Preconditioned, Adaptive, Multipole-Accelerated Iterative Methods for Three-Dimensional First-Kind Integral Equations of Potential Theory

This paper presents a preconditioned, Krylov-subspace iterative algorithm, where a modified multipole algorithm with a novel adaptation scheme is used to compute the iterates for solving dense matrix problems generated by Galerkin or collocation schemes applied to three-dimensional, first-kind, integral equations that arise in potential theory. A proof is given that this adaptive algorithm reduces both matrix-vector product computation time and storage to order N, and experimental evidence is given to demonstrate that the combined preconditioned, adaptive, multipole-accelerated (PAMA) method is nearly order N in practice. Examples from engineering applications are given to demonstrate that the accelerated method is substantially faster than standard algorithms on practical problems.