Banded target matrices and recursive FSAI for parallel preconditioning

Abstract

Abstract. In this paper we propose a parallel preconditioner for the CG solver based on successive applications of the FSAI preconditioner. We first compute an FSAI factor Gout for coefficient matrix A, and then another FSAI preconditioner is computed for either the preconditioned matrix S = GoutAG T out or a sparse approximation of S. This process can be iterated to obtain a sequence of triangular factors whose product forms the final preconditioner. Numerical results onto large SPD matrices arising from geomechanical models account for the efficiency of the proposed preconditioner which provides a reduction of the iteration number and of the CPU time of the iterative phase with respect to the original FSAI preconditioner.

Fsaipack

“…The final G implicitly inherits a large number of nonzeroes, even though each G k is quite sparse. This idea was proposed by Kolotilina and Yeremin [1999] and later developed by Wang and Zhang [2003] and Bergamaschi and Martinez [2012]. It looks quite similar to multilevel preconditioning (see e.g., Saad and Suchomel [2002] and Janna et al [2009]).…”

“…It is based upon an algorithm that for each preconditioner row adaptively chooses the most significant terms to be retained. Another interesting strategy consists of building the preconditioner recursively, hence computing a dense and high-quality FSAI as the product of a few sparse factors [Bergamaschi and Martinez 2012].…”

Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

“…The initial Newton vector is obtained after a small number of iterations of a conjugate gradient procedure for the minimization of the Rayleigh quotient (DACG, [16]). As the initial preconditioner we choose RFSAI [17], which is built by recursively applying the (factorized sparse approximate inverse) FSAI preconditioner developed in [18]. We elected a factorized approximate inverse preconditioner (AIP) since it is more naturally parallelizable than preconditioners based on ILU factorizations.…”

“…Choice of the Initial Preconditioner. Following the developments in [17], we propose an implicit enlargement of the sparsity pattern using a banded target matrix ; the lower factor of the FSAI preconditioner is obtained by minimizing ‖ − ‖ over the set of matrices having a fixed sparsity pattern. Denoting with out the result of this minimization, we compute explicitly the preconditioned matrix = out out and then evaluate a second FSAI factor in for .…”

“…This RFSAI-recursive FSAI-procedure can be iterated a number of times to yield a preconditioner written as a product of several triangular factors. This preconditioner has shown its efficiency in the acceleration of the PCG onto realistic geomechanical problems of very large size [17].…”