Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

作者：Fernandez L; Novotny A A; Prakash R^*

来源：Numerical Functional Analysis and Optimization, 2018, 39(9): 937-966.

DOI：10.1080/01630563.2018.1432645

• Modified Helmholtz equation
• noniterative reconstruction method
• potential inverse problem
• topological derivatives
• topology optimization

This paper deals with an inverse potential problem posed in two dimensional space whose 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 measurements of the associated potential. Since the inverse problem, we are dealing with, 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 is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies 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 expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.

• 出版日期2018