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Constraint Preconditioning for Indefinite Linear Systems
 SIAM J. Matrix Anal. Appl
, 2000
"... . The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and ..."
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Cited by 110 (14 self)
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. The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions. Key words. preconditioning, indenite matrices, Krylov subspace methods AMS subject classications. 65F10, 65F15, 65F50 1. Introduction. In this paper, we are concerned with investigating a new class of preconditioners for indenite systems of linear equations of a sort which arise in constrained optimization as well as in leastsquares, saddlepoint and Stokes problems. We attempt to solve the indenite linear system A B T B 0  {z } A x 1 x...
Preconditioning For The SteadyState NavierStokes Equations With Low Viscosity
 SIAM J. SCI. COMPUT
, 1996
"... We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single ei ..."
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Cited by 59 (10 self)
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We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to one, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.
Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems
 SIAM J. Sci. Comput
, 1999
"... In this paper, a block LU preconditioner for saddle point problems is presented. ..."
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Cited by 5 (2 self)
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In this paper, a block LU preconditioner for saddle point problems is presented.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2000; 00:1–6 Prepared using nlaauth.cls [Version: 2002/09/18 v1.02] Analysis of block matrix preconditioners
"... In this paper, we describe and analyze several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linearquadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term w ..."
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In this paper, we describe and analyze several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linearquadratic 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 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 ⊂ R2 or R3, and a preconditioner independent of h and α when Ω ⊂ R2. 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 c © 2000 John Wiley & Sons, Ltd. key words: Optimal control, elliptic Neumann problem, augmented Lagrangian, saddle point