Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement

IF 10.8 1区 数学 Q1 MATHEMATICS, APPLIED
SIAM Review Pub Date : 2024-11-07 DOI:10.1137/23m1556629
Daan Huybrechs, Lloyd N. Trefethen
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引用次数: 0

Abstract

SIAM Review, Volume 66, Issue 4, Page 683-693, November 2024.
In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hp\kern .7pt$-mesh refinement for solution of \pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables $s = \log(x)$, and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in $hp\kern .7pt$-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.
西格蒙德函数、奇异点的多尺度解析和 $hp$ 网格细化
SIAM 评论》,第 66 卷第 4 期,第 683-693 页,2024 年 11 月。 在这篇简短的概念性论文中,我们注意到密切相关的数学适用于四种不同的情况:(1) 平滑函数的 sigmoidal 和 RBF 近似,(2) 具有奇点的解析函数的有理近似,(3) $hp\kern .7pt$ 网格细化以求解 \pdes,以及 (4) 双指数(DE)和广义高斯正交。这些关系从变量 $s = \log(x)$ 的变化开始,为多个领域的新分析和新方法提供了可能性。关于(2)和(3),我们发现这两个问题在奇点附近都具有相同的 "线性渐变 "效应--在有理近似中是簇状极点,在$hp\kern .7pt$网格细化中是多项式阶数。关于 (4),我们注意到这里也出现了渐减效应,而变量的变化解释为我们揭示了为什么 DE 方法和广义高斯方法能有效积分任意奇点。
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来源期刊
SIAM Review
SIAM Review 数学-应用数学
CiteScore
16.90
自引率
0.00%
发文量
50
期刊介绍: Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter. Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.
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