Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

TL;DR

This paper introduces the first globally convergent numerical method for a challenging coefficient inverse problem in wave equations, proving uniqueness and demonstrating high accuracy with noisy data.

Contribution

It develops a new convexification-based numerical method for a specific inverse problem, establishing global convergence and uniqueness within an approximate model.

Findings

The method achieves high reconstruction accuracy with noisy data.

A new approximate mathematical model is validated by computational experiments.

Global convergence is proven without requiring a good initial guess.

Abstract

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the $n-$d, $n\geq 2$ wave equation with the unknown potential in the most challenging case when the $\delta -$ function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an $n-$d ($n=2,3$) extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.