Convergence of a Two-Level Ideal Algorithm for a Parametric Shape Optimization Model Problem

The numerical approximation of the solution of a PDE is generally obtained with the resolution of a system of equations (linear or nonlinear) that comes from the discretization of the PDE on a given domain. The resulting system may be stiff, partly due to the approximation of differential operators, and therefore makes the iterative methods harder to converge. In order to overcome this difficulty, the classical (or geometrical) multigrid strategies aim at preconditioning this system through the use of coarser representations (grids). Equivalently, the numerical treatment of an optimization problem is potentially subject to stiffness difficulties. In the framework of a parametric shape optimization problem, hierarchical representations can be used to enhance the multivel strategies to this context. In this paper, by analogy with the Poisson equation (elliptic linear PDE), which is the typical example for linear multigrid methods, we address a convex parametric shape optimization model problem. We describe the ideal cycle of a two-level algorithm adapted to shape optimization problems relying on appropriate transfer operators (prolongation and restriction). The efficiency of a multigrid strategy is ensured by a mesh-independent convergence rate. With the help of a symbolic calculus software we show that this is indeed the case (we derive a convergence rate which is independent of the dimension of the parametric representation). Moreover this rate is ``small'' (smaller than the convergence rate of basic iterative methods such as Jacobi, Gauss-Seidel, etc.). Numerical examples are worked out and corroborate the theoretical results