Low-Rank Kernel Matrix Approximation Using Skeletonized Interpolation With Endo- or Exo-Vertices
For practitioners of multiphysics simulations needing efficient kernel matrix compression, this work provides a practical comparison of pivot selection strategies, though the results are incremental and domain-specific.
The paper studies four pivot selection strategies for Skeletonized Interpolation to compress kernel matrices, finding that maximally-dispersed vertices achieve near-optimal ranks with small initial sets, outperforming Chebyshev grids, sphere points, and random vertices on three LS-DYNA benchmarks.
The efficient compression of kernel matrices, for instance the off-diagonal blocks of discretized integral equations, is a crucial step in many algorithms. In this paper, we study the application of Skeletonized Interpolation to construct such factorizations. In particular, we study four different strategies for selecting the initial candidate pivots of the algorithm: Chebyshev grids, points on a sphere, maximally-dispersed and random vertices. Among them, the first two introduce new interpolation points (exo-vertices) while the last two are subsets of the given clusters (endo- vertices). We perform experiments using three real-world problems coming from the multiphysics code LS-DYNA. The pivot selection strategies are compared in term of quality (final rank) and efficiency (size of the initial grid). These benchmarks demonstrate that overall, maximally-dispersed vertices provide an accurate and efficient sets of pivots for most applications. It allows to reach near-optimal ranks while starting with relatively small sets of vertices, compared to other strategies.