Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks
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Numerische Mathematik
Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks Owe Axelsson1,2 · János Karátson3,4 Received: 6 February 2020 / Revised: 21 July 2020 © The Author(s) 2020
Abstract Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear. Mathematics Subject Classification 65F08 · 65F10
1 Introduction Iterative solution methods are widely used for the solution of linear and linearized systems of equations. For early references, see [1–3]. A key aspect is then to use a proper preconditioning, that is a matrix that approximates the given matrix accurately but is still much cheaper to solve systems with and which results in tight eigenvalue bounds of the preconditioned matrix, see e.g. [4–6]. This should hold irrespective
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János Karátson [email protected]
1
Institute of Geonics of the Czech Academy of Sciences, Ostrava, Czech Republic
2
Department of Information Technology, Uppsala University, Uppsala, Sweden
3
MTA-ELTE Numerical Analysis and Large Networks Research Group, Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary
4
Department of Analysis, Technical University, Budapest, Hungary
123
O. Axelsson, J. Karátson
of the dimension of the system and thus allow a fast large scale modelling. Thereby preconditioners that exploit matrix structures can have considerate advantage. Differential operators or matrices on coupled two-by-two block form with square blocks, or which have been reduced to such a form from a more general block form, arise in various applications. The simplest example is a complex valued system, (A + i B)(x + i y) = f + ig, where A, B, x, y, f and g are real valued, which in order to avoid complex arithmetics, is rewritten in the real valued form,
A −B B A
x f = , y g
that is, where no complex arithmetics is needed for its solution. For examples of use of iterative solution methods in this context, see e.g. [7–10]. As we shall see, much more important examples arise for instance when solving optimal control problems for partial differential equations. After di
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