A New GPU Algorithm to Compute a Level Set-Based Analysis for the Parallel Solution of Sparse Triangular Systems

Abstract

A myriad of problems in science and engineering, involve the solution of sparse triangular linear systems. They arise frequently as part of direct and iterative solvers for linear systems and eigenvalue problems, and hence can be considered as a key building block of sparse numerical linear algebra. This is why, since the early days, their parallel solution has been exhaustively studied, and efficient implementations of this kernel can be found for almost every hardware platform. In the GPU context, the most widespread implementation of this kernel is the one distributed in NVIDIA CUSPARSE library, which relies on a preprocessing stage to aggregate the unknowns of the triangular system into level sets. This determines an execution schedule for the solution of the system, where the level sets have to be processed sequentially while the unknowns that belong to one level set can be solved in parallel. One of the disadvantages of the CUSPARSE implementation is that this preprocessing stage is often extremely slow in comparison to the runtime of the solving phase. In this work, we present a parallel GPU algorithm that is able to compute the same level sets as CU S PARSE but takes significantly less runtime. Our experiments on a set of matrices from the SuiteSparse collection show acceleration factors of up to 44×. Additionally, we provide a routine capable of solving a triangular linear system on the same pass used to calculate the level sets, yielding important performance benefits.