A combination technique for optimal control problems constrained by random PDEs

TL;DR

This paper introduces a combination technique that efficiently solves optimal control problems constrained by random PDEs by linearly combining solutions from low-fidelity models, achieving high accuracy with reduced computational cost.

Contribution

The paper presents a novel combination technique using mixed differences of spatial and stochastic approximations for efficient solution of random PDE-constrained optimal control problems.

Findings

Achieves accuracy comparable to fine tensor product methods

Reduces computational cost significantly

Validates effectiveness through numerical experiments

Abstract

We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the (possible) convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.