THE USE OF GRAPH THEORY IN A PARALLEL MULTIFRONTAL METHOD FOR SEQUENCES OF UNSYMMETRIC PATTERN SPARSE MATRICES

Multifrontal matrix factorization methods used for solving large, sparse systems of linear equations decompose sparse matrices into overlapping dense submatrices which can be represented by vertices with relationships between submatrices shown via various types of edges. This paper describes the use of graph theory in a new parallel, distributed memory multifrontal method for the LU factorization of sequences of matrices with an identical, unsymmetric pattern. The directed acyclic graphs formed by these vertices and the various edge sets are used to structure the computations, schedule the parallel factorization, and provide a robust capability to dynamically change the pivot ordering to maintain numerical stability. Pivot reordering determines necessary permutations based on a path analysis of two component edge sets. The path properties represented by these edge sets deene the impacts of these permutations on the structures of the submatrices and the number of nonzeros in the matrix factors. Transitive reductions of these edge sets provide the communications paths needed for parallel implementation.