Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

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.

Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems

“…Special iterative solvers for discrete optimality systems such as (3) should be not only robust with respect to (wrt) the mesh refinement quantified by the discretization parameter h but also wrt the regularization parameter that can be quite small depending on the cost that we are willing to pay. Such kind of h and robust preconditioned iterative methods have been proposed and investigated in [1,5,35,37,43]; see also [2,14,34,36,40], for handling control and state constraints, and the references therein. Alternatively, we can use all-at-once multigrid methods to solve saddle-point problems such as (3) efficiently; see, e.g., [38] and the review paper [10].…”

“…For fixed , discretization error estimates can be found, e.g., in [19]. There is a huge number of publications on efficient preconditioned iterative solvers for symmetric, but indefinite systems in general; see, e.g., the unified approach proposed in [42], the survey paper [8], the review article [31], the books [15] and [6], the more recent papers [1,3,4,33], and the literature cited therein. Special iterative solvers for discrete optimality systems such as (3) should be not only robust with respect to (wrt) the mesh refinement quantified by the discretization parameter h but also wrt the regularization parameter that can be quite small depending on the cost that we are willing to pay.…”