SIAM Review, Volume 64, Issue 2, Page 341-342, May 2022.
This issue features three Research Spotlight articles. The first of these is entitled “Variance and Covariance of Distributions on Graphs" and is coauthored by Karel Devriendt, Samuel Martin-Gutierrez, and Renaud Lambiotte. Given a distribution on the graph (that is, a function $p$ from the set of nodes to $[0,1]$ such that the sum of all values is one) the key ingredient in the authors' definition of the variance and covariance is a notion of distance between the nodes of a graph. Although their definitions of (co)variance are valid for different choices of distance, the authors focus on a metric called the effective resistance. The effective resistance resembles geodesic distance in that it reflects the length of the paths between a pair of nodes but differs in that it takes into account all paths, and their lengths, between a node pair. Conveniently, the resulting variance and covariance values can be calculated via evaluation of a quadratic product and matrix trace. In support of the newly introduced measures, the balance of the paper is devoted to the application of the new definitions in practice and to the explanation of the conceptual correspondence between their (co)variance measures and some known graph characteristics. Citing the benefits of the framework induced by their (co)variance measures, the authors leave the reader with suggestions for future application in fields such as neuroscience, economics, and social networks. Competitive tournaments are an integral part of daily life, from schoolyard games to professional sports, to politics and biology. Authors Alexander Strang, Karen C. Abbott, and Peter J. Thomas tackle the difficult problem of quantifying competition in tournament modeling. Their article, “The Network HHD: Quantifying Cyclic Competition in Trait-Performance Models of Tournaments,” outlines how to adapt the discrete Helmholtz--Hodge decomposition (HHD), first introduced in the literature as a method for ranking objects from incomplete and imbalanced data, to study competitive tournaments characterized by intransitivity, the presence of which provides a challenge in ranking. Specifically, an intransitive tournament is one in which there is no clear global ranking of all competitors, and it corresponds to a cycle in the model. The authors show that the HHD arises in the context of representing a generic tournament as a weighted sum of so-called perfectly transitive and perfectly cyclic components. Since a trait-performance model assumes that “the probability that one competitor beats another is a function of their traits” the goal becomes that of identifying which traits and performance functions influence the weights in the HHD. This is addressed with the theoretical results in section 4. Schematics and graphs complement the discussion. Using the code made available by the authors, the interested reader may wish to try an analysis of a tournament for themselves. The third article, “Bilinear Optimal Control of an Advection-Reaction-Diffusion System," authored by Roland Glowinski, Yongcun Song, Xiaoming Yuan, and Hangrui Yue, offers both theoretical and computational insights to solving the mathematical problem specified in the title. Readers may appreciate the space devoted to the motivation and introduction to the problem as well as to the description of the associated difficulties both in deriving existence results and in numerical computation of solutions. Following a few preliminaries, the authors derive a proof of existence of optimal controls for the problem, labeled in the introduction as BCP, without the special case assumptions used elsewhere in the literature. The balance of the article is devoted to issues associated with the numerical implementation. Although their proposed nested conjugate gradient method is straightforward to present on paper, several obstacles arise in the implementation, and the authors tackle each of these in turn. For computational efficiency, for example, an inexact line search is proposed to replace the computationally intractable exact line search; preconditioned CG is employed for two internal subproblems that arise. The choice of the time and space discretizations is explained as well. There are many technical details to consume, but a key point for the readers is well summarized by the authors just before the detailed experiments, namely, “despite its apparent complexity, the nested CG method” given here “is easy to implement" with aspects that are amenable to parallelizability. Misha E. Kilmer Section Editor Misha.Kilmer@tufts.edu

中文翻译：

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研究聚光灯

SIAM 评论，第 64 卷，第 2 期，第 341-342 页，2022 年 5 月。
本期以三篇研究聚焦文章为特色。其中第一个题为“图上分布的方差和协方差”，由 Karel Devriendt、Samuel Martin-Gutierrez 和 Renaud Lambiotte 合着。给定图上的分布（即，来自集合的函数 $p$节点到 $[0,1]$ 使得所有值的总和为 1）作者定义方差和协方差的关键要素是图节点之间的距离概念。尽管他们对 (co ）方差对于不同的距离选择有效，作者关注称为有效阻力的度量。有效阻力类似于测地线距离，因为它反映了一对节点之间的路径长度，但不同之处在于它考虑了所有路径及其长度，节点对之间。方便地，所得方差和协方差值可以通过二次乘积和矩阵迹线的评估来计算。为了支持新引入的度量，本文的其余部分致力于新定义在实践中的应用，以及解释它们的（协）方差度量与一些已知的图特征之间的概念对应关系。作者引用了他们的（协）方差测量所带来的框架的好处，为读者提供了未来在神经科学、经济学和社交网络等领域的应用的建议。竞技比赛是日常生活中不可或缺的一部分，从校园游戏到职业体育，再到政治和生物。作者 Alexander Strang、Karen C. Abbott 和 Peter J. Thomas 解决了锦标赛建模中量化竞争的难题。他们的文章“The Network HHD: Quantifying Cyclic Competition in Trait-Performance Models of Tournaments”概述了如何调整离散 Helmholtz-Hodge 分解 (HHD)，该分解首次在文献中被引入，作为对不完整和不平衡对象进行排序的方法数据，以研究以非传递性为特征的竞争性锦标赛，其存在对排名提出了挑战。具体来说，不及物锦标赛是所有参赛者没有明确的全球排名的锦标赛，它对应于模型中的一个循环。作者表明，HHD 是在将通用锦标赛表示为所谓的完美传递和完美循环分量的加权和的背景下出现的。由于特质绩效模型假设“一个竞争对手击败另一个竞争对手的概率是其特质的函数”，因此目标变成了确定哪些特质和绩效函数影响 HHD 中的权重。第 4 节中的理论结果解决了这一问题。示意图和图表补充了讨论。使用作者提供的代码，感兴趣的读者可能希望自己尝试对锦标赛进行分析。第三篇文章“对流-反应-扩散系统的双线性最优控制”由 Roland Glowinski、宋永存、袁晓明和岳航瑞撰写，为解决标题中指定的数学问题提供了理论和计算方面的见解。读者可能会欣赏专门介绍问题的动机和介绍的空间，以及在推导存在结果和解的数值计算中相关困难的描述。经过一些初步准备，作者得出了该问题存在最优控制的证明，在引言中标记为 BCP，而没有文献中其他地方使用的特殊情况假设。本文的其余部分专门讨论与数值实现相关的问题。尽管他们提出的嵌套共轭梯度方法很容易在纸上呈现，但在实施过程中出现了一些障碍，作者依次解决了这些障碍。例如，为了计算效率，提出了一种不精确的线搜索来代替计算上难以处理的精确线搜索；预处理 CG 用于出现的两个内部子问题。还解释了时间和空间离散化的选择。有许多技术细节要消耗，但作者在详细实验之前为读者很好地总结了一个关键点，即“尽管看起来很复杂，但嵌套 CG 方法”这里给出的“易于实现”与方面可并行化。Misha E. Kilmer 部分编辑 Misha.Kilmer@tufts.edu