An inverse obstacle problem with a single pair of Cauchy data: Laplace's equation case

TL;DR

This paper introduces a novel hybrid sampling and iterative method to uniquely recover a polygonal obstacle and constant conductivity in Laplace's equation from minimal boundary data, demonstrating efficiency through numerical tests.

Contribution

It presents a new hybrid approach combining sampling and iterative schemes for inverse obstacle problems with single Cauchy data, ensuring uniqueness and practical reconstruction.

Findings

Proved uniqueness under certain boundary data assumptions.

Developed an efficient hybrid reconstruction method.

Validated the approach with numerical examples.

Abstract

This paper is concerned with an inverse obstacle problem for the Laplace's equation. The aim is to recover the constant conductivity coefficient in the equation and the boundary of a Dirichlet polygonal obstacle from a single pair of Cauchy data. Uniqueness results are established under some a priori assumptions on the input boundary value data. A domain-defined sampling method, based on the factorization method originating from inverse acoustic scattering, has been proposed to recover both the constant conductivity coefficient and the polygonal obstacle. A hybrid strategy, which combines the sampling method and iterative scheme, is employed {\color{hgh}to reconstruct} the location and shape of the obstacle. Numerical examples indicate that our method is efficient.