通过Devito中的参数化时间分片优化偏微分方程的有限差分方法
Optimising finite-difference methods for PDEs through parameterised time-tiling in Devito
摘要：
差分法是求解偏微分方程的一种常用方法。在一个大的问题集中，近似值可能需要几天或几周的时间来评估，但大部分计算可能发生在一个循环嵌套中。研究人员的建模过程也不简单，需要将微分方程模型转换为模板内核，然后单独优化。Devito是一种旨在加快速度并消除这一繁琐过程中的错误的工具，用于有效地使用有限差分方法。在这项工作中，我们实现时间平铺，循环嵌套优化，在Devito产生的运行时间减少高达45%，至少20%的声波方程家族，广泛用于Devito的目标域的地震成像的整个stenetry。我们提出了一个估计的算术强度下的时间平铺和模型来预测模板计算的运行时改进。我们还考虑推广的时间瓦片不完美的循环巢，一个不太广泛研究的问题。
英文摘要：
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.
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