cuPentBatch—A batched pentadiagonal solver for NVIDIA GPUs

Abstract

We introduce cuPentBatch -our own pentadiagonal solver for NVIDIA GPUs. The development of cuPentBatch has been motivated by applications involving numerical solutions of parabolic partial differential equations, which we describe. Our solver is written with batch processing in mind (as necessitated by parameter studies of various physical models). In particular, our solver is directed at those problems where only the right-hand side of the matrix changes as the batch solutions are generated. As such, we demonstrate that cuPentBatch outperforms the NVIDIA standard pentadiagonal batch solver gpsvInterleavedBatch for the class of physically-relevant computational problems encountered herein. Program SummaryProgram Title: cuPentBatch https://github.com/munstermonster/cuPentBatch Licensing Provision: Apache License 2.0

cuSten — CUDA finite difference and stencil library

“…In this section we show how the cuSten library can be used as part of a larger program that the authors developed using the cuPentBatch [17] solver, a batched pentadiagonal matrix solver. We also provide a benchmark at the end of the section to show how cuSten performs versus a serial implementation.…”

A Batched GPU Methodology for Numerical Solutions of Partial Differential Equations

“…In this paper we consider batched numerical solutions of systems of the form Ax = b on a GPU, in particular where A is a tridiagonal or pentadiagonal matrix. Batched solutions of such tri/pentadiagonal matrix systems are becoming increasingly prevalent methods for tackling a variety of problems on GPUs which offer a high level of parallelism [1,2,3,4].…”

“…We emphasize that the methodology in this article may find application wherever GPUs are useful for accelerating established computational procedures. For example, in Fluid Mechanics, there has been a broad application of GPUs where solutions of Poisson's equation are commonly required [7,8], tsunami modelling and simulation [9], numerical linear algebra [10,11], batch solving of 1D partial differential equations [4], ADI methods for 2D simulations [5,6] and modelling mesoscopic-scale flows using Lattice-Boltzmann methods [12]. Also, in the field of gravitational wave data analysis, much work is done simplifying complex waveforms and analyses to achieve results in a physically reasonable and desirable amount of time.…”

“…We also note that, while there are options for single A, multiple b, tridiagonal and pentadiagonal matrices in cuSPARSE, these options overwrite the LHS data leading to a necessary correction step having to be performed when using these in iteration resulting in additional computational cost. Thus we propose the implementation of two solvers, one implementing the Thomas Algorithm for tridiagonal matrices and the other implementing the pentadiagonal equivalent as presented in [4,20], each with a single global copy of the A matrix and multiple b matrices. We will benchmark these implementations against cuThomasBatch [3] (implemented as gtsvInterleavedBatch in the cuSPARSE library) and existing work by the authors cuPentBatch [4], for tridiagonal and pentadiagonal matrices respectively, which are the existing state of the art for the algorithms we are interested in.…”

“…Thus we propose the implementation of two solvers, one implementing the Thomas Algorithm for tridiagonal matrices and the other implementing the pentadiagonal equivalent as presented in [4,20], each with a single global copy of the A matrix and multiple b matrices. We will benchmark these implementations against cuThomasBatch [3] (implemented as gtsvInterleavedBatch in the cuSPARSE library) and existing work by the authors cuPentBatch [4], for tridiagonal and pentadiagonal matrices respectively, which are the existing state of the art for the algorithms we are interested in. We point the user to the papers [3] and [4] for existing benchmark comparisons of these algorithms with multiple A matrices with the cuSPARSE library and comparisons with serial/OpenMP implementations.…”