Convexification of a 3-D coefficient inverse scattering problem

TL;DR

This paper develops a globally convergent convexification method for a 3D inverse scattering problem using Carleman weights, demonstrating effective numerical performance without smallness assumptions or iterative tail function updates.

Contribution

It introduces a new convexification approach for 3D inverse scattering that guarantees global convergence without smallness constraints or tail function iterations.

Findings

Method converges globally for the 3D Helmholtz inverse problem.

Numerical tests show good performance of the algorithm.

No smallness assumption or tail function iteration needed.

Abstract

A version of the so-called "convexification" numerical method for a coefficient inverse scattering problem for the 3D Hemholtz equation is developed analytically and tested numerically. Backscattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.