Quasi-Monte Carlo integration for feedback control under uncertainty

TL;DR

This paper develops a quasi-Monte Carlo approach for efficiently computing feedback control laws in uncertain parabolic PDE systems, leveraging parametric regularity to outperform traditional Monte Carlo methods.

Contribution

It introduces a novel analysis of parametric regularity for Riccati-based feedback operators, enabling the application of higher-order QMC methods to control problems with infinite-dimensional uncertainty.

Findings

QMC methods achieve better error rates than Monte Carlo in this context.

The Riccati feedback operator depends analytically on parameters under certain conditions.

First study of Banach-space-valued integration by higher-order QMC methods.

Abstract

A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen--Lo\`eve expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC methods.