GiMMiK - Generating Bespoke Matrix Multiplication Kernels for Accelerators: Application to High-Order Computational Fluid Dynamics
File(s)
Published version
Accepted version
Author(s)
Wozniak, BD
Witherden, FD
Russell, FP
Vincent, PE
Kelly, PHJ
Type
Journal Article
Abstract
Matrix multiplication is a fundamental linear algebra routine
ubiquitous in all areas of science and engineering. Highly optimised BLAS libraries (cuBLAS and clBLAS on GPUs) are the most popular choices for an implementation of the General Matrix Multiply (GEMM) in software. In this paper we present GiMMiK - a generator of bespoke matrix multiplication kernels for the CUDA and OpenCL platforms. GiMMiK exploits a prior knowledge of the operator matrix to generate highly performant code. The performance of GiMMiK's kernels is particularly apparent in a block-bypanel type of matrix multiplication, where the block matrix is typically small (e.g. dimensions of 96 × 64). Such operations are characteristic to our motivating application in PyFR - an implementation of Flux Reconstruction schemes for high-order fluid flow simulations on mixed unstructured meshes. GiMMiK fully unrolls the matrix-vector product and embeds matrix entries directly in the code to benefit from the use of the constant cache and compiler optimisations. Further, it reduces the number of floating-point operations by removing multiplications by zeros. Together with the ability of our kernels to avoid the poorly optimised cleanup code, executed by library GEMM, we are able to outperform cuBLAS on two NVIDIA GPUs: GTX 780 Ti and Tesla K40c. We observe speedups of our kernels over cuBLAS GEMM of up to 9.98 and 63.30 times for a 294 × 1029 99% sparse PyFR matrix in double precision on the Tesla K40c and GTX 780 Ti correspondingly. In single precision, observed speedups reach 12.20 and 13.07 times for a 4 × 8 50% sparse PyFR matrix on the two aforementioned cards. Using GiMMiK as the matrix multiplication kernel provider allows us to achieve a speedup of up to 1.70 (2.19) for a simulation of an unsteady flow over a cylinder executed with PyFR in double (single) precision on the Tesla K40c. All results were generated with GiMMiK version 1.0.
ubiquitous in all areas of science and engineering. Highly optimised BLAS libraries (cuBLAS and clBLAS on GPUs) are the most popular choices for an implementation of the General Matrix Multiply (GEMM) in software. In this paper we present GiMMiK - a generator of bespoke matrix multiplication kernels for the CUDA and OpenCL platforms. GiMMiK exploits a prior knowledge of the operator matrix to generate highly performant code. The performance of GiMMiK's kernels is particularly apparent in a block-bypanel type of matrix multiplication, where the block matrix is typically small (e.g. dimensions of 96 × 64). Such operations are characteristic to our motivating application in PyFR - an implementation of Flux Reconstruction schemes for high-order fluid flow simulations on mixed unstructured meshes. GiMMiK fully unrolls the matrix-vector product and embeds matrix entries directly in the code to benefit from the use of the constant cache and compiler optimisations. Further, it reduces the number of floating-point operations by removing multiplications by zeros. Together with the ability of our kernels to avoid the poorly optimised cleanup code, executed by library GEMM, we are able to outperform cuBLAS on two NVIDIA GPUs: GTX 780 Ti and Tesla K40c. We observe speedups of our kernels over cuBLAS GEMM of up to 9.98 and 63.30 times for a 294 × 1029 99% sparse PyFR matrix in double precision on the Tesla K40c and GTX 780 Ti correspondingly. In single precision, observed speedups reach 12.20 and 13.07 times for a 4 × 8 50% sparse PyFR matrix on the two aforementioned cards. Using GiMMiK as the matrix multiplication kernel provider allows us to achieve a speedup of up to 1.70 (2.19) for a simulation of an unsteady flow over a cylinder executed with PyFR in double (single) precision on the Tesla K40c. All results were generated with GiMMiK version 1.0.
Date Issued
2016-01-18
Date Acceptance
2015-12-21
Citation
Computer Physics Communications, 2016, 202, pp.12-22
ISSN
0010-4655
Publisher
Elsevier
Start Page
12
End Page
22
Journal / Book Title
Computer Physics Communications
Volume
202
Copyright Statement
© 2016 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
License URL
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (E
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/I006761/1
EP/I00677X/1
EP/I012036/1
EP/L000407/1
EP/K027379/1
Subjects
Matrix Multiplication
GPU
CUDA
Code Generation
GEMM
GiMMiK
cuBLAS
Publication Status
Published