A stochastic Galerkin method for optimal Dirichlet boundary control problems with uncertain data

TL;DR

This paper introduces a stochastic Galerkin method to solve elliptic Dirichlet boundary control problems with uncertain data, providing error estimates, preconditioning strategies, and numerical validation.

Contribution

It develops a novel stochastic Galerkin framework for boundary control problems with randomness, including error analysis and efficient solvers.

Findings

Error estimates for control and state variables

Effective preconditioners for large linear systems

Numerical experiments confirming method efficiency

Abstract

The paper deals with a stochastic Galerkin approximation of elliptic Dirichlet boundary control problems with random input data. The expectation of a tracking cost functional with the deterministic constrained control is minimized. Error estimates are derived for the control variable in $L^2(\partial \mathcal D)$-norm and state variable in $L^2(\Omega\times\mathcal D)$-norm. To solve large linear systems, appropriate preconditioners are proposed for both unconstrained and constrained scenarios. To illustrate the validity and efficiency of the proposed approaches, some numerical experiments are performed.