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Addressing the envelope reduction of sparse matrices using a genetic programming system

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Abstract

Large sparse symmetric matrix problems arise in a number of scientific and engineering fields such as fluid mechanics, structural engineering, finite element analysis and network analysis. In all such problems, the performance of solvers depends critically on the sum of the row bandwidths of the matrix, a quantity known as envelope size. This can be reduced by appropriately reordering the rows and columns of the matrix, but for an \(N\times N\) matrix, there are \(N!\) such permutations, and it is difficult to predict how each permutation affects the envelope size without actually performing the reordering of rows and columns. These two facts compounded with the large values of \(N\) used in practical applications, make the problem of minimising the envelope size of a matrix an exceptionally hard one. Several methods have been developed to reduce the envelope size. These methods are mainly heuristic in nature and based on graph-theoretic concepts. While metaheuristic approaches are popular alternatives to classical optimisation techniques in a variety of domains, in the case of the envelope reduction problem, there has been a very limited exploration of such methods. In this paper, a Genetic Programming system capable of reducing the envelope size of sparse matrices is presented and evaluated against four of the best-known and broadly used envelope reduction algorithms. The results obtained on a wide-ranging set of standard benchmarks from the Harwell–Boeing sparse matrix collection show that the proposed method compares very favourably with these algorithms.

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Notes

  1. This, in fact, means that their corresponding graphs have more than one component and in such a scenario, studying each component of the system separately is more likely to result in an efficient analysis. It is probable that in the original process of generating these 6 matrices from their actual numerical values, a rounding error occurred leading to some entries being ignored.

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Authors and Affiliations

  1. School of Engineering-Emerging Technologies, University of Tabriz, 5166614761, Tabriz, Iran

    Behrooz Koohestani

  2. School of Computer Science and Electronic Engineering, University of Essex, Colchester, CO4 3SQ, UK

    Riccardo Poli

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  1. Behrooz Koohestani
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  2. Riccardo Poli
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Correspondence to Behrooz Koohestani.

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Koohestani, B., Poli, R. Addressing the envelope reduction of sparse matrices using a genetic programming system. Comput Optim Appl 60, 789–814 (2015). https://doi.org/10.1007/s10589-014-9688-2

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  • DOI: https://doi.org/10.1007/s10589-014-9688-2

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