Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

TL;DR

This paper develops a quasi-Monte Carlo method for solving parabolic PDE-constrained optimal control problems under uncertainty, effectively handling high-dimensional integrals involving risk measures with improved accuracy over traditional Monte Carlo methods.

Contribution

The paper introduces a tailored QMC approach for high-dimensional stochastic integrals in PDE-constrained control problems with risk measures, achieving near-linear error rates regardless of stochastic dimension.

Findings

QMC method outperforms Monte Carlo in accuracy and efficiency

Error rate is essentially linear under moderate assumptions

Numerical results confirm the method's effectiveness

Abstract

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem -- and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.