NDLI: Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zulehner, Walter Schberl, Joachim
Copyright Year	2007
Abstract	We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented Lagrangian approach is usually chosen. However, the adjustment of the involved parameters is a critical issue. We will present a different approach that is not based on such an explicit augmentation technique. For the considered class of symmetric and indefinite preconditioners, assumptions are presented that lead to symmetric and positive definite problems with respect to a particular scalar product. Therefore, conjugate gradient acceleration can be used. An important class of applications are optimal control problems. It is typical for such problems that the cost functional contains an extra regularization parameter. For control problems with elliptic state equations and distributed control, a special indefinite preconditioner for the discretized problem is constructed, which leads to convergence rates of the preconditioned conjugate gradient method that are not only independent of the mesh size but also independent of the regularization parameter. Numerical experiments are presented for illustrating the theoretical results.
Starting Page	752
Ending Page	773
Page Count	22
File Format	PDF
ISSN	08954798
DOI	10.1137/060660977
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	3
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-07-04
Access Restriction	Subscribed
Subject Keyword	conjugate gradient methods optimal control problems indefinite preconditioners PDE-constrained optimization problems Iterative methods for linear systems Newton-type methods KKT systems saddle point problems Conditioning of matrices
Content Type	Text
Resource Type	Article
Subject	Analysis

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