Multilevel methods and fast linear solvers for PDE-constrained optimization under uncertainty.

Tommaso Vanzan (Politecnico di Torino)

Abstract: In recent years there has been an increasing interest in optimization problems constrained by random partial differential equations, since the latter are effective mathematical tools to take into account intrinsic randomness or partial knowledge of systems under study.
The solution of such problems requires an extremely high computational cost, and thus motivates a very active area of research.
In particular, the use of multilevel/sparse techniques is not trivial in this context, since a direct multilevel/sparse approximation of the objective functional involves negative weights, which may eventually lead to a loss of convexity.

In the first part of the talk, we present a novel framework to use multilevel and sparse quadrature formulae, that still preserves the properties (e.g., convexity) of the continuous problem. Our approach consists in solving a sequence of optimization problems, each discretized with different levels of accuracy of the physical and probability spaces. The final approximation of the minimizer is obtained in a postprocessing step, by suitably combining the adjoint variables computed on the different levels.

In the second part, I will give an overview of selected contributions to fast inner solvers tailored for the extremely large KKT systems, based on algebraic, operator preconditioning and multigrid strategies.