Acceleration of tensor-product operations for high-order finite element methods

TL;DR

This paper focuses on optimizing GPU implementations of tensor-product operators used in high-order finite element methods, providing a performance model and optimization strategies to enhance computational efficiency.

Contribution

It introduces a detailed optimization approach and a performance model for tensor-product operators in finite element methods on GPUs, addressing low arithmetic intensity challenges.

Findings

Achieved near-peak GPU performance for tensor-product operators.

Developed a performance model calibrated with empirical data.

Identified key optimization strategies for low arithmetic intensity kernels.

Abstract

This paper is devoted to GPU kernel optimization and performance analysis of three tensor-product operators arising in finite element methods. We provide a mathematical background to these operations and implementation details. Achieving close-to-the-peak performance for these operators requires extensive optimization because of the operators' properties: low arithmetic intensity, tiered structure, and the need to store intermediate results inside the kernel. We give a guided overview of optimization strategies and we present a performance model that allows us to compare the efficacy of these optimizations against an empirically calibrated roofline.