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

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.

Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems

“…Inverse source problems in wave propagations in a single domain are well studied (see, e.g., [13,15,17,21,22]). In contrast, there are no results in the literature for inverse hyperbolic problems on separate intervals.…”

“…In addition, the solution may be unstable [10][11][12][13][14]. The ill-posedness is usually overcome by using regularization methods [10,12,[14][15][16][17]. Another often used approach is to transform the inverse problem to a well-posed direct problem [14,[18][19][20].…”

“…Recently, many investigations have been devoted to inverse problems for PDEs, in particular, inverse problems for hyperbolic-type equations. Inverse problems for HPDEs in the connected domain are well studied in the literature (see, e.g., [13,15,17,21] and references therein).…”

“…The authors of [15] introduce a two-dimensional wave equation on a double connected domain and study the inverse problem of recovering the interior boundary curve from overspecified data on the exterior boundary. The inverse problem is formulated as a system of boundary integral equations.…”

“…The motivation is explained by the capacity of the hyperbolic equations to describe a large number of problems in elasticity, thermoelasticity and wave propagation (see, e.g., [2][3][4][5][6]12,18,22]). Also, it comes from the theoretical and numerical challenges of the present specific problem (see, e.g., [5,[8][9][10][11][12]15,17,18,21,23]).…”