Cache-Efficient Multigrid Algorithms

Abstract

Multigrid is widely used as an efficient solver for sparse linear systems arising from the discretization of elliptic boundary value problems. Linear relaxation methods such as Gauss–Seidel and Red–Black Gauss–Seidel form the principal computational component of multigrid, and thus affect its efficiency. In the context of multigrid, these iterative solvers are executed for a small number of iterations (2–8). We exploit this property of the algorithm to develop a cache-efficient multigrid method, by focusing on improving the memory behavior of the linear relaxation methods. The efficiency in our cache-efficient linear relaxation algorithm comes from two sources: reducing the number of data cache and TLB misses, and reducing the number of memory references by keeping values registerresident. Our optimizations are applicable to multigrid applied to linear systems arising from constant coefficient elliptic PDEs on structured grids. Experiments on five modern computing platforms show a performance improvement of 1.15–2.7 times over a standard implementation of Full Multigrid V-Cycle.