A fast hierarchical direct solver for singular integral equations defined on disjoint boundaries and application to fractal screens
Abstract
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.