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,
$\mathrm {Z}^{\mathrm {os}}$
, based on their row- and column-sampling, to ensure the accuracy of their low-rank approximation (LRA), is presented. The misrepresentation arises when auxiliary grid-based interaction matrices,
$\mathbf {G}$
, of smaller dimensions are used to replace
$\mathrm {Z}^{\mathrm {os}}$
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,
$\tau _{\mathcal {G}}$
, for
$\mathbf {G}$
, 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
$\mathbf {G}$
and
$\mathrm {Z}^{\mathrm {os}}$
, 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,
$\bar {\tau }$
, is then used for
$\tau _{\mathcal {G}}$
across all algorithm levels. Representative examples validate the method and show its effectiveness.