Sparse Preconditioned Iterative Methods for Dense Linear Systems

Two sparse preconditioned iterative methods are presented to solve dense linear systems arising in the solution of two dimensional boundary integral equations. In the rst method, the sparse preconditioner is constructedsimply by choosing a small block of elements in the coeecient matrix of a dense linear system. The two-grid method falls into this category when the dense linear system arises from the Nystr om method for a second kind boundary integral equation. In the second method, the sparse preconditioner is obtained through condensation of the coeecient matrix by discrete Fourier transforms, which can be implemented eeciently using fast Fourier transforms. Both iterative methods involve only O(N 2) arithmetic operations per iteration, and converge rapidly when the dense linear systems arise from quadrature methods for boundary integral equations arising in two dimensional problems. Our numerical experiments demonstrate the computational eeciency of each method.