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Analysis of block matrix preconditioners for elliptic optimal control problems
Author(s) -
Mathew T. P.,
Sarkis M.,
Schaerer C. E.
Publication year - 2007
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.526
Subject(s) - preconditioner , mathematics , augmented lagrangian method , saddle point , positive definite matrix , conjugate gradient method , linear system , schur complement , rate of convergence , discretization , acceleration , iterative method , matrix (chemical analysis) , algorithm , mathematical optimization , mathematical analysis , computer science , geometry , computer network , channel (broadcasting) , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , classical mechanics , composite material
In this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear‐quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter α is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for Ω ⊂ R 2 or R 3 , and a preconditioner independent of h and α when Ω ⊂ R 2 . Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h . In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright © 2007 John Wiley & Sons, Ltd.