Tight Bounds for Low Dimensional Star Stencils in the External Memory Model

Abstract

Stencil computations on low dimensional grids are kernels of many scientific applications including finite difference methods used to solve partial differential equations. On typical modern computer architectures such stencil computations are limited by the performance of the memory subsystem, namely by the bandwidth between main memory and the cache. This work considers the computation of star stencils, like the 5-point and 7-point stencil, in the external memory model. The analysis focuses on the constant of the leading term of the non-compulsory I/Os. Optimizing stencil computations is an active field of research, but so far, there has been a significant gap between the lower bounds and the performance of the algorithms. In two dimensions, matching constants for lower and upper bounds are provided closing a gap of 4. In three dimensions, the bounds match up to a factor of $\sqrt{2}$ improving the known results by a factor of 2$\sqrt{3}\sqrt{B}$, where B is the block (cache line) size of the external memory model. For higher dimensions n, the presented lower bounds improve the previously known by a factor between 4 and 6 leaving a gap of $\sqrt[n-1]{n!} \thickapprox{{n} \over{e}}$.