An inverse source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

TL;DR

This paper introduces a novel numerical approach for inverse source problems in hyperbolic equations, utilizing a Volterra integral and the quasi-reversibility method, with proven Lipschitz-like convergence and successful numerical reconstructions.

Contribution

It develops a new method combining Volterra integral equations and quasi-reversibility for hyperbolic inverse problems, with convergence analysis using Carleman estimates.

Findings

Convergence of the regularized solution with Lipschitz-like rate.

Numerical tests show accurate source reconstructions.

Method applicable to general hyperbolic equations.

Abstract

We propose in this paper a new numerical method to solve an inverse source problem for general hyperbolic equations. This is the problem of reconstructing sources from the lateral Cauchy data of the wave field on the boundary of a domain. In order to achieve the goal, we derive an equation involving a Volterra integral, whose solution directly provides the desired solution of the inverse source problem. Due to the presence of such a Volterra integral, this equation is not in a standard form of partial differential equations. We employ the quasi-reversibility method to find its regularized solution. Using Carleman estimates, we show that the obtained regularized solution converges to the exact solution with the Lipschitz-like convergence rate as the measurement noise tends to 0. This is one of the novelties of this paper since currently, convergence results for the quasi-reversibility method are only valid for purely differential equations. Numerical tests demonstrate a good reconstruction accuracy.