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

TL;DR

This paper develops a method to reliably estimate solutions to risk-neutral optimal control problems governed by linear elliptic PDEs with random inputs, using Monte Carlo and finite element methods, supported by theoretical tail bounds and numerical validation.

Contribution

It introduces an exponential tail bound for the error between finite dimensional and original solutions, ensuring reliable approximation of the risk-neutral control problem.

Findings

Exponential tail bounds for solution errors

Finite element and Monte Carlo methods effectively approximate the control problem

Numerical simulations confirm theoretical error estimates

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.