Discretization and regularization for the reconstruction of inhomogeneities by scattering measurements

TL;DR

This paper develops a discretized and regularized variational approach for reconstructing inhomogeneities from acoustic scattering measurements, providing convergence analysis for parameter choices to approximate the true solution.

Contribution

It introduces a fully discrete variational framework with regularization for inverse scattering problems and analyzes parameter selection for accurate reconstruction.

Findings

Convergence of the discrete regularized solution to the true inhomogeneity.

Guidelines for choosing measurement, regularization, and discretization parameters.

Validation of the approach through theoretical analysis.

Abstract

We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem with regularization. Such a problem depends on a variety of parameters, that is, the number of measurements, the regularization parameter and the discretization parameter, namely the size of the mesh on which we discretize the unknown coefficients of the Helmholtz type equation modelling our physical system. We show, through a convergence analysis, that one can carefully choose these parameters in such a way that the solution to this discrete regularized minimum problem is a good approximation of the looked-for solution to the inverse problem.