On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains

Roman Chapko,Leonidas Mindrinos

doi:10.3390/math10050750

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 Fréchet 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.

Highlights

The inverse problem of reconstructing part of a boundary of an object from overdetermined measurements on the accessible part of the boundary has attracted a great deal of attention in different research areas because of its importance in various applications [1–4].This problem is related to the solution of partial differential equations (PDEs) and, because of its non-linearity and ill-posedness, is rather complicated in both theoretical and numerical aspects.Most numerical methods for such kind of problems provide iterative methods with regularization techniques
A special potential representation of the solution led to a sequence of non-linear integral equations. We extend this approach to an inverse boundary problem for a hyperbolic PDE
We extended the integral equation method to the inverse hyperbolic problem of the reconstruction of the interior boundary curve given the Cauchy data on the exterior boundary of a doubly connected planar domain

Summary

Introduction

The inverse problem of reconstructing part of a boundary of an object from overdetermined measurements on the accessible part of the boundary has attracted a great deal of attention in different research areas because of its importance in various applications [1–4] This problem is related to the solution of partial differential equations (PDEs) and, because of its non-linearity and ill-posedness, is rather complicated in both theoretical and numerical aspects. The regularized Newton method is used, which requires in every step the numerical solution of the direct problem These well-posed time-dependent direct problems are reduced to integral equations using heat potentials.

Semi-Discretization in Time

A Boundary Integral Equation Method

The Iterative Scheme

Numerical Implementation

Numerical Results

Conclusions