2 (R) = W m 2 W m 1 W m V m ψ mn x l = 0 m, n; l = 0, 1,, K x l = n c n φ mn (x) pointwise Wavelets in Scattering Calculations p.20/43

21 Unit Translates 2 Daubechies 2 Scaling Functions Translated Wavelets in Scattering Calculations p.21/43

22 Constant Function 2 Summing Daubechies 2 Wavelets to Represent a Constant Function Wavelets in Scattering Calculations p.22/43

23 Linear Function 6 Summing Daubechies 2 Wavelets to Represent a Linear Function Wavelets in Scattering Calculations p.23/43

24 Wavelet Numerical Analysis The fractal nature of wavelets makes standard numerical techniques inefficient. The scaling equation replaces all numerical methods. The key elements of wavelet numerical analysis are the compactness of the basis functions and the ability to exactly compute moments: x m φ := (x m, φ) = x m φ(x)dx Wavelets in Scattering Calculations p.24/43

25 Moments and Scaling The normalization condition gives: x 0 φ = 1; The scaling equation gives: x m φ = (x m, φ) = (Dx m, Dφ) = 1 2 m+1/2 2K 1 l=0 h l (x m, T l φ) these equations recursively determine all moments in terms of the scaling coefficients. Wavelets in Scattering Calculations p.25/43

26 Moments and Scaling Similar methods can be used to get EXACT values for (x l, φ mn ) (x l, ψ mn ) ( dl φ mn dx l, φ m n ) (dl φ mn dx l, ψ m n ) (dl ψ mn dx l, φ m n ) (dl ψ mn dx l, ψ m n {x k, w k } N k=1 ; (φ mn, φ m l φ m l ) φ(x)p 2N 1 (x)dx = N P 2N 1 (x l )w l as well as many other useful quantities; including quantities involving finite limits of integration. l=1 Wavelets in Scattering Calculations p.26/43

27 One-Point Quadrature For the Daubechies wavelets x 1 2 φ = x2 φ This means that P (x)φ(x)dx = P ( x 1 φ ) is exact for P (x) = a + bx + cx 2 This is a useful quadrature rule for scaling functions with small support (fine resolution). Wavelets in Scattering Calculations p.27/43

28 Scattering Singularities 1 I n := ( x ± i0 +, T n φ) I n := (D 1 x ± i0 +, DT n φ) = (D 1 x ± 0 +, T 2n Dφ) = 2K 1 2 l=0 h l I 2n+l iπ = a a 1 x ± i0 + = I n + endpoint terms I n = 1 n m=0 ( 1) m n m (xm, φ) 1 n m max m=0 ( 1) m n m xm n large Wavelets in Scattering Calculations p.28/43

29 Scattering Singularities These linear equations can be solved for I n ; The partial moments needed to treat endpoint integrals can be determined exactly. The method can also be applied to integrate the logarithmic singularities that appear in associated Legendre functions. Wavelets in Scattering Calculations p.29/43

30 Integrals over the Singularity Example of calculated singular I n s for the Daubechies K = 3 scaling function: K=3 I ± k I 1 ± i I 2 ± ±i I 3 ± i I 4 ± i Wavelets in Scattering Calculations p.30/43

31 Wavelet Transform The Wavelet transform is the real orthogonal transformation that relates the scaling function basis on V m to the wavelet basis on W m+1 W m+2 W m+k V m+k The matrix elements of a smooth kernel in the wavelet basis is the sum of a sparse matrix and a matrix with small norm: K mn = φ n (x)k(x, y)φ m (y)dxdy = S mn + mn < ɛ The wavelet approximation is to ignore the small matrix. Wavelets in Scattering Calculations p.31/43

32 Wavelet Transform There is a fast algorithm for implementing the wavelet transform that treats the coefficients h l and g l as coefficients of a filter. There is an algorithm ( the Kessler Algorithm ) that can implement the wavelet transform without storing the entire matrix. Wavelets in Scattering Calculations p.32/43

33 Solving the L-S Equation Choose a finest resolution = 1/2 j V j (we use K = 3). Transform [0, ] to a finite interval with the singularity at zero. Expand the solution in the scaling basis on Vj. Use the one point quadrature for the regular integrals and the In for the singular integrals. Use the fast-wavelet transform to transform to the equivalent wavelet basis. Discard terms with small matrix elements. Solve the resulting sparse-matrix linear equation. Invert the solution using the fast wavelet transform. Insert the solution vector back in the integral equation using the one-point quadrature rule and the I n. The resulting solution does NOT require the computation of the basis functions. Wavelets in Scattering Calculations p.33/43

34 Model - Malfliet-Tjon V H = p2 2m + V V (r) = λ 1 e µ 1r r + λ 2 e µ 2r r 1/2m λ 1 µ 1 λ 2 µ MeV fm MeV fm 1.55 fm MeV fm 3.11 fm 1 example: s-wave half on-shell K-matrix Wavelets in Scattering Calculations p.34/43

35 Structure of equation: f(x) = g(x) + K(x, y) f(y)dy y f = f n φ n (x) x n = x 1 φn f m = g(x m ) + n ( K(xm, y n ) K(x m, 0) ) y n + K(x m, 0)I n ) f n f(x) = g(x) + n ( K(x, yn ) K(x, 0) y n + K(x, 0)I n f n Wavelets in Scattering Calculations p.35/43

36 Transformed Kernel Wavelets in Scattering Calculations p.36/43

37 Transformed K-matrix u b Wavelets in Scattering Calculations p.37/43

38 Why does it work? Consider the expansion of f(x) in the wavelet basis f(x) = n= d n φ mn (x) + m n= l=m k c ln ψ ln (x); c ln = 2 l (n+2k 1) 2 l n ψ ln (x)f(x)dx c ln vanishes if f(x) can be represented by a polynomial of degree K on [2 l n, 2 l (n + 2K 1)]. Wavelets in Scattering Calculations p.38/43

39 K = 3, E = 10 MeV -J N series on-shell interpolated on-shell Wavelets in Scattering Calculations p.39/43

40 K = 3, E = 80MeV -J N series on-shell interpolated on shell Wavelets in Scattering Calculations p.40/43

41 Sparse Matrix Convergence K=3, E=10 MeV, J=-7 ɛ percent on-shell value on-shell error mean-square error Wavelets in Scattering Calculations p.41/43

42 Sparse Matrix Convergence K=3, E=80, MeV J=-7 ɛ percent on-shell value on-shell error mean-square error Wavelets in Scattering Calculations p.42/43

43 Conclusions Wavelet bases can be used to accurately solve the equations of scattering theory in momentum space. A new type of numerical analysis, called wavelet numerical analysis, which is based on scaling and support properties of the basis functions, is used for accurate numerical calculations. Wavelet numerical analysis leads to an accurate treatment of the scattering singularities. The fast wavelet transform coupled with the Kessler Algorithm leads to a sparse-matrix representation of the kernel requiring a minimal amount of storage. It automatically identifies irrelevant basis functions. Our calculations show that a 96% reduction in the size of the matrix results in mean square error of about 1 part in We are currently testing the method for two-body scattering without partial waves. Other promising applications include relativistic few-body equations and the numerical solution of renormalization group equations. Wavelets in Scattering Calculations p.43/43