A Parallel Sparse Hybrid Solver and Its Relations to Graphs and Hypergraphs

In this whitepaper, we review the state-of-the-art hybrid solver, which uses generalized form DS factorization, for solving system
of equations of the form Ax = f, and this solver’s relations to graphs and hypergraphs. We investigate two different reordering
strategies for the DS factorization preconditioning scheme: reordering via graph partitioning (GP) and reordering via hypergraph
partitioning (HP).In the GP scheme, the partitioning objective of minimizing the edge cutsize corresponds to minimizing the total
number of nonzeros in the off-diagonal blocks of the reordered matrix. In the HP scheme, the partitioning objective of
minimizing the cutsize, according to the cut-net metric, corresponds to minimizing the total number of nonzero columns in the
off-diagonal blocks of the reordered matrix. In both of the two schemes, partitioning constraint of maintaining balance on the part
weights corresponds to maintaining balance on the nonzero counts of the diagonal blocks of the reordered matrix. The
partitioning objective of GP relates to minimizing the number of nonzeros in the reduced system, whereas the partitioning
objective of HP exactly models minimizing the size of the reduced system. We tested the performance of two partitioning
schemes on a wide range of matrices for 4-, 8-, 16-, 32-, and 64-way permutations. Results showed that HP scheme performs
better than GP scheme in terms of solution times.