On the use of graph Laplacians in the integral equation modeling of complex and multiscale problems

Integral Equations are widely used in Computational Electromagnetics for solving radiation and scattering problems. When solved with the Boundary Element Method (BEM), integral equations give rise to dense linear systems. When the system´s dimensionality is large, fast iterative or direct solvers have to be used and the BEM matrix has to be well-conditioned to ensure numerical stability. In fact, in solving the matrix associated linear system, the matrix condition number (the ratio of the matrix largest and smallest singular value) is related to iterative solvers´ convergence rates and to the error sensitivity of the solution: the highest the condition number, the highest the convergence time and the error sensitivity. Unfortunately, many commonly used formulations suffer from severe ill-conditionings especially for large, multiscale, and complex problems. The problem complexity of current applications is steadily and rapidly increasing. For this reason, the impact of well-conditioned formulations on state-of-the-art computational technology is destined to be more and more predominant and pervasive.