The variational approach to mollification for inverse problems

Abstract

Mollification is a standard way to approximate functions in various senses. In the past two or three decades, the notion of mollification has also been used to solve ill-posed linear equations. A corner stone in the development of this idea is probably the well-known approximate inverses, introduced in 1990 by Louis and Maass. In the specific field of deconvolution, Lannes et al. initiated a variational formulation of a similar regularization principle. Until recently, this variational approach lacked generality and some mathematical foundation: generality since it was designed for deconvotution (or Fourier synthesis) only, and mathematical foundation since no result on the behavior of the solution when letting the natural regularization parameter go to zero had been obtained (or even questioned). Our work over the past few years focused in part on these aspects. The purpose of this talk is twofold: one the one hand, we will give an overview of the aforementioned extension and theoretical foundation of the variational approach to mollification, and on the other hand, we shall describe some recent results on approximate intertwining relationships of operators, which are central to the implementation of the methodology in the general case. [1] A. K. Louis & P. Maass, A mollifier method for linear operator equations of the first kind, Inverse Problems, 1990. [2] A. Lannes et al., Stabilized reconstruction in signal and image processing, J. Mod. Opt., 1987.