The Effect of Various Sparsity Structures on Parallelism and Algorithms to Reveal Those Structures

Structured sparse matrices can greatly benefit parallel numerical methods in terms of parallel performance and convergence. In this chapter, we present combinatorial models for obtaining several different sparse matrix forms. There are four basic forms we focus on: singly-bordered block-diagonal form, doubly-bordered block-diagonal form, nonempty off-diagonal block minimization, and block diagonal with overlap form. For each of these forms, we first present the form in detail and describe what goals are sought within the form, and then examine the combinatorial models that attain the respective form while targeting the sought goals, and finally explain in which aspects the forms benefit certain parallel numerical methods and their relationship with the models. Our work focuses especially on graph and hypergraph partitioning models in obtaining the mentioned forms. Despite their relatively high preprocessing overhead compared to other heuristics, they have proven to model the given problem more accurately and this overhead can be often amortized due the fact that matrix structure does not change much during a typical numerical simulation. This chapter presents a number of models and their relationship with parallel numerical methods.