Low-rank structure for interactions between Fourier indices

Determine whether underlying low-rank structures exist for the curve integral operators that describe interactions between two different Fourier indices in the Fourier-transformed boundary integral equations for three-dimensional acoustic scattering by general smooth surfaces, with the goal of developing fast solvers for the resulting linear systems.

Background

Fourier transformation in the azimuthal variable converts the surface boundary integral equations into coupled curve integral equations. For general smooth surfaces, interactions between different Fourier indices do not vanish, so assembling and solving the resulting linear systems remains computationally demanding.

The conclusion identifies the existence of exploitable low-rank structure in these cross-Fourier-index interactions as an unresolved question. Such a structure could enable fast iterative or direct solvers; the authors contrast this with axisymmetric surfaces, where the corresponding interactions vanish entirely.