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

Roman Chapko, Leonidas Mindrinos

TL;DR

This work addresses reconstructing the interior boundary $\Gamma_1$ of a cavity in a doubly connected planar domain from exterior Cauchy data for the 2D wave equation. It introduces a two-step dimension-reduction framework that first applies a Laguerre transform in time to produce a sequence of stationary boundary-value problems, then reformulates the solution via modified single-layer potentials into a nonlinear boundary integral system solved by a Newton-type iteration using the Fréchet derivative. The method derives explicit fundamental solutions $\Phi_n$ from Bessel functions, solves the well-posed linear subsystems with Nyström discretization, and treats the ill-posed parts with Tikhonov regularization, updating the interior boundary through a linearized Fréchet-derivative step. Numerical tests on apple-shaped contours demonstrate accurate, albeit initial-condition-sensitive, reconstructions from both exact and noisy data, validating the practicality of the approach for 2D hyperbolic inverse boundary problems and suggesting extensions to three dimensions.

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.

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

TL;DR

This work addresses reconstructing the interior boundary of a cavity in a doubly connected planar domain from exterior Cauchy data for the 2D wave equation. It introduces a two-step dimension-reduction framework that first applies a Laguerre transform in time to produce a sequence of stationary boundary-value problems, then reformulates the solution via modified single-layer potentials into a nonlinear boundary integral system solved by a Newton-type iteration using the Fréchet derivative. The method derives explicit fundamental solutions from Bessel functions, solves the well-posed linear subsystems with Nyström discretization, and treats the ill-posed parts with Tikhonov regularization, updating the interior boundary through a linearized Fréchet-derivative step. Numerical tests on apple-shaped contours demonstrate accurate, albeit initial-condition-sensitive, reconstructions from both exact and noisy data, validating the practicality of the approach for 2D hyperbolic inverse boundary problems and suggesting extensions to three dimensions.

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.
Paper Structure (9 sections, 37 equations, 2 figures)

This paper contains 9 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A two-step approach for dimension reduction
  3. Semi-discretization in time
  4. A boundary integral equation method
  5. The iterative scheme
  6. Numerical implementation
  7. Numerical results
  8. Conclusions
  9. Acknowledgements

Figures (2)

  • Figure 2: Reconstructions of the boundary $\Gamma_1$ of the rounded rectangle for exact data (left) and data with $3\%$ noise (right).
  • Figure 3: Reconstructions of the apple-shaped boundary $\Gamma_1$ for exact data (left) and data with $3\%$ noise (right).

Theorems & Definitions (1)

  • Definition 1