Shape optimization method for an inverse geometric source problem and stability at critical shape

Abstract

This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

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

“…and u N is the solution of Neumann problem (4). The cost function J KV measures the gap of energy between the solutions of the Dirichlet and Neumann problems corresponding to the data given above.…”

“…Remark 1. The new method allows us to define the cost function J in the whole domain Ω\ω which brings advantages of robustness in the reconstruction such as the Kohn-Vogelius cost function J KV compared to the Least Squares fitting J LS which is defined only on the boundary ∂Ω (see [1,2,4]). Compared to the Kohn-Vogelius method, the latter requires two problems to be solved at each iteration, however the new method (CCBM), needs a single complex problem to be solved.…”

On the numerical approximation of some inverse problems governed by nonlinear delay differential equation

“…A particular interest has been established mathematical models describing the immune response during infectious diseases which are formulated as systems of nonlinear delaydifferential equations (DDEs) characterized by multiple constant delays, moderate size, and stiffness [4,5]. This paper deals with a topic that has become increasingly relevant in current research: inverse problems [1][2][3]20,24].…”

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

“…The problem in consideration is also known in the literature as the Alt-Caffarelli problem [3]. It originates from the description of free surfaces for ideal fluids [30], but copious industrial applications leading to similar formulations to (2) arises in many other related contexts, see [9,28,29]. We mention that, in this paper, we do not tackle the question of existence of optimal shape solutions for the proposed shape optimization problem.…”

“…The methods of shape optimization is a well-established tool to solve free boundary problems, and in the case of (2), the method can be applied in several ways. The usual strategy is to choose one of the boundary conditions on the free boundary to obtain a well-posed state equation and then track the remaining boundary data in L 2 (Σ) (see [24,25,35,37,42,55,56]) or utilize the Dirichlet energy functional as a shape functional (see [23,34]).…”