A fast algorithm for particle simulations
Journal of Computational Physics
A parallel hashed Oct-Tree N-body algorithm
Proceedings of the 1993 ACM/IEEE conference on Supercomputing
A parallel adaptive fast multipole method
Proceedings of the 1993 ACM/IEEE conference on Supercomputing
Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods
SIAM Journal on Scientific Computing
The fast multipole method: numerical implementation
Journal of Computational Physics
The Fast Multipole Method I: Error Analysis and Asymptotic Complexity
SIAM Journal on Numerical Analysis
Proceedings of the 18th annual international conference on Supercomputing
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Dense operators for preconditioning sparse linear systems have traditionally been considered infeasible due to their excessive computational and memory requirements. With the emergence of techniques such as block low-rank approximations and hierarchical multipole approximations, the cost of computing and storing these preconditioners has reduced dramatically. This paper describes the use of multipole operators as parallel preconditioners for sparse linear systems. Hierarchical multipole approximations of explicit Green's functions are effective preconditioners due to their bounded-error properties. By enumerating nodes in proximity preserving order, one can achieve high parallel efficiency in computing matrix-vector products with these dense preconditioners. The benefits of the approach are illustrated on the Poisson problem and the generalized Stokes problem arising in incompressible fluid flow simulations. Numerical experiments show that the multipole-based techniques are effective preconditioners that can be parallelized efficiently on multiprocessing platforms.