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

<Para ID="Par1">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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$N\times N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mrow> </math> </EquationSource> </InlineEquation> matrix, there are <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$N!$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>N</mi> <mo>!</mo> </mrow> </math> </EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>N</mi> </math> </EquationSource> </InlineEquation> 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. Copyright Springer Science+Business Media New York 2015