Publication Date:
2018-05-21
Description:
The importance of Schur-complement-based preconditioners is well established for classical saddle point problems in $mathbb{R}^N imes mathbb{R}^M$. In this paper we extend these results to multiple saddle point problems in Hilbert spaces $X_1imes X_2 imes cdots imes X_n$. 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.
Print ISSN:
0272-4979
Electronic ISSN:
1464-3642
Topics:
Mathematics
Permalink