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.
Highlights
Iterative solution methods are widely used for the solution of linear and linearized systems of equations
It was found that the square block matrix, PRESB preconditioning method has superior properties compared to them and to most other methods
In this paper we present the major properties of the PRESB preconditioner on operator level, with short derivations
Summary
Iterative solution methods are widely used for the solution of linear and linearized systems of equations. As we shall see, for many applications sharp eigenvalue bounds for the preconditioned operator can be derived, which are only influenced to a minor extent by the inner solver so one can even use a Chebyshev iterative acceleration method. It was found that the square block matrix, PRESB preconditioning method has superior properties compared to them and to most other methods It is most robust, it leads to a small condition number of the preconditioned matrix which holds uniformly with respect to both problem and method parameters, and sharp eigenvalue bounds can be derived. In this paper we present the major properties of the PRESB preconditioner on operator level, with short derivations This includes presentation of a typical class of optimal control problems in Sect.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have