GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
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
An Efficient Iterative Method for the Generalized Stokes Problem
SIAM Journal on Scientific Computing
Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods
SIAM Journal on Scientific Computing
Multipole-based preconditioners for large sparse linear systems
Parallel Computing - Parallel matrix algorithms and applications (PMAA '02)
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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. In our prior work [15], we have demonstrated the use of multipole-based techniques as effective parallel preconditioners for sparse linear systems. At one extreme, multipole-based preconditioners behave as dense (bounded interaction) matrices (multipole degree 0), while at the other extreme, they are represented entirely as series expansions. In this paper, we show that: (i) merely truncating the kernel of the integral operator generating the preconditioner leads to poor convergence properties; (ii) far-field interactions, in the form of multipoles, are critical for rapid convergence; (iii) the importance and required accuracy of far-field interactions varies with the complexity of the problem; and (iv) the preconditioner resulting from a judicious mix of near and far-field interactions yields excellent convergence and parallelization properties. Our experimental results are illustrated on the Poisson problem and the generalized Stokes problem arising in incompressible fluid flow simulations.