Showing papers by "Antoine Laurain published in 2013"
30 Nov 2013
TL;DR: In this article, the authors describe the advantages of the distributed shape derivative in terms of generality, easeiness of computation and numerical implementation, and give several examples of numerical applications such as the inverse conductivity problem and the level set method.
Abstract: The well-known structure theorem of Hadamard-Zol\'esio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with shapes having a lower regularity.
It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method.
18 citations
TL;DR: In this paper, the shape optimization problem is formulated as a shape optimization, where the interface between the fluorescent inclusion and the background constitutes the unknown shape and the associated inverse problem is stabilized by a priori properties or information.
Abstract: Fluorescence tomography is a non-invasive imaging modality that reconstructs fluorophore distributions inside a small animal from boundary measurements of the fluorescence light. The associated inverse problem is stabilized by a priori properties or information. In this paper, cases are considered where the fluorescent inclusions are well separated from the background and have a spatially constant concentration. Under these a priori assumptions, the identification process may be formulated as a shape optimization problem, where the interface between the fluorescent inclusion and the background constitutes the unknown shape. In this paper, we focus on the computation of the so-called topological derivative for fluorescence tomography which could be used as a stand-alone tool for the reconstruction of the fluorophore distributions or as the initialization in a level-set-based method for determining the shape of the inclusions.
17 citations
TL;DR: It is proved that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual.
Abstract: In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ?u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0---1 values, which amounts to solving a topology optimization problem with volume constraint.
6 citations