On the numerical solution of a hyperbolic inverse boundary value problem in bounded domains

TL;DR

This paper presents a numerical method combining Laguerre transform and integral equations to accurately reconstruct cavity boundaries in bounded domains from wave equation data, with stability achieved through Tikhonov regularization.

Contribution

It introduces a novel iterative scheme that linearizes the inverse boundary problem and employs specialized quadrature and regularization techniques for stable solutions.

Findings

Accurate boundary reconstructions demonstrated in numerical tests

Method effectively stabilizes ill-conditioned systems

Combines Laguerre transform with boundary integral equations

Abstract

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fr\'echet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.