Robust Grid-Based Computation of H-Matrix Blocks' Low-Rank Approximation for Vector Basis Functions

A fast method to assess and correct the domain misrepresentation of impedance matrix blocks, Zos, based on their row- and columnsampling, to ensure the accuracy of their low-rank approximation (LRA), is presented. The misrepresentation arises when auxiliary grid-based interaction matrices, G, of smaller dimensions are used to replace Zos to accelerate their LRA. Specifically, a recently introduced fast multilevel LRA algorithm, which uses truncated singluar-value decompositions of matrices corresponding to interactions between basis-functions and proxy grids, is shown to lose error controllability when used for vector basis/testing functions. A pre-processing stage that correctly and automatically sets the algorithm's truncation threshold, τG, for G, such that the domains of their truncated versions represent those of the blocks to the desired accuracy, is proposed. The method first analyzes small, randomly sampled portions of G and Zos, increasing the sample size until their spectrums converge; it then uses the converged spectra to gradually increase the rank and lower the truncation threshold, until the desired error level is achieved. The resulting truncation threshold, τ, is then used for τG across all algorithm levels. Representative examples validate the method and show its effectiveness.