Secondary Re-Ordering Methods for Parallel Partitioned Sparse Inverse Solutions

This paper compares ordering algorithms to reduce the number of serial steps required for parallel repeat solutions. These algorithms are applied after the initial application of a fill-reducing algorithm, and are called "secondary re-ordering" algorithms. One algorithm described in this paper optimally reduces the number of no-fill partitions in the partitioned inverse of the LDL^T factors of A. The algorithm computes the minimum number of subsets in the partition over all permutations of L which preserve the lower triangular structure of the matrix. The paper considers also a tree rotations method to reduce the height of the elimination tree. This paper describes experiments on the behavior of these algorithms when used in combination with a variety of primary ordering algorithms: minimum degree, multiple minimum degree, minimum length and nested dissection. Numerical results on power system and finite element matrices are presented. The main conclusion is that the primary re-ordering is more important than the secondary re-ordering method in increasing parallelism.