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

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.