Efficient out-of-core algorithms for linear relaxation using blocking covers

When a numerical computation fails to fit in the primary memory of a serial or parallel computer, a so-called “out-of-core” algorithm must be used which moves data between primary and secondary memories. In this paper, we study out-of-core algorithms for sparse linear relaxation problems in which each iteration of the algorithm updates the state of every vertex in a graph with a linear combination of the states of its neighbors. We give a general method that can save substantially on the I/O traffic for many problems. For example, our technique allows a computer with M words of primary memory to perform T=Ω(M^1/5) cycles of a multigrid algorithm for a two-dimensional elliptic solver over an n-point domain using only Θ(nT/M^1/5) I/O transfers, as compared with the naive algorithm which requires Ω(nT) I/O´s