impact of far-field interactions on performance of multipole-based preconditioners for 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. 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.
preconditioning stokes problem iterative methods multipole methods