Wavelets As Galerkin Basis Aleksandr Yakovlev Department of Computational Physics St Petersburg State University.

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Presentation transcript:

Wavelets As Galerkin Basis Aleksandr Yakovlev Department of Computational Physics St Petersburg State University

Contents 1.Introduction 2.Galerkin Method 3.Multiresolution analysis 4.Multiresolution analysis and Galerkin Method 5.Solution for an example equations 5.1 Finite differences method 5.2 Wavelet Galerkin solution 5.3 Incorporation of boundary conditions 5.4 Offsetting boundary conditions to control error 5.5 Comparison of results

Introduction 1.Differential equations ODE PDE constant coefficients variable coefficients 2.Domain and boundary conditions Dirichlet Neuman Cyclic 3.Method Galerkin method 4.Improvements Use Wavelet basis Increase resolution Increase order

The Galerkin Method

Multiresolution Analysis

Daubechies D6 scaling function

Multiresolution Analysis and Galerkin method

Finite difference solution to DE

Finite difference solution to DE (1)

Wavelet Galerkin Method

Wavelet Galerkin Method (1)

Wavelet Galerkin Method (2)

Wavelet Galerkin Method (3)

Wavelet Galerkin Method (4)

Incorporation of boundary conditions

Offsetting boundary conditions to control error

D12 wavelet coefficients of the delta function

Error in wavelet solution to boundary value problem

Offsetting of boundary sources to control error

Error in wavelet solution to boundary value problem with offset sources

Comparsion of results

Decay in error of wavelet and finite difference solutions with increasing sample size

Variation of computation time with increasing sample size

Acknowlegements Prof. S.Yu. Slavyanov, St Petersburg State Univercity Prof. A.V. Tsiganov, St Petersburg State Univercity Prof. S.L. Yakovlev, St Petersburg State Univercity And other colleagues of mine from the Department of Computational Physics