SIAM Review, Volume 66, Issue 4, Page 681-681, November 2024.
Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change of variables $s = \log(x)$ connects different areas of computational mathematics. Authors Daan Huybrechs and Lloyd “Nick” Trefethen show new relationships between smooth approximation, rational approximation theory, adaptive mesh refinement, and numerical quadrature. For example, the authors show that this change of variables can be naturally tied to a “linear tapering” effect near singularities, which is a common feature in both rational approximation and $hp$-mesh refinement. Through a number of effective examples, the authors illustrate the power of these relationships across areas that have seen relatively independent lines of development. In doing so, the authors suggest opportunities for developing and analyzing new methods by leveraging the new connections, including mesh refinement strategies, techniques for multivariate approximation, and hybrid approaches that combine the strengths of disparate methods. How well can information be recovered from water waves? This question is at the heart of this issue's second research spotlight, “Feynman's Inverse Problem.” Author Adrian Kirkeby is motivated by a thought experiment posed by the physicist and iconoclast Richard Feynman wherein an insect floating in a swimming pool wants to determine where and when others have jumped into the pool, causing the waves the insect observes. Kirkeby constructs and analyzes a linear 2D-3D system of partial differential equations (PDEs) for the forward model. Leveraging the nonlocality of this system of PDEs, Kirkeby shows conditions under which the insect can determine the source of the waves---in fact, uniquely---simply by observing the wave amplitude and water velocity in any small area of the surface. This model is then extended to capture settings where noisy observations and observations at a finite number of time and space points are collected, and establishes stability properties and error bounds for the reconstruction. The paper concludes with illustrative numerical experiments based on a nonharmonic Fourier inversion method. Kirkeby also highlights several avenues for future research, noting that inverse problems for water or other surface waves have received less attention than those involving acoustic or electromagnetic waves. As an added bonus, the referenced video of Feynman is not to be missed.

中文翻译：

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 研究聚焦


SIAM 评论，第 66 卷，第 4 期，第 681-681 页，2024 年 11 月。

对数变换在数据科学、数学和工程学中被广泛使用，但它们仍然可以揭示看似不相关的学科之间令人惊讶的联系。本期的第一个研究聚焦“Sigmoid 函数、奇点的多尺度分辨率和 $hp$-Mesh 细化”阐明了变量 $s = \log（x）$ 的变化如何连接计算数学的不同领域。作者 Daan Huybrechs 和 Lloyd “Nick” Trefethen 展示了平滑近似、有理近似理论、自适应网格细化和数值正交之间的新关系。例如，作者表明，变量的这种变化可以自然地与奇点附近的“线性锥度”效应相关联，这是有理近似和 $hp$ 网格细化中的共同特征。通过许多有效的例子，作者说明了这些关系在相对独立的发展路线领域中的力量。在此过程中，作者提出了利用新连接开发和分析新方法的机会，包括网格细化策略、多元近似技术以及结合不同方法优势的混合方法。从水波中恢复信息的能力如何？这个问题是本期第二篇研究焦点“费曼逆问题”的核心。作者 Adrian Kirkeby 的灵感来自物理学家和反传统者理查德·费曼 （Richard Feynman） 提出的一个思想实验，其中一只漂浮在游泳池中的昆虫想要确定其他人在何时何地跳入游泳池，从而引起昆虫观察到的波浪。 Kirkeby 为正演模型构建并分析了偏微分方程 （PDE） 的线性 2D-3D 系统。利用这种偏微分方程系统的非局域性，Kirkeby 展示了昆虫可以确定波源的条件---事实上，只需通过观察表面任何小区域的波幅和水速---就可以独特地确定波的来源。然后，该模型被扩展为捕获噪声观测值和收集有限数量时间和空间点的观测值的设置，并为重建建立稳定性属性和误差边界。本文最后以基于非谐波傅里叶反演法的说明性数值实验作为结论。Kirkeby 还强调了未来研究的几种途径，指出水或其他表面波的逆问题比涉及声波或电磁波的问题受到的关注较少。作为额外的奖励，Feynman 的参考视频不容错过。