Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems

TL;DR

This paper extends Schur complement preconditioning techniques to multi-saddle point problems in Hilbert spaces with block tridiagonal Hessians, providing bounds on condition numbers and applications to optimal control problems.

Contribution

It generalizes Schur complement preconditioners to multiple saddle point problems in Hilbert spaces with explicit condition number bounds.

Findings

Derived sharp bounds for condition numbers independent of operators

Established new existence results for optimal control problems

Constructed efficient preconditioners for discretized optimality systems

Abstract

The importance of Schur complement based preconditioners are well-established for classical saddle point problems in $\mathbb{R}^N \times \mathbb{R}^M$. In this paper we extend these results to multiple saddle point problems in Hilbert spaces $X_1\times X_2 \times \cdots \times X_n$. For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems.