低秩核矩阵近似：基于骨架化插值的内或外顶点方法
Low-Rank Kernel Matrix Approximation Using Skeletonized Interpolation With Endo- or Exo-Vertices
AI总结：
本文研究了利用骨架化插值构造核矩阵分解的方法，通过比较四种顶点选择策略，发现最大分散顶点在精度和效率上表现优异，能以较小的初始网格实现接近最优的秩。
摘要：
核矩阵的有效压缩，例如离散积分方程的非对角块，是许多算法中的关键步骤。本文研究了骨架化插值在构造此类分解中的应用。特别地，我们研究了四种选择算法初始候选主元的策略：切比雪夫网格、球面上的点、最大分散点和随机点。其中前两种引入新的插值点（外顶点），后两种是给定簇的子集（内顶点）。我们使用三个来自多物理代码LS-DYNA的现实问题进行实验。通过比较这些策略在质量（最终秩）和效率（初始网格大小）方面的表现，这些基准测试表明，最大分散点在大多数应用中提供了准确且高效的主元集。这使得在相对较小的顶点集合下就能达到接近最优的秩，相较于其他策略。
英文摘要：
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.
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