Distributed ℋ2-matrices for non-local operators

Abstract

ℋ2-matrices can be used to approximate dense n × n matrices resulting from the discretization of certain non-local operators (e.g., Fredholm-type integral operators) in script O sign(n k) units of storage, where k is a parameter controlling the accuracy of the approximation. Since typically k ≪ n holds, this representation is much more efficient than the conventional representation by a two-dimensional array. For very large problem dimensions, the amount of available storage becomes a limiting factor for practical algorithms. A popular way to provide sufficiently large amounts of storage at relatively low cost is to use a cluster of inexpensive computers that are connected by a network. This paper presents a method for managing an ℋ2-matrix on a distributed-memory cluster that can be proven to be of almost optimal parallel efficiency.