Tommaso Vanzan: Sparse and multilevel approximations for PDE-constrained optimization under uncertainty

Abstract:

In this talk we are concerned with the minimization of the expected value of a convex functional constrained by a random partial differential equation. 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.

We here present a novel and alternative framework for using multilevel and sparse quadrature formulae (presented in [Nobile, Vanzan, A combination technique for optimal control problems constrained by random PDEs, SIAM/ASA J. on UQ., 2024] and [Nobile, Vanzan, Multilevel quadrature formulae for the optimal control of random PDEs, arXiv:2407.06678, 2024]), 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. We will discuss a complete convergence analysis for multilevel quadrature formulae, and present numerical experiments confirming the better computational complexity of our multilevel approach, even beyond the theoretical assumptions.