Research Spotlights

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.