2,483 research outputs found

    Sparse approximate inverses and target matrices

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    If P has a prescribed sparsity and minimizes the Frobenius norm ||I-PA||F it is called a sparse approximate inverse of A. It is well known that the computation of such a matrix P is via the solution of independent linear least squares problems for the rows separately (and therefore in parallel). In this paper we consider the choice of other norms, and introduce the idea of `target' matrices. A target matrix, T, is readily inverted and thus forms part of a preconditioner when ||T-PA|| is minimized over some appropriate sparse matrices P. The use of alternatives to the Frobenius norm which maintain parallelizability whilst discussed in early literature does not appear to have been exploited.\ud \ud This work was supported by the Engineering and Physical Sciences Research council and NAG Ltd

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

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    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 MM. 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 MM. The proposed criterion enables us to make MM 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 MM 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

    Scalable Task-Based Algorithm for Multiplication of Block-Rank-Sparse Matrices

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    A task-based formulation of Scalable Universal Matrix Multiplication Algorithm (SUMMA), a popular algorithm for matrix multiplication (MM), is applied to the multiplication of hierarchy-free, rank-structured matrices that appear in the domain of quantum chemistry (QC). The novel features of our formulation are: (1) concurrent scheduling of multiple SUMMA iterations, and (2) fine-grained task-based composition. These features make it tolerant of the load imbalance due to the irregular matrix structure and eliminate all artifactual sources of global synchronization.Scalability of iterative computation of square-root inverse of block-rank-sparse QC matrices is demonstrated; for full-rank (dense) matrices the performance of our SUMMA formulation usually exceeds that of the state-of-the-art dense MM implementations (ScaLAPACK and Cyclops Tensor Framework).Comment: 8 pages, 6 figures, accepted to IA3 2015. arXiv admin note: text overlap with arXiv:1504.0504

    Scalable iterative methods for sampling from massive Gaussian random vectors

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    Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian ran- dom vectors that are parameterised by the inverse of their covariance matrix, is a fundamental problem in computational statistics. In this paper, we show how we can exploit arbitrarily accu- rate approximations to a GMRF to speed up Krylov subspace sampling methods. We also show that these methods can be used when computing the normalising constant of a large multivariate Gaussian distribution, which is needed for both any likelihood-based inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particu- lar, we show that when the precision matrix is a perturbation of a (block) circulant matrix, it is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
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