Model selection for inverse problems: Best choice of basis functions and model order selection

A complete solution for an inverse problem needs five main steps: choice of basis functions for discretization, determination of the order of the model, estimation of the hyperparameters, estimation of the solution, and finally, characterization of the proposed solution. Many works have been done for the three last steps. The first two have been neglected for a while, in part due to the complexity of the problem. However, in many inverse problems, particularly when the number of data is very low, a good choice of the basis functions and a good selection of the order become primary. In this paper, we first propose a complete solution within a Bayesian framework. Then, we apply the proposed method to an inverse elastic electron scattering problem. INTRODUCTION In a very general linear inverse problem, the relation between the data y = [y1, · · · ,ym] and the unknown function f(.) is yi = ∫ hi(r)f(r)dr, i= 1, · · · ,m, (1) where hi(r) is the system response for the data yi. We assume here that the hi(r) are known perfectly. The first step for any numerical processing is the choice of a basis function bj(r) and an order k, in such a way to be able to write