Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems
Provides theoretical foundations and practical preconditioners for a class of optimization problems, but the extension is incremental.
The paper extends Schur complement preconditioners from classical saddle point problems to multiple saddle point problems in Hilbert spaces, deriving sharp condition number bounds independent of the operators. Applied to optimal control problems, it yields new existence results and efficient preconditioners.
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.