Convergence of a two-level ideal algorithm for a parametric shape inverse model problem

Abstract

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of Multigrid (MG) methods. In this article, by analogy with the Poisson equation, which is the typical example for linear MG methods, we address a parametric shape inverse problem. We describe the ideal cycle of a two-level algorithm adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we show that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design are realized. Finally, some connections with the direct and inverse Broyden–Fletcher–Goldfarb–Shanno preconditioning methods are shown.