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Preprint 36-2003
Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. Part I: Theory
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Author(s) : Eric de Sturler , Jörg Liesen
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 36-2003
- 65F10 Iterative methods for linear systems
- 65F15 Eigenvalues, eigenvectors
- 65D18 Computer graphics and computational geometry
We analyze the properties of both classes of preconditioned matrices, in particular their spectrum. Using analytical results we show that the related system matrix has the more favorable spectrum, which in many applications translates into faster convergence for Krylov subspace methods. We show that fast convergence depends mainly on the quality of the splitting, a topic for which a substantial body of theory exists. Our analysis also provides a number of new relations between block-diagonal preconditioners and constraint preconditioners. For constrained problems, solving the the related system produces iterates that satisfy the constraints exactly, just as for systems with a constraint preconditioner. Finally, for the Lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a Newton iteration. Our theoretical results are confirmed by numerical experiments on a constrained optimization problem.
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