Combination preconditioning of saddle point systems for positive definiteness

Abstract

Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski ( ˙ Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self-adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to a nonstandard inner product always allow a MINREStype method (W-PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product a more efficient CG-like method (W-PCG) can be reliably used. We establish eigenvalue expressions for Krzyzanowski preconditioners and show that for a specific ˙ choice of parameters, although the Krzyzanowski preconditioned saddle point matrix is self-adjoint with ˙ respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only W-PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, W-PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems

A New Analysis of Block Preconditioners for Saddle Point Problems

“…Some previous analyzes focus on the conditions to have the preconditioned matrix positive definite in a non standard inner product, and develop related conjugate gradient like methods; see, e.g., [11,28,36]. Here we offer a complementary viewpoint, giving estimates that vary continuously in function of the main parameters (1.3), (1.4), without any restriction on these latter.…”

“…Now, it is clear the the scaling of M A plays an important role for block triangular and Uzawa preconditioners. With µ ≤ 1 , we have the appealing result of Corollary 4.5, but, on the other hand, if one rescales the preconditioner for A to have µ ≥ 1 , all eigenvalues are real which may also be attractive, allowing to use conjugate gradient methods in nonstandard inner products [11,28,36]. To investigate this, we rescaled the algebraic multigrid preconditioner by a factor α , entailing that µ ≈ 0.4 α and µ = α .…”

GMRES with multiple preconditioners

“…have been proposed and explored for saddle-point problems, where the constants α 1 , α 2 are fixed; see in particular [31], [32], [48]. One can interpret the first step of MPGMRES, for t = 2, as choosing the optimal values of α 1 , α 2 in (2.12) so that the residual r (1) is minimal.…”

On the Eigenvalues and Eigenvectors of Block Triangular Preconditioned Block Matrices

“…Two popular preconditioners are the block lower triangular matrix [8,16,17] 2) and block upper triangular matrix [6,13,15,16,20]…”

“…This may be the case when, for example, we precondition to achieve self-adjointness and positive definiteness with respect to a nonstandard inner product [8,14,17]. However, in many situations the eigenvectors will also have an effect on convergence.…”