SIAM Review, Volume 66, Issue 3, Page 479-479, May 2024.
Equitable distribution of geographically dispersed resources presents a significant challenge, particularly in defining quantifiable measures of equity. How can we optimally allocate polling sites or hospitals to serve their constituencies? This issue's first Research Spotlight, “Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites," addresses these questions by demonstrating the application of topological data analysis to identify holes in resource accessibility and coverage. Authors Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter employ persistent homology, a technique that tracks the formation and disappearance of these holes as spatial scales vary. To make matters concrete, the authors consider a case study on access to polling sites and use a non-Euclidean distance that accounts for both travel and waiting times. In their case study, the authors use a weighted Vietoris--Rips filtration based on a symmetrized form of this distance and limit their examination to instances where the approximations underlying the filtration are less likely to lead to approximation-based artifacts. Details, as well as source code, are provided on the estimation of the various quantities, such as travel time, waiting time, and demographics (e.g., age, vehicle access). The result is a homology class that “dies" at time $t$ if it takes $t$ total minutes to cast a vote. The paper concludes with an exposition of potential limitations and future directions that serve to encourage additional investigation into this class of problems (which includes settings where one wants to deploy different sensors to cover a spatial domain) and related techniques. What secrets lurk within? From flaws in human-made infrastructure to materials deep beneath the Earth's land and ocean surfaces to anomalies in patients, our next Research Spotlight, “When Data Driven Reduced Order Modeling Meets Full Waveform Inversion," addresses math and methods to recover the unknown. Authors Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, and Jörn Zimmerling show how tools from numerical linear algebra and reduced-order modeling can be brought to bear on inverse wave scattering problems. Their setup encapsulates a wide variety of sensing modalities, wherein receivers emit a signal (such as an acoustic wave) and a time series of wavefield measurements is subsequently captured at one or more sources. Full waveform inversion refers to the recovery of the unknown “within" and is typically addressed via iterative, nonlinear equations/least-squares solvers. However, it is often plagued by a notoriously nonconvex, ill-conditioned optimization landscape. The authors show how some of the challenges typically encountered in this inversion can be mitigated with the use of reduced-order models. These models employ observed data snapshots to form lower-dimensional, computationally attractive approximations. A key to the paper's developments is a unification of several Galerkin projection--based models and ensuring that these approximation models benefit the inversion. The latter is achieved by having the reduced-order models aimed at capturing “internal waves," and only later addressing the resulting mismatch with the measured data. Through several illustrations, the authors demonstrate how the approach can be deployed. The paper concludes with open questions in the intersection of inverse problems and reduced-order models.

中文翻译：

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 研究热点


《SIAM 评论》，第 66 卷，第 3 期，第 479-479 页，2024 年 5 月。

地理上分散的资源的公平分配提出了重大挑战，特别是在定义可量化的公平衡量标准方面。我们如何才能最佳地分配投票站或医院来为其选区提供服务？本期的第一个研究热点“资源覆盖率的持久同源性：投票站访问的案例研究”通过演示拓扑数据分析的应用来识别资源可访问性和覆盖率中的漏洞来解决这些问题。作者 Abigail Hickok、Benjamin Jarman、迈克尔·约翰逊 (Michael Johnson)、罗家杰 (Jiajie Luo) 和梅森·A·波特 (Mason A. Porter) 采用了持久同源性，这是一种随着空间尺度变化而跟踪这些洞的形成和消失的技术。为了使问题具体化，作者考虑了一个关于进入投票站的案例研究，并使用了在他们的案例研究中，作者使用基于该距离的对称形式的加权 Vietoris-Rips 过滤，并将他们的检查限制在过滤基础近似值较小的情况。提供了有关各种量估计的详细信息以及源代码，例如旅行时间、等待时间和人口统计数据（例如年龄、车辆通行）。结果是一个同源类，如果投票总共需要 $t$ 分钟，那么它会在 $t$ 时间“死亡”。本文最后阐述了潜在的局限性和未来的方向，这些限制和未来方向有助于鼓励对此类的进一步研究问题（包括想要部署不同传感器来覆盖空间域的设置）以及相关技术中潜藏着哪些秘密？ 从人造基础设施的缺陷到地球陆地和海洋表面深处的材料，再到患者的异常，我们的下一个研究焦点“当数据驱动的降阶建模遇到全波形反演”将讨论恢复未知事物的数学和方法。 Liliana Borcea、Josselin Garnier、Alexander V. Mamonov 和 Jörn Zimmerling 展示了如何使用数值线性代数和降阶建模工具来解决逆波散射问题，他们的装置封装了各种传感模式，其中接收器发射信号。随后在一个或多个源处捕获信号（例如声波）和波场测量的时间序列。全波形反演是指恢复“内部”未知数，并且通常通过迭代、非线性方程/最小化来解决。平方解算器。然而，它经常受到众所周知的非凸、病态优化环境的困扰。作者展示了如何通过使用降阶模型来缓解反演中通常遇到的一些挑战。这些模型利用观察到的数据快照来形成低维、计算上有吸引力的近似值。本文发展的一个关键是统一了几个基于伽辽金投影的模型，并确保这些近似模型有利于反演。后者是通过建立旨在捕获“内波”的降阶模型来实现的，然后才解决由此产生的与测量数据的不匹配问题。通过几个插图，作者演示了如何部署该方法。论文最后以开放式的方式得出结论：逆问题和降阶模型的交叉问题。