Vector-aware space cuts in stencil computations

Abstract

This thesis presents work on how cache-efficient serial and parallel stencil computations can be optimized to efficiently use vector instructions on modern general-purpose processors. A stencil computation iteratively updates each point of a d-dimensional grid based on a function of itself and its nearby neighbors. Stencil computations have a wide range of applications from simulation to machine learning, and are often time-intensive. Although a stencil can be computed by repeatedly looping over its d- dimensional grid, previous studies have shown that various parallel and cache-efficient algorithms, including recursive cache-oblivious algorithms, can compute stencils more efficiently. However, these recursive cache-oblivious algorithms can disrupt vectorization and increase the number of instructions required for the computation. Vector-aware space cuts can be used to preserve vectorization in serial and parallel cache-oblivious algorithms for stencil computations, thereby improving their performance. I present improved theoretical analyses of the commonly used and efficient parallel cache-oblivious stencil algorithm due to Frigo and Strumpen to account for vector operations, and we show that vector-aware space cuts improve the asymptotic work -- total computation -- of the algorithm by a factor of the maximum vector width. I present an improved analysis of the algorithm's span -- length of a longest path of dependencies in the computation -- to be O(rhw1 / lg(2(r??1))), where r is the number of trapezoids created in a parallel cut. I show that these theoretical improvements are borne out in practice by demonstrating up to a 1:61 X speed-up in the serial stencil computation and a 1:08 X speed-up in the parallel computation.