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.

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
• Public URL: https://durham-repository.worktribe.com/output/1303781
• Related Public URLs: https://arxiv.org/pdf/1903.08243.pdf