Probability-of-failure-based optimization for random PDEs through concentration-of-measure inequalities

¹ Department of Applied Mathematics and Statistics, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain
² Computational Mechanics and Scientific Computing Group, Technical University of Cartagena, Cartagena, Spain

^* Corresponding author: f.periago@upct.es

Received: 27 May 2022
Accepted: 22 October 2023

Abstract

Control and optimization problems constrained by partial differential equations (PDEs) with random input data and that incorporate probabilities of failure in their formulations are numerically extremely challenging, since the computational cost of estimating the tails of a probability distribution is prohibitive in many situations encountered in real-life engineering problems. In addition, probabilities of failure are often discontinuous and include huge flat regions where gradients vanish. Based on the McDiarmid concentration-of-measure inequality, this paper proposes a new functional which provides a tight and smooth bound for the probability of a given random functional of exceeding a prescribed threshold parameter. Hence, this approach relieves the above-mentioned difficulties in the case where the solution map is convex with respect to the random parameter, as in the case of a deterministic differential operator and the random parameter appearing linearly in the right-hand side term. Well-posedness of the corresponding optimal control problem is established and the viability of the proposed method is numerically illustrated by two benchmarks examples arising in topology optimization and optimal control theory.