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 first method, the sparse preconditioner is constructed simply by choosing a small block of elements in the coefficient matrix of a dense linear system. The two-grid method falls into this category when the dense linear system arises from the Nystrom method for a second kind boundary integral equation. In the second method, the sparse preconditioner is obtained through condensation of the coefficient matrix by discrete Fourier transforms, which can be implemented efficiently 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. The author’s numerical experiments demonstrate the computational efficiency of each method.