Improved Balanced Incomplete Factorization

Abstract

Abstract. In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008Comput., 30 ( ), pp. 2302Comput., 30 ( -2318. The improvements are based on a deeper understanding of the Inverse Sherman-Morrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems.

Preconditioners based on the ISM factorization

“…Numerical results in [10] show the same that was already true for BIF. The method is robust and the dependence of the dropping tolerance is not critical.…”

“…To get the missed factor U it is proposed in [10] to compute the ISM factorization of A T . To fix notation let…”

“…The first one is to replace the way that the pivots are computed. Instead of to compute them as in (10), that is as sr i , they are computed as z T i A T Az i . The second one follows the ideas of Tismenetsky in [18], by storing additional entries that are used only in computations, but finally discarded a better preconditioner is computed, see Algorithm 2.2 in [11].…”

Nova implementação do precondicionador AINV em acelerador NVIDIA utilizando a biblioteca CUSP

“…Por isso, nasúltimas décadas, os métodos que aproximam diretamente a inversa de A, sendo portanto bons candidados a precondicionadores, têm atraído bastante atenção [8,10]. A mais importante vantagem destes métodosé que sua aplicação consiste simplesmente no produto matriz-vetor.…”

An overview of approximate inverse methods

“…This article is a computational comparison of some preconditioners alternatives based on approximations of inverse [3,4,7,8,9,10,13,14]. We have implemented sequential codes, in Matlab and/or C and tested these codes by solving some academic problems.…”