Olver and I recently developed a fast and stable algorithm for the solution of singular integral equations. It is a new systematic approach for converting singular integral equations into almost-banded and block-banded systems of equations. The structures of these systems lend themselves to fast direct solution via the adaptive QR factorization. However, as the number of disjoint boundaries increases, the computational effectiveness deteriorates and specialized linear algebra is required.
Our starting point for specialized linear algebra is an alternative algorithm based on a recursive block LU factorization recently developed by Aminfar, Ambikasaran, and Darve. This algorithm specifically exploits the hierarchically off-diagonal low-rank structure arising from coercive singular integral operators of elliptic partial differential operators. 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 takes a fraction of the original time. Our fast direct solver allows for the exploration of reduced-basis problems, where the boundary density for any incident plane wave can be represented by a periodic Fourier series whose coefficients are in turn expanded in weighted Chebyshev or ultraspherical bases.
A fractal antenna uses a self-similar design to achieve excellent broadband performance. Similarly, a fractal screen uses a fractal such as a Cantor set to screen electromagnetic radiation. Hewett, Langdon, and Chandler-Wilde have shown recently that the density on the nth convergent to a fractal screen converges to a non-zero element in the suitable Sobolev space, resulting in a physically observable and persistent scattered field as n tends to infinity. We use our hierarchical solver to show numerical results for prefractal screens.
