A study of vectorization for matrix-free finite element methods

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 vectoriz...

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Bibliographic Details
Published in:The international journal of high performance computing applications 2020-11, Vol.34 (6), p.629-644
Main Authors: Sun, Tianjiao, Mitchell, Lawrence, Kulkarni, Kaushik, Klöckner, Andreas, Ham, David A, Kelly, Paul HJ
Format: Article
Language:English
Subjects:
Assembly
Finite element analysis
Finite element method
Kernels
Mathematical analysis
Matrix algebra
Matrix methods
Operators (mathematics)
Polynomials
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Summary: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.
ISSN:1094-3420
1741-2846
DOI:10.1177/1094342020945005