Generalized Source Integral Equation Calderón Preconditioning in Two-Dimensions

Generalized source integral equations (GSIEs) are surface integral equations (SIEs) that make use of a modified Green's function kernel to increase the rank deficiency of moment matrix off-diagonal blocks. Their modified kernels include auxiliary contributions that can be attributed to sources internal to the scatterer surface. These are designed to approximately cancel the surface source radiation into the scatterer and, by that, reduce the broadside components of the interactions and make dominant end-fire components. This reduction of the effective dimensionality of the interactions translates to slower scaling of the rank with the electrical dimension of the objects, improved low-rank (LR) compression, and better asymptotic scaling of the performance of LR approximation-based fast solvers. Various mechanisms for producing the auxiliary contributions have been proposed, including induced currents on reflective shields, equivalent extinction sources on 'absorbing' shields, and multipoles. Common to all formulation is their derivation as a modification of the conventional electric field integral equation (EFIE). As such, they may inherit the various ill-conditioning mechanisms that plague the EFIE, such as the dense-discretization and high-frequency breakdowns, as well as non-uniqueness at resonance frequencies. Our recent work on the topic focuses on the analytical and numerical investigation of these issues and the development of cures thereof.