Optimisations for quadrature representations of finite element tensors through automated code generation

TL;DR

This paper explores automated code generation optimizations for finite element matrix computations using quadrature, demonstrating improved performance for complex variational forms through various strategies.

Contribution

It introduces new optimization strategies for code generation of finite element matrices, especially for complex forms, enhancing performance and scalability.

Findings

Optimizations significantly improve runtime performance for complex forms

Automated code generation reduces manual coding effort and errors

Different representations have trade-offs in code size and generation time

Abstract

We examine aspects of the computation of finite element matrices and vectors which are made possible by automated code generation. Given a variational form in a syntax which resembles standard mathematical notation, the low-level computer code for building finite element tensors, typically matrices, vectors and scalars, can be generated automatically via a form compiler. In particular, the generation of code for computing finite element matrices using a quadrature approach is addressed. For quadrature representations, a number of optimisation strategies which are made possible by automated code generation are presented. The relative performance of two different automatically generated representations of finite element matrices is examined, with a particular emphasis on complicated variational forms. It is shown that approaches which perform best for simple forms are not tractable for more complicated problems in terms of run time performance, the time required to generate the code or the size of the generated code. The approach and optimisations elaborated here are effective for a range of variational forms.