Level-based Blocking for Sparse Matrices: Sparse Matrix-Power-Vector Multiplication

• Main Authors: Alappat, Christie L, Hager, Georg, Schenk, Olaf, Wellein, Gerhard
• Format: Journal Article
• Language: English
• Published: 03-05-2022
• Subjects: Computer Science - Distributed, Parallel, and Cluster Computing; Computer Science - Numerical Analysis; Computer Science - Performance; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2205.01598

The multiplication of a sparse matrix with a dense vector (SpMV) is a key
component in many numerical schemes and its performance is known to be severely
limited by main memory access. Several numerical schemes require the
multiplication of a sparse matrix polynomial with a dense vector, which is
typically implemented as a sequence of SpMVs. This results in low performance
and ignores the potential to increase the arithmetic intensity by reusing the
matrix data from cache. In this work we use the recursive algebraic coloring
engine (RACE) to enable blocking of sparse matrix data across the polynomial
computations. In the graph representing the sparse matrix we form levels using
a breadth-first search. Locality relations of these levels are then used to
improve spatial and temporal locality when accessing the matrix data and to
implement an efficient multithreaded parallelization. Our approach is
independent of the matrix structure and avoids shortcomings of existing
"blocking" strategies in terms of hardware efficiency and parallelization
overhead. We quantify the quality of our implementation using performance
modelling and demonstrate speedups of up to 3$\times$ and 5$\times$ compared to
an optimal SpMV-based baseline on a single multicore chip of recent Intel and
AMD architectures. As a potential application, we demonstrate the benefit of
our implementation for a Chebyshev time propagation scheme, representing the
class of polynomial approximations to exponential integrators. Further
numerical schemes which may benefit from our developments include $s$-step
Krylov solvers and power clustering algorithms.