A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

Abstract

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.

Imaging of small penetrable obstacles based on the topological derivative method

“…Next, we briefly point out the main challenges faced in the solution of the inverse scattering problem investigated here in comparison to the inverse potential problem previously studied in the paper by authors (Fernandez et al, 2019). In fact, the new ideas are used in the asymptotic analysis involved in obtaining the topological derivatives of the cost functional for the inverse scattering problem because:…”

“…The inverse potential problem analyzed in the paper by Fernandez et al (2019) involves only real-valued functions, which are solutions of boundary value problems governed by a coercive partial differential equation named as modified Helmholtz equation. In this way, the estimate of auxiliary states was simpler, which is not the case in the current setting.…”

“…Building on the original ideas introduced in the previous paper by the authors (Fernandez et al, 2019) in the context of an inverse potential problem, this study aims to solve an inverse scattering problem by proposing a reconstruction method based on the concept of topological derivatives. The key novelties of this paper are as follows:…”

Reconstruction algorithms of an inverse geometric problem for the modified Helmholtz equation

“…We consider an inverse geometric problem for the modified Helmholtz equation to determine unknown obstacles from a single pair of boundary Cauchy data on an exterior surface . Over the past few decades, there has been extensive studies in numerical methods for the modified Helmholtz equation (Fernandez et al 2018(Fernandez et al , 2019Hua et al 2017), which arises in many important fields of physics and engineering (Lesnic and Bin-Mohsin 2012;Liang and Subramaniam 1997;Callihan and Wood 2012;Bakker and Kuhlman 2011;Politis et al 2002). In this paper, what we are interested in using reconstruction algorithms to detect the salient features of unknown obstacles within a body.…”

“…Ammari and Uhlmann (2004) proved that the knowledge of the partial Cauchy data for the Schrödinger equation on any open subset of the boundary determines uniquely the potential q provided that q is known in a neighborhood of the bound. Fernandez et al (2018Fernandez et al ( , 2019 employed noniterative reconstruction method to deal with an inverse geometric problem governed by the modified Helmholtz equation. Bin-Mohsin and ; Lesnic and Bin-Mohsin (2012) applied the meshless method of fundamental solutions to determine an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from Cauchy data for the modified Helmholtz equation.…”