Model-based autotuning of discretization methods in numerical simulations of partial differential equations

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We present an autotuning approach for compile-time optimization of numerical discretization methods in simulations of partial differential equations. Our approach is based on data-driven regression of performance models for numerical methods. We use these models at compile time to automatically determine the parameters (e.g., resolution, time step size, etc.) of numerical simulations of continuum spatio-temporal models in order to optimize the tradeoff between simulation accuracy and runtime. The resulting autotuner is developed for the compiler of a Domain-Specific Language (DSL) for numerical simulations. The abstractions in the DSL enable the compiler to automatically determine the performance models and know which discretization parameters to tune. We demonstrate that this high-level approach can explore a large space of possible simulations, with simulation runtimes spanning multiple orders of magnitude. We evaluate our approach in two test cases: the linear diffusion equation and the nonlinear Gray-Scott reaction–diffusion equation. The results show that our model-based autotuner consistently finds configurations that outperform those found by state-of-the-art general-purpose autotuners. Specifically, our autotuner yields simulations that are on average 4.2x faster than those found by the best generic exploration algorithms, while using 16x less tuning time. Compared to manual tuning by a group of researchers with varying levels of expertise, the autotuner was slower than the best users by not more than a factor of 2, whereas it was able to significantly outperform half of them.