Multipole-based preconditioners for large sparse linear systems

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. 2003 Elsevier B.V. All rights reserved.