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Fast Stencil Computations using Fast Fourier Transforms
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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 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 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× to 8.5× faster for aperiodic stencil problems.more » « less
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Modern vector processors support a wide variety of instructions for fixed-point digital signal processing. These instructions support a proliferation of rounding, saturating, and type conversion modes, and are often fused combinations of more primitive operations. While these are common idioms in fixed-point signal processing, it is difficult to use these operations in portable code. It is challenging for programmers to write down portable integer arithmetic in a C-like language that corresponds exactly to one of these instructions, and even more challenging for compilers to recognize when these instructions can be used. Our system, Pitchfork, defines a portable fixed-point intermediate representation, FPIR, that captures common idioms in fixed-point code. FPIR can be used directly by programmers experienced with fixed-point, or Pitchfork can automatically lift from integer operations into FPIR using a term-rewriting system (TRS) composed of verified manual and automatically-synthesized rules. Pitchfork then lowers from FPIR into target-specific fixed-point instructions using a set of target-specific TRSs. We show that this approach improves runtime performance of portably-written fixed-point signal processing code in Halide, across a range of benchmarks, by geomean 1.31× on x86 with AVX2, 1.82× on ARM Neon, and 2.44× on Hexagon HVX compared to a standard LLVM-based compiler flow, while maintaining or improving existing compile times.more » « less
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3409964.3461803
