A new spectral method approach to singular integral equations

Abstract: 

I will describe a new spectral method approach for solving univariate singular integral equations on a union of multiple disjoint boundaries. Singular integral equations have a rich history in classical applications such as acoustic scattering for electromagnetics and seismic imaging, fracture mechanics, fluid dynamics, and beam physics.

The new spectral method uses several remarkable properties of Chebyshev polynomials including the geometric convergence of expansions with analyticity, explicit formulae for their Hilbert and Cauchy transforms, and fast low-rank bivariate approximation to integral kernels. Chebyshev and ultraspherical polynomials are utilized to convert singular integral operators into numerically banded infinite-dimensional operators. The resulting system can be solved in O(n) operations using an adaptive QR factorization, where n is the optimal number of unknowns needed to resolve the solution to high pre-determined accuracy. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. For fast and spectrally accurate numerical evaluation of the scattered field in the whole complex plane, analytical formulae for the single- and double-layer operators are derived. These formulae allow for accurate numerical evaluation arbitrarily close to the boundary. Applications include the mathematics of the Faraday cage and acoustic scattering for the Helmholtz and gravity Helmholtz equations.

When the number of disjoint boundaries increases, the effectiveness of banded linear algebra is diminished. The starting point for more specialized linear algebra is an alternative algorithm based on hierarchical block diagonalization. This algorithm specifically exploits the hierarchically off-diagonal low-rank structure in singular integral operators implied by the strong admissibility criterion satisfied by asymptotically smooth integral operators. While close in spirit to the Fast Multipole Method of Greengard and Rokhlin, we apply it to the banded representation of singular integral operators instead of discretizations arising from quadrature rules. The hierarchical solver involves a pre-computation phase independent of the right-hand side. Once this pre-computation factorizes the operator, the solution of many right-hand sides can be performed with reduced complexity. Applications include fractal screens and near-resonance involving multiple disjoint boundaries.