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

• DOI: 10.1137/060660977
• Publication Date: 2007-07-03T22:00:19Z
• Authors: Joachim Schöberl; Walter Zulehner
• Keywords: Saddle point; Augmented Lagrangian method; Regularization

We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block is the positivity (1,1) on kernel (2,1) block. Many solution methods, however, require that satisfied everywhere. To enforce everywhere, an augmented Lagrangian approach usually chosen. However, adjustment involved parameters a critical issue. will present different not based explicit augmentation technique. For considered class symmetric and preconditioners, assumptions are presented lead to positive definite problems with respect particular scalar product. Therefore, conjugate gradient acceleration can be used. An important applications optimal control problems. It typical for cost functional contains extra regularization parameter. elliptic state equations distributed control, special preconditioner discretized problem constructed, which leads convergence rates preconditioned method only independent mesh size but also Numerical experiments illustrating theoretical results.