Robust Preconditioners for Multiple Saddle Point Problems and Applications to Optimal Control Problems

TL;DR

This paper develops robust block diagonal preconditioners for multiple saddle point problems, especially in PDE-constrained optimal control, ensuring stability and efficiency across varying regularization parameters, with practical applications to heat and wave equations.

Contribution

It characterizes block structured norms for well-posedness and constructs alpha-robust preconditioners applicable to a broad class of PDE-constrained optimal control problems.

Findings

Preconditioners are effective for heat and wave control problems.

Numerical experiments confirm the robustness of the proposed methods.

The approach ensures stability across different regularization parameters.

Abstract

In this paper we consider multiple saddle point problems with block tridiagonal Hessian in a Hilbert space setting. Well-posedness and the related issue of preconditioning are discussed. We give a characterization of all block structured norms which ensure well-posedness of multiple saddle point problems as a helpful tool for constructing block diagonal preconditioners. We subsequently apply our findings to a general class of PDE-constrained optimal control problems containing a regularization parameter $\alpha$ and derive $\alpha$-robust preconditioners for the corresponding optimality systems. Finally, we demonstrate the generality of our approach with two optimal control problems related to the heat and the wave equation, respectively. Preliminary numerical experiments support the feasibility of our method.