Estimating the parallelism in the solution of sparse triangular linear systems

Abstract

Most direct methods to solve sparse linear systems, as well as preconditioners for iterative methods, factorize the problem to operate on sparse triangular matrices. The solution of sparse triangular linear systems is then one of the fundamental building blocks of numerical methods, which has motivated extensive research dedicated to achieve efficient algorithms for this operation in parallel hardware platforms. However, as it often occurs with sparse problems, the parallel performance of each method depends heavily on the nonzero pattern of the matrix. In this sense, observing these characteristics can allow predicting which method is better suited for each problem.In the case of sparse triangular matrices, one of the most important constrains to parallelism, is the amount of data dependencies during forward or backward substitution. This is related to the number of level-sets, i.e. groups of independent rows in the matrix, but the high cost of computing this number makes its utilization impractical.In this work, we propose different strategies to approximate the number of level-sets through inexpensive procedures, and provide implementations of the heuristics for CPU and GPU.