Singular Integration by Interpolation (SIBI) for Integral Equations via Tensor Trains

An efficient approach for populating the near weakly-singular and near singular integral interactions in the Boundary Element Method (BEM) solution of the Electric (EFIE) and Magnetic Field Integral equation (MFIE) on flat triangular meshes is outlined. The singular integration problem is recast into a multidimensional interpolation problem, where the quadruple singular integral, corresponds to a six-dimensional function, which is interpolated by a set of pre-computed universal (frequency, materials independent) look-up tables. In a previous version of SIBI this interpolation was performed via multidimensional sparse grids. In this paper, a low-rank tensor, the Tensor Train (TT) decomposition is used. The proposed method is compared to singularity subtraction, as well as Sparse Grids SIBI. The proposed method offers improved accuracy and computational cost.