More than 10 years of research related to the development of efficient GPU routines for the sparse matrix-vector product (SpMV) have led to several realizations, each with its own strengths and weaknesses. In this work, we review some of the most relevant efforts on the subject, evaluate a few prominent routines that are publicly available using more than 3000 matrices from different applications, and apply machine learning techniques to anticipate which SpMV realization will perform best for each sparse matrix on a given parallel platform. Our numerical experiments confirm the methods offer such varied behaviors depending on the matrix structure that the identification of general rules to select the optimal method for a given matrix becomes extremely difficult, though some useful strategies (heuristics) can be defined. Using a machine learning approach, we show that it is possible to obtain unexpensive classifiers that predict the best method for a given sparse matrix with over 80% accuracy, demonstrating that this approach can deliver important reductions in both execution time and energy consumption.

Sparse triangular linear systems are ubiquitous in a wide range of science and engineering fields, and represent one of the most important building blocks of Sparse Numerical Lineal Algebra methods. For this reason, their parallel solution has been subject of exhaustive study, and efficient implementations of this kernel can be found for almost every hardware platform. However, the strong data dependencies that serialize a great deal of the execution and the load imbalance inherent to the triangular structure poses serious difficulties for its parallel performance, specially in the context of massively- parallel processors such as GPUs. To this day, the most widespread GPU implementation of this kernel is the one distributed in NVIDIA CUSPARSE library, which relies on a preprocessing stage to determine the parallel execution schedule. Although the solution phase is highly efficient, this strategy pays the cost of constant synchronizations with the CPU. In this work, we present a synchronization-free GPU al- gorithm to solve sparse triangular linear systems for the CSR format. The experimental evaluation shows performance improvements over CUSPARSE and a recently proposed synchronization-free method for the CSC matrix format.

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.