Block preconditioners for symmetric indefinite linear systems

Abstract

SUMMARYThis paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a Krylov subspace method for solving symmetric indefinite linear systems. The focus is on large-scale real world problems where block approximations are a practical necessity. The main illustration is the performance of the block diagonal, constrained, and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large-scale finite element discretization of Biot's consolidation equations. This system of equations is of fundamental importance to geomechanics. Numerical studies show that simple diagonal approximations to the (1,1) block K and inexpensive approximations to the Schur complement matrix S may not always produce the most spectacular time savings when K is explicitly available, but is able to deliver reasonably good results on a consistent basis. In addition, the block diagonal preconditioner with a negative (2,2) block appears to be reasonably competitive when compared to the more complicated ones. These observation are expected to remain valid for coefficient matrices whereby the (1,1) block is sparse, diagonally significant (a notion weaker than diagonal dominance), moderately well-conditioned, and has a much larger block size than the (2,2) block.

Using constraint preconditioners with regularized saddle-point problems

“…In the former case, (23) and (24) may be simplified to . Since Q is orthogonal, the general eigenvalue problem (26) is equivalent to considering…”

“…where G ∈ R n×n is some symmetric matrix, have been considered (for example, see [3,4,5,8,18,23].) When C = 0, (2) is commonly known as a constraint preconditioner [2,16,17,19].…”

A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations

“…There are also recent numerical results showing that SQMR combined with a symmetric indefinite preconditioner is more effective than a positive definite preconditioner (e.g. References [3][4][5][6][7]). …”

“…Reference [8]) that we will apply to the proposed modified SSOR preconditioner in this paper is applicable only under the left-right preconditioning framework for SQMR. Given a preconditioner of the form P = P L P R , the left-right preconditioned system is as follows: In the last few years, several effective symmetric indefinite preconditioners such as MJ [4], GJ [5] and block constrained preconditioner P c [6] [7] have been proposed for solving large linear systems given in (1) using SQMR. The preconditioners, GJ and P c , were developed based on the block structure of A.…”

“…ã m+j,m+j = αã m+j,m+j (9) End Following the analysis of Toh et al [5], a block constrained preconditioner P c , was derived from A by replacing K by diag(K) in [6] as:…”