Fast Multipole Preconditioners for Sparse Matrices Arising from Elliptic
Equations
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by
Huda Ibeid, Rio Yokota, Jennifer Pestana, David Keyes
2016
Abstract
Among optimal hierarchical algorithms for the computational solution of
elliptic problems, the Fast Multipole Method (FMM) stands out for its
adaptability to emerging architectures, having high arithmetic intensity,
tunable accuracy, and relaxable global synchronization requirements. We
demonstrate that, beyond its traditional use as a solver in problems for which
explicit free-space kernel representations are available, the FMM has
applicability as a preconditioner in finite domain elliptic boundary value
problems, by equipping it with boundary integral capability for satisfying
conditions at finite boundaries and by wrapping it in a Krylov method for
extensibility to more general operators. Here, we do not discuss the well
developed applications of FMM to implement matrix-vector multiplications within
Krylov solvers of boundary element methods. Instead, we propose using FMM for
the volume-to-volume contribution of inhomogeneous Poisson-like problems, where
the boundary integral is a small part of the overall computation. Our method
may be used to precondition sparse matrices arising from finite
difference/element discretizations, and can handle a broader range of
scientific applications. Compared with multigrid methods, it is capable of
comparable algebraic convergence rates down to the truncation error of the
discretized PDE, and it offers potentially superior multicore and distributed
memory scalability properties on commodity architecture supercomputers.
Compared with other methods exploiting the low rank character of off-diagonal
blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may
reduce the amount of communication because it is matrix-free and exploits the
tree structure of FMM. We describe our tests in reproducible detail with freely
available codes and outline directions for further extensibility.
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