The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

TL;DR

This paper introduces a numerical approach combining Fourier series expansion and the quasi-reversibility method to solve an inverse source problem for hyperbolic equations, especially in scenarios with wave reflection where traditional methods fail.

Contribution

The paper develops a novel numerical method using Fourier series and quasi-reversibility for inverse hyperbolic problems with wave reflection, including convergence proof and numerical validation.

Findings

Convergence of the quasi-reversibility method as noise decreases

Significance of the special orthogonal basis in the solution

Numerical validation of the proposed approach

Abstract

We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo- acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthogonal basis of $L^2$. Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.