Convexification method for a coefficient inverse problem and its performance for experimental backscatter data for buried targets

TL;DR

This paper introduces a globally convex numerical method for 3D coefficient inverse problems using Carleman weights, demonstrating effective reconstruction of buried targets from experimental backscatter data.

Contribution

The paper develops a novel convexification approach for inverse problems, leveraging Carleman estimates to ensure global convergence and improve reconstruction accuracy.

Findings

Method successfully reconstructs buried objects from experimental data.

Global strict convexity ensures convergence to the true solution.

Numerical results validate the method's effectiveness.

Abstract

We present in this paper a novel numerical reconstruction method for solving a 3D coefficient inverse problem with scattering data generated by a single direction of the incident plane wave. This inverse problem is well-known to be a highly nonlinear and ill-posed problem. Therefore, optimization-based reconstruction methods for solving this problem would typically suffer from the local-minima trapping and require strong a priori information of the solution. To avoid these problems, in our numerical method, we aim to construct a cost functional with a globally strictly convex property, whose minimizer can provide a good approximation for the exact solution of the inverse problem. The key ingredients for the construction of such functional are an integro-differential formulation of the inverse problem and a Carleman weight function. Under a (partial) finite difference approximation, the global strict convexity is proven using the tool of Carleman estimates. The global convergence of the gradient projection method to the exact solution is proven as well. We demonstrate the efficiency of our reconstruction method via a numerical study of experimental backscatter data for buried objects.