Communication complexity of the fast multipole method and its algebraic variants

R. Yokota, G. Turkiyyah, and D. Keyes
Supercomputing Frontiers and Innovations, 1, pp. 63-84, (2014)

Communication complexity of the fast multipole method and its algebraic variants

Keywords

Communication complexity, Hierarchical low-rank approximation, Fast multipole methods, H-matrices, Sparse solvers

Abstract

​A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for efficient implementation of many linear algebraic operations of scientific computing at the exascale. The combination provides asymptot- ically optimal computational and communication complexity and applicability to large classes of operators that commonly arise in scientific computing applications. A convergence of the mathe- matical theories of the fast multipole and H-matrix methods has been underway for over a decade. We recap this mathematical unification and describe implementation aspects of a hybrid of these two compelling hierarchical algorithms on hierarchical distributed-shared memory architectures, which are likely to be the first to reach the exascale. We present a new communication complexity estimate for fast multipole methods on such architectures. We also show how the data structures and access patterns of H-matrices for low-rank operators map onto those of fast multipole, leading to an algebraically generalized form of fast multipole that compromises none of its architecturally ideal properties.

Code

DOI: 10.14529/jsfi140104

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