research-article

A Fast Algorithm for Aperiodic Linear Stencil Computation using Fast Fourier Transforms

Authors:

Rezaul Chowdhury,

Pramod Ganapathi,

Yimin ZhuAuthors Info & Claims

ACM Transactions on Parallel Computing, Volume 10, Issue 4

Article No.: 22, Pages 1 - 34

https://doi.org/10.1145/3606338

Published: 14 December 2023 Publication History

Abstract

Stencil computations are widely used to simulate the change of state of physical systems across a multidimensional grid over multiple timesteps. The state-of-the-art techniques in this area fall into three groups: cache-aware tiled looping algorithms, cache-oblivious divide-and-conquer trapezoidal algorithms, and Krylov subspace methods.

In this article, we present two efficient parallel algorithms for performing linear stencil computations. Current direct solvers in this domain are computationally inefficient, and Krylov methods require manual labor and mathematical training. We solve these problems for linear stencils by using discrete Fourier transforms preconditioning on a Krylov method to achieve a direct solver that is both fast and general. Indeed, while all currently available algorithms for solving general linear stencils perform Θ(NT) work, where N is the size of the spatial grid and T is the number of timesteps, our algorithms perform o(NT) work.

To the best of our knowledge, we give the first algorithms that use fast Fourier transforms to compute final grid data by evolving the initial data for many timesteps at once. Our algorithms handle both periodic and aperiodic boundary conditions and achieve polynomially better performance bounds (i.e., computational complexity and parallel runtime) than all other existing solutions.

Initial experimental results show that implementations of our algorithms that evolve grids of roughly 10⁷ cells for around 10⁵ timesteps run orders of magnitude faster than state-of-the-art implementations for periodic stencil problems, and 1.3× to 8.5× faster for aperiodic stencil problems.

Code Repository: https://github.com/TEAlab/FFTStencils

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Cited By

Han HLi KCui WBai DZhang YYuan LChen YZhang YCao TYang M(2025)FlashFFTStencil: Bridging Fast Fourier Transforms to Memory-Efficient Stencil Computations on Tensor Core UnitsProceedings of the 30th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming10.1145/3710848.3710897(355-368)Online publication date: 28-Feb-2025
https://dl.acm.org/doi/10.1145/3710848.3710897

Index Terms

A Fast Algorithm for Aperiodic Linear Stencil Computation using Fast Fourier Transforms
1. Computing methodologies
2. Mathematics of computing
  1. Mathematical software
    1. Solvers

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Published In

cover image ACM Transactions on Parallel Computing

ACM Transactions on Parallel Computing Volume 10, Issue 4

December 2023

138 pages

ISSN:2329-4949

EISSN:2329-4957

DOI:10.1145/3637451

Editor:
David A. Bader
New Jersey Institute of Technology, USA

Issue’s Table of Contents

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Publication History

Published: 14 December 2023

Online AM: 24 July 2023

Accepted: 27 June 2023

Received: 16 February 2022

Published in TOPC Volume 10, Issue 4

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Han HLi KCui WBai DZhang YYuan LChen YZhang YCao TYang M(2025)FlashFFTStencil: Bridging Fast Fourier Transforms to Memory-Efficient Stencil Computations on Tensor Core UnitsProceedings of the 30th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming10.1145/3710848.3710897(355-368)Online publication date: 28-Feb-2025
https://dl.acm.org/doi/10.1145/3710848.3710897

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