A new coupled complex boundary method (CCBM) for an inverse obstacle problem

Abstract

In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion ω contained in a larger bounded domain Ω via boundary measurements on ∂Ω. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [16]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion ω. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain ω. We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.

On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems

“…To analyze the shape Hessian (30), we will write it into an equivalent expression (see [22,23,24,25,26]) and adapt the method already used in [1,2]. We first introduce the operators L : [53,Sec.…”

“…In this work, we are primarily interested in solving the free boundary problem (FBP) (1) through the novel application of the so-called coupled complex boundary method or CCBM in solving stationary FBPs through the context of shape optimization. For simplicity of discussion, we will consider the prototypical case of (1) in two spatial dimensions popularly known as the exterior Bernoulli problem wherein f ≡ 0, g ≡ 1, and h = λ, where λ < 0 is a fixed constant in (1). That is, we consider the problem of finding a pair (Ω, u) := (Ω, u(Ω)) that solves the system −∆u = 0 in Ω, u = 1 on Γ, u = 0 and ∂ n u = λ on Σ.…”

“…In this paper, we want to offer yet another shape optimization reformulation of (2) which, of course, also applies to (1). The idea is somewhat similar to [52,63], but applies the concept of complex PDEs.…”

“…It is later on applied to an inverse conductivity problem with one measurement in [32] and also to parameter identification in elliptic problems in [65]. Much more recently, CCBM was also applied in solving inverse obstacle problems by Afraites in [1]. To the best of our knowledge -as the method has not been applied yet to solving FBPs in previous investigations -this is the first time that CCBM will be explored to deal with stationary FBPs, particularly as a numerical resolution to the exterior Bernoulli problem.…”

Shape reconstruction for advection-diffusion problems by shape optimization techniques: The case of constant velocity

“…For other reconstruction problems, but without the presence of the advection term, we refer readers, for example, to [17,18,20] for inverse scattering problems and [8,7,13,14] for inverse boundary problems involving the (time-dependent) heat equation. More related studies can also be found in [1,2,3] and the references therein.…”