MG/OPT and MLMC for Robust Optimization of PDEs

TL;DR

This paper introduces a combined MG/OPT and MLMC algorithm for efficiently solving robust PDE-constrained control problems with uncertainty, demonstrating improved convergence and reduced computational cost through hierarchical multilevel strategies.

Contribution

It develops a novel integrated MG/OPT and MLMC framework for robust PDE control, leveraging hierarchical structures to enhance efficiency and convergence.

Findings

Reduced number of samples on expensive levels

Faster convergence to the optimal solution

Significant computational time savings observed

Abstract

An algorithm is proposed to solve robust control problems constrained by partial differential equations with uncertain coefficients, based on the so-called MG/OPT framework. The levels in this MG/OPT hierarchy correspond to discretization levels of the PDE, as usual. For stochastic problems, the relevant quantities (such as the gradient) contain expected value operators on each of these levels. They are estimated using a multilevel Monte Carlo method, the specifics of which depend on the MG/OPT level. Each of the optimization levels then contains multiple underlying multilevel Monte Carlo levels. The MG/OPT hierarchy allows the algorithm to exploit the structure inherent in the PDE, speeding up the convergence to the optimum. In contrast, the multilevel Monte Carlo hierarchy exists to exploit structure present in the stochastic dimensions of the problem. A statement about the rate of convergence of the algorithm is proven, and some additional properties are discussed. The performance of the algorithm is numerically investigated for three test cases. A reduction in the number of samples required on expensive levels and therefore in computational time can be observed.