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Applications of Mathematics

, Volume 62, Issue 6, pp 537–559 | Cite as

Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications

  • Owe Axelsson
Article
  • 73 Downloads

Abstract

Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods.

Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks.

Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.

Keywords

preconditioning Schur complement transformation optimal control implicit time integration 

MSC 2010

65F08 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Institute of Geonics of the Czech Academy of SciencesOstravaCzech Republic

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