Approximate Solutions to Integral Equations by Wavelet Decomposition Methods

Abstract

We construct approximate solutions to Inverse Problems associated to equations of the form Af = g where A is an integral operator. For a given f , the Forward Problem consists in calculating its image through A, while the Inverse Problem looks for f for a given g. In order to solve the Inverse Problem we project the data into finite dimensional subspaces of wavelets in the context of a multiresolution analysis and solve the Foward Problem for each element of the basis by means of a Galerkin-type scheme. From these computations, we can accurately build a solution to the Inverse Problem based on properties of the chosen wavelets and suitable hyphotesis on the operator. We present examples related to fractional calculus.