Fast Stencil Computations using Fast Fourier Transforms

Abstract

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.

Gamify Stencil Dwarf on Cloud for Democratizing Scientific Computing

“…State-of-the-arts Although recent studies on Stencil Dwarf only exhibit their own absolute performance without comparison in experiments [37,66,1], we reproduce artifacts exhaustively and compare the performance of different state-of-the-arts for a comprehensive analysis. Two classic vectorization methods (Auto Vectorization [35] and Data Reorganization [64]) are employed first as a standard baseline on CPUs.…”

“…C(VR) Pluto [7] Diamond [9] AutoVec. C(VR) Folding [34] Polyhedral Folding C(VR) Tetris (CPU) Tessellate Skewed 2 C(VR) Brick [66] Brick Scatter G(CCU) AN5D [37] Temporal -G(CCU) Tetris (GPU) Checkerboard Trapezoid G(TCU) Tetris Polymorphic Template C(VR) + G(TCU) 1 For better clarity, CPU, GPU, Vector Register, CUDA Core Units and Tensor Core Units are abbreviated with C, G, VR, CCU and TCU respectively. 2 Notice we use a keyword as an abbreviation for some algorithms.…”

Fast Option Pricing using Nonlinear Stencils

“…The value computed for the root node is the required option value. Straightforward iterative implementation of this method runs in time Θ 𝑇 2 . This method can be used for both European and American options.…”

“…A stencil is called linear if it computes the value of a cell at time step 𝑡 as a fixed linear combination of cell values at time steps before 𝑡, otherwise, it is called nonlinear. For 1D linear stencils Ahmad et al [2] provide FFT-based algorithms that take O (𝑇 log𝑇 ) time for periodic grids and O 𝑇 log 2 𝑇 time for aperiodic grids, assuming the size of the input grid to be Θ (𝑇 ).…”

“…They have the same computational complexity of Θ (𝑁𝑇 ) where 𝑁 is the size of the grid. Recently, Ahmad et al [2] presented an FFT-based algorithm that is the first algorithm to perform linear stencil computations in 𝑜 (𝑁𝑇 ) computational complexity.…”