Optimising finite-difference methods for PDEs through parameterised time-tiling in Devito

TL;DR

This paper introduces parameterised time-tiling in Devito, significantly improving the efficiency of finite-difference PDE solvers by reducing runtime and simplifying the modeling process for seismic imaging applications.

Contribution

It implements time-tiling optimization in Devito, provides an estimator for arithmetic intensity, and models runtime improvements, including generalization to imperfect loop nests.

Findings

Runtime decreased by up to 45% with time-tiling.

At least 20% performance gain across acoustic wave equation stencils.

Effective prediction of runtime improvements using the proposed model.

Abstract

Finite-difference methods are widely used in solving partial differential equations. In a large problem set, approximations can take days or weeks to evaluate, yet the bulk of computation may occur within a single loop nest. The modelling process for researchers is not straightforward either, requiring models with differential equations to be translated into stencil kernels, then optimised separately. One tool that seeks to speed up and eliminate mistakes from this tedious procedure is Devito, used to efficiently employ finite-difference methods. In this work, we implement time-tiling, a loop nest optimisation, in Devito yielding a decrease in runtime of up to 45%, and at least 20% across stencils from the acoustic wave equation family, widely used in Devito's target domain of seismic imaging. We present an estimator for arithmetic intensity under time-tiling and a model to predict runtime improvements in stencil computations. We also consider generalisation of time-tiling to imperfect loop nests, a less widely studied problem.