论文标题
关于经验插值法和重力波替代的稳定性和准确性
On the stability and accuracy of the Empirical Interpolation Method and Gravitational Wave Surrogates
论文作者
论文摘要
减少基础和经验插值方法（EIM）方法的结合在许多学科中都取得了出色的结果。特别是，在引力波（GW）科学中，这些结果包括构建非侵入性替代模型的GWS到快速参数估计，增加了减少阶四个四倍体的使用。这些替代物具有与爱因斯坦方程的超级计算机模拟基本上没有区别的显着特征，但是可以在标准笔记本电脑上以每个多极模式的milisecond进行评估。在本文中，我们阐明了对GW科学实践中最初引入和使用的对EIM的普遍误解。也就是说，我们证明每次迭代时的EIM选择插值节点，以使相关的Vandermonde型矩阵尽可能可逆。不一定像有时会想到的那样优化其插值的条件或准确性。实际上，我们还介绍了EIM的两个新变体，它们嵌套了，它们在调理和Lebesgue常数方面确实进行了优化，并通过数值实验与使用GWS的原始EIM进行比较。我们的分析和数值结果表明，解决原始EIM，条件和Lebesgue常数之间存在微妙的关系，与严格近似理论和相关领域的积极研究相吻合。
The combination of the Reduced Basis and the Empirical Interpolation Method (EIM) approaches have produced outstanding results in many disciplines. In particular, in gravitational wave (GW) science these results range from building non-intrusive surrogate models for GWs 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 the order of a milisecond per multipole mode on a standard laptop. In this article we clarify a common misperception of the EIM as originally introduced and used in practice in GW science. Namely, we prove that the EIM at each iteration chooses the interpolation nodes so as to make the related Vandermonde-type matrix as invertible as possible; not necessarily optimizing its conditioning or accuracy of the interpolant as is sometimes thought. In fact, we introduce two new variations of the EIM, nested as well, which do optimize with respect to conditioning and the Lebesgue constant, respectively, and compare them through numerical experiments with the original EIM using GWs. Our analyses and numerical results suggest a subtle relationship between solving for the original EIM, conditioning, and the Lebesgue constant, in consonance with active research in rigorous approximation theory and related fields.
