SIAM Review, Volume 65, Issue 2, Page 437-437, May 2023.
As highlighted by Tisseur and Meerbergen in SIAM Review, 43 (2001), pp. 235--286, nonlinear eigenvalue problems arise in diverse applications such as acoustics of high-speed trains, the study of elastic materials, fluid mechanical control, and pedestrian-induced structural vibrations. In this issue's first Research Spotlights article, “Contour Integral Methods for Nonlinear Eigenvalue Problems: A Systems Theoretic Approach,” authors Michael C. Brennan, Mark Embree, and Serkan Gugercin illuminate new algorithmic approaches for these challenging problems. The authors reimagine Loewner matrix methods and demonstrate how significant gains in accuracy can be achieved with relatively little additional computational expense over conventional approaches. The article ties together several active areas of applied mathematics. For example, the authors seek to avoid explicitly computing an inverse map in the contour integral by employing randomized sketching-like techniques. The authors' intention in making these connections is to open up for further research and refinement a broad family of algorithms for the solution of challenging nonlinear eigenvalue problems. Susceptible-Infectious-Recovered (SIR) and related compartmental models for disease spread have received heightened attention by a global community seeking to better query and understand epidemic dynamics. “The Role of Directionality, Heterogeneity, and Correlations in Epidemic Risk and Spread,” the second Research Spotlights article, highlights key differences from treating such models via directed connections (that distinguish between risk and spread) rather than via simple undirected graph abstractions. Authors Antoine Allard, Cristopher Moore, Samuel V. Scarpino, Benjamin M. Althouse, and Laurent Hébert-Dufresne explore how the in-degree and out-degree distributions for random directed graphs jointly affect model dynamics. By using results from branching processes, this model allows one to examine characteristics such as epidemic size and the probability that a single infection generates a large-scale epidemic. The authors provide several examples to illustrate when established asymptotic theory holds and when it does not, highlighting important implicit assumptions regarding the Poisson distribution of risk as well as the independence of risk and spread.

中文翻译：

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研究聚焦

SIAM Review，第 65 卷，第 2 期，第 437-437 页，2023 年 5 月。
正如 Tisseur 和 Meerbergen 在 SIAM Review, 43 (2001), pp. 235--286 中强调的那样，非线性特征值问题出现在各种应用中，例如高速列车的声学、弹性材料的研究、流体机械控制和行人-引起的结构振动。在本期的第一篇 Research Spotlights 文章“非线性特征值问题的轮廓积分方法：系统理论方法”中，作者 Michael C. Brennan、Mark Embree 和 Serkan Gugercin 阐述了解决这些具有挑战性的问题的新算法方法。作者重新构想了 Loewner 矩阵方法，并展示了与传统方法相比，如何通过相对较少的额外计算费用来显着提高准确性。这篇文章将应用数学的几个活跃领域联系在一起。例如，作者试图通过使用类似随机草图的技术来避免在等高线积分中显式计算逆映射。作者建立这些联系的目的是为进一步研究和改进广泛的算法系列开辟道路，以解决具有挑战性的非线性特征值问题。易感-传染-恢复 (SIR) 和相关的疾病传播区室模型受到了寻求更好地查询和了解流行病动态的全球社区的高度关注。“方向性、异质性和相关性在流行病风险和传播中的作用”，第二篇 Research Spotlights 文章强调了通过有向连接（区分风险和传播）而不是通过简单的无向图抽象处理此类模型的主要区别。作者 Antoine Allard、Cristopher Moore、Samuel V. Scarpino、Benjamin M. Althouse 和 Laurent Hébert-Dufresne 探讨了随机有向图的入度和出度分布如何共同影响模型动力学。通过使用分支过程的结果，该模型允许人们检查诸如流行病规模和单一感染产生大规模流行病的概率等特征。作者提供了几个例子来说明已建立的渐近理论何时成立，何时不成立，强调了关于风险泊松分布以及风险和利差的独立性的重要隐含假设。和 Laurent Hébert-Dufresne 探索了随机有向图的入度和出度分布如何共同影响模型动力学。通过使用分支过程的结果，该模型允许人们检查诸如流行病规模和单一感染产生大规模流行病的概率等特征。作者提供了几个例子来说明已建立的渐近理论何时成立，何时不成立，强调了关于风险泊松分布以及风险和利差的独立性的重要隐含假设。和 Laurent Hébert-Dufresne 探索了随机有向图的入度和出度分布如何共同影响模型动力学。通过使用分支过程的结果，该模型允许人们检查诸如流行病规模和单一感染产生大规模流行病的概率等特征。作者提供了几个例子来说明已建立的渐近理论何时成立，何时不成立，强调了关于风险泊松分布以及风险和利差的独立性的重要隐含假设。