A direct solver with O(N) complexity for integral equations on one-dimensional domains

TL;DR

This paper introduces a linear-time direct solver for integral equations on one-dimensional domains, leveraging low-rank structures in the matrix to efficiently invert systems arising from discretizations.

Contribution

It presents a novel O(N) complexity algorithm for direct inversion of boundary integral equations, applicable to non-oscillatory and oscillatory kernels with efficiency considerations.

Findings

Achieves O(N) complexity for non-oscillatory kernels

Efficient for long and intermediate wave-lengths in oscillatory kernels

Relates to H and H^2 matrix techniques and Hierarchically Semi-Separable matrices

Abstract

An algorithm for the direct inversion of the linear systems arising from Nystrom discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes' equations. The scaling coefficient suppressed by the "big-O" notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H and H^2 matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.