Robust Dropping Criteria for F-norm Minimization Based Sparse Approximate Inverse Preconditioning

Dropping tolerance criteria play a central role in Sparse Approximate Inverse preconditioning. Such criteria have received, however, little attention and have been treated heuristically in the following manner: If the size of an entry is below some empirically small positive quantity, then it is set to zero. The meaning of "small" is vague and has not been considered rigorously. It has not been clear how dropping tolerances affect the quality and effectiveness of a preconditioner $M$. In this paper, we focus on the adaptive Power Sparse Approximate Inverse algorithm and establish a mathematical theory on robust selection criteria for dropping tolerances. Using the theory, we derive an adaptive dropping criterion that is used to drop entries of small magnitude dynamically during the setup process of $M$. The proposed criterion enables us to make $M$ both as sparse as possible as well as to be of comparable quality to the potentially denser matrix which is obtained without dropping. As a byproduct, the theory applies to static F-norm minimization based preconditioning procedures, and a similar dropping criterion is given that can be used to sparsify a matrix after it has been computed by a static sparse approximate inverse procedure. In contrast to the adaptive procedure, dropping in the static procedure does not reduce the setup time of the matrix but makes the application of the sparser $M$ for Krylov iterations cheaper. Numerical experiments reported confirm the theory and illustrate the robustness and effectiveness of the dropping criteria.Comment: 27 pages, 2 figure

Preconditioners based on the ISM factorization

In this paper we survey our work on preconditioners based on the Inverse Sherman-Morrison factorization. The most important theoretical results are also summarized and some numerical conclusions are provided.This work was supported by the Spanish Ministerio de Economia y Competitividad under grant MTM2014-58159-P and by the project 13-06684S of the Grant Agency of the Czech Republic.Bru García, R.; Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J.; Tuma, M. (2015). Preconditioners based on the ISM factorization. Universitatea "Ovidius" Constanta. Analele. Seria Matematica. 23(3):17-27. doi:10.1515/auom-2015-0044S172723

Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations

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Optimizing two-level preconditionings for the conjugate gradient method

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A posteriori error estimates in L2-norm for the least squares finite element method applied to a first order system of differential equations

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