Reliable Error Estimates for Optimal Control of Linear Elliptic PDEs with Random Inputs

Abstract

We discretize a risk-neutral optimal control problem governed by a linear elliptic partial differential equation with random inputs using a Monte Carlo sample-based approximation and a finite element discretization, yielding finite dimensional control problems. We establish an exponential tail bound for the distance between the finite dimensional problems' solutions and the risk-neutral problem's solution. The tail bound implies that solutions to the risk-neutral optimal control problem can be reliably estimated with the solutions to the finite dimensional control problems. Numerical simulations illustrate our theoretical findings.

26 pages, 11 figures

References in corpus (2)

• Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis
• Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

Cited by in corpus (2)

• Criticality measure-based error estimates for infinite dimensional optimization
• Consistency of sample-based stationary points for infinite-dimensional stochastic optimization