Balanced incomplete factorization preconditioner with pivoting

[EN] In this work we study pivoting strategies for the preconditioner presented in Bru (SIAM J Sci Comput 30(5):2302-2318, 2008) which computes the LU factorization of a matrix A. This preconditioner is based on the Inverse Sherman Morrison (ISM) decomposition [Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula. Bru (SIAM J Sci Comput 25(2):701-715, 2003), that using recursion formulas derived from the Sherman-Morrison formula, obtains the direct and inverse LU factors of a matrix. We present a modification of the ISM decomposition that allows for pivoting, and so the computation of preconditioners for any nonsingular matrix. While the ISM algorithm at a given step computes only a new pair of vectors, the new pivoting algorithm in the k-th step also modifies all the remaining vectors from k + 1 to n. Thus, it can be seen as a right looking version of the ISM decomposition. The results of numerical experiments with ill-conditioned and highly indefinite matrices arising from different applications show the robustness of the new algorithm, since it is able to solve problems that are not possible to solve otherwise.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The work was supported by Conselleria de Innovacion, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana (CIAICO/2021/162).Marín Mateos-Aparicio, J.; Mas Marí, J. (2023). Balanced incomplete factorization preconditioner with pivoting. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 117(1). https://doi.org/10.1007/s13398-022-01334-1117

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Utilización del paralelismo multihebra en el precondicionado y la resolución iterativa de sistemas lineales dispersos

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