The combination of the Reduced Basis (RB) and the Empirical Interpolation Method (EIM) approaches have produced outstanding results in gravitational wave (GW) science and in many other disciplines. In GW science in particular, these results range from building non-intrusive surrogate models for gravitational waves to fast parameter estimation adding the use of Reduced Order Quadratures. These surrogates have the salient feature of being essentially indistinguishable from or very close to supercomputer simulations of the Einstein equations but can be evaluated in less than a second on a laptop. In this note we analyze in detail how the EIM at each iteration attempts to choose the interpolation nodes so as to make the related Vandermonde-type matrix "as invertible as possible" as well as attempting to optimize the conditioning of its inversion and minimizing a Lebesgue-type constant for accuracy. We then compare through numerical experiments the EIM performance with fully optimized nested variations. We also discuss global optimal solutions through Fekete nodes. We find that in these experiments the EIM is actually close to optimal solutions but can be improved with small variations and relative low computational cost.

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关于经验插值法和引力波替代品的稳定性和准确性

约简基 (RB) 和经验插值法 (EIM) 方法的结合在引力波 (GW) 科学和许多其他学科中产生了出色的结果。特别是在 GW 科学中，这些结果的范围从构建引力波的非侵入式替代模型到快速参数估计，增加了降阶正交的使用。这些替代品的显着特点是与爱因斯坦方程的超级计算机模拟基本上无法区分或非常接近，但可以在不到一秒钟的时间内在笔记本电脑上进行评估。在这篇笔记中，我们详细分析了 EIM 在每次迭代时如何尝试选择插值节点，从而使相关的 Vandermonde 型矩阵“尽可能可逆” 以及尝试优化其反演的条件并最小化 Lebesgue 型常数以提高准确性。然后，我们通过数值实验将 EIM 性能与完全优化的嵌套变化进行比较。我们还通过 Fekete 节点讨论全局最优解。我们发现，在这些实验中，EIM 实际上接近于最优解，但可以通过小的变化和相对较低的计算成本来改进。