Robust preconditioners for PDE-constrained optimization with limited observations

TL;DR

This paper develops and analyzes preconditioners for PDE-constrained optimization problems with limited observation data, such as boundary-only observations, ensuring robustness against regularization and mesh size variations.

Contribution

It introduces novel preconditioners tailored for PDE-constrained optimization with limited data, maintaining uniform condition numbers regardless of regularization and mesh size.

Findings

Preconditioners are robust with respect to regularization parameter.

Preconditioners are robust with respect to mesh size.

Numerical results confirm theoretical robustness.

Abstract

Regularization robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume that observation data is available throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. In this paper we propose and analyze preconditioners for PDE-constrained optimization problems with limited observation data, e.g. observations are only available at the boundary of the solution domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the condition number of the preconditioned optimality system is uniformly bounded, independently of the size of these two parameters. We first consider a prototypical elliptic control problem and thereafter more general PDE-constrained optimization problems. Our theoretical findings are illuminated by several numerical results.