Fast Stencil Computations using Fast Fourier Transforms

TL;DR

This paper introduces novel algorithms using fast Fourier transforms to perform linear stencil computations more efficiently, significantly reducing computational work and runtime compared to existing methods, for both periodic and aperiodic boundary conditions.

Contribution

The paper presents the first algorithms that leverage FFTs to compute stencil evolutions in a single step, achieving sublinear work and better performance bounds than prior approaches.

Findings

Algorithms perform o(NT) work, improving over Θ(NT) of previous methods.

Experimental results show orders of magnitude speedup for large grids and timesteps.

Effective for both periodic and aperiodic boundary conditions.

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 paper, 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 DFT preconditioning on a Krylov method to achieve a direct solver which is both fast and general. Indeed, while all currently available algorithms for solving general linear stencils perform $\Theta(NT)$ 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 $10^7$ cells for around $10^5$ timesteps run orders of magnitude faster than state-of-the-art implementations for periodic stencil problems, and 1.3$\times$ to 8.5$\times$ faster for aperiodic stencil problems.