Properties of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization Problems
Complex systems in science and engineering can often be modeled with partial differential equations (PDEs) with uncertain parameters. In order to improve the design of such systems, we formulate a risk-neutral optimization problem with PDE constraints, an infinite-dimensional stochastic program. We apply the sample average approximation (SAA) to the risk-neutral PDE-constrained optimization problem and analyze the consistency of the SAA optimal value and solutions. Our analysis exploits hidden compactness in PDE-constrained optimization problems, allowing us to construct deterministic, compact sets containing the solutions to the risk-neutral problem and those to the SAA problems. Exploiting further problem structure, we establish nonasymptotic sample size estimates using the covering number approach, thereby we shed light on the computational resources needed to obtain accurate solutions.
Johannes Milz is research associate at the Technical University of Munich. Johannes' research broadly lies in optimization under uncertainty with a current focus on complexity analysis of and algorithmic design for PDE-constrained optimization problems under uncertainty. He received his doctorate in applied mathematics from the Technical University of Munich in 2021.