A study of vectorization for matrix-free finite element methods

Sun, Tianjiao; Mitchell, Lawrence; Kulkarni, Kaushik; Klöckner, Andreas; Ham, David A; Kelly, Paul H.J.

doi:10.1177/1094342020945005

A study of vectorization for matrix-free finite element methods

Sun, Tianjiao; Mitchell, Lawrence; Kulkarni, Kaushik; Klöckner, Andreas; Ham, David A; Kelly, Paul H.J.

Authors

Tianjiao Sun

Lawrence Mitchell

Kaushik Kulkarni

Andreas Klöckner

David A Ham

Paul H.J. Kelly

Abstract

Vectorization is increasingly important to achieve high performance on modern hardware with SIMD instructions. Assembly of matrices and vectors in the finite element method, which is characterized by iterating a local assembly kernel over unstructured meshes, poses difficulties to effective vectorization. Maintaining a user-friendly high-level interface with a suitable degree of abstraction while generating efficient, vectorized code for the finite element method is a challenge for numerical software systems and libraries. In this work, we study cross-element vectorization in the finite element framework Firedrake via code transformation and demonstrate the efficacy of such an approach by evaluating a wide range of matrix-free operators spanning different polynomial degrees and discretizations on two recent CPUs using three mainstream compilers. Our experiments show that our approaches for cross-element vectorization achieve 30% of theoretical peak performance for many examples of practical significance, and exceed 50% for cases with high arithmetic intensities, with consistent speed-up over (intra-element) vectorization restricted to the local assembly kernels.

Citation

Sun, T., Mitchell, L., Kulkarni, K., Klöckner, A., Ham, D. A., & Kelly, P. H. (2020). A study of vectorization for matrix-free finite element methods. International Journal of High Performance Computing Applications, 34(6), 629-644. https://doi.org/10.1177/1094342020945005

Journal Article Type	Article
Acceptance Date	Jun 16, 2020
Online Publication Date	Jul 31, 2020
Publication Date	2020-11
Deposit Date	Apr 16, 2019
Publicly Available Date	Jul 23, 2020
Journal	International Journal of High Performance Computing Applications
Print ISSN	1094-3420
Electronic ISSN	1741-2846
Publisher	SAGE Publications
Peer Reviewed	Peer Reviewed
Volume	34
Issue	6
Pages	629-644
DOI	https://doi.org/10.1177/1094342020945005
Related Public URLs	https://arxiv.org/pdf/1903.08243.pdf

Files

Accepted Journal Article (669 Kb)
PDF

Copyright Statement
This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).