SIAM Review, Volume 65, Issue 4, Page 917-917, November 2023.
The metric dimension $\beta(G)$ of a graph $G = (V,E)$ is the smallest cardinality of a subset $S$ of vertices such that all other vertices are uniquely determined by their distances to the vertices in the resolving set $S$. Finding the metric dimension of a graph is an NP-hard problem. Determining whether the metric dimension is less than a given value is NP-complete. In the first article in the Survey and Review section of this issue, “Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications,” Richard C. Tillquist, Rafael M. Frongillo, and Manuel E. Lladser provide an exhaustive introduction to metric dimension. The overview of its vital results includes applications in game theory, source localization in transmission processes, and preprocessing in the computational analysis of biological sequence data. The paper is worth reading for a broad audience. The second Survey and Review article, by Ludovic Chamoin and Frédéric Legoll, is “An Introductory Review on A Posteriori Error Estimation in Finite Element Computations.” It is devoted to basic concepts and tools for verification methods that provide computable and mathematically certified error bounds and also addresses the question on the localization of errors in the spatial domain. The focus of this review is on a particular method and problem, namely, a conforming finite element method for linear elliptic diffusion problems. The tools of dual analysis and the concept of equilibrium enable a unified perspective on different a posteriori error estimation methods, e.g., flux recovery methods, residual methods, and duality-based constitutive relation error methods. Other topics considered are goal-oriented error estimation, computational costs, and extensions to other finite element schemes and other mathematical problems. While the presentation is self-contained, it is assumed that the reader is familiar with finite element methods. The text is written in an interdisciplinary style and aims to be useful for applied mathematicians and engineers.

中文翻译：

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调查与回顾

SIAM Review，第 65 卷，第 4 期，第 917-917 页，2023 年 11 月。
图 $G = (V,E)$ 的度量维度 $\beta(G)$ 是子集 $S$ 的最小基数使得所有其他顶点都由它们到解析集 $S$ 中的顶点的距离唯一确定。求图的度量维度是一个 NP 难题。确定度量维度是否小于给定值是 NP 完全的。在本期调查与评论部分的第一篇文章“了解离散空间中的情况：公制维度及其应用的调查”中，Richard C. Tillquist、Rafael M. Frongillo 和 Manuel E. Lladser提供对公制尺寸的详尽介绍。其重要结果的概述包括博弈论中的应用、传输过程中的源定位以及生物序列数据计算分析中的预处理。这篇论文值得广大读者阅读。第二篇调查和评论文章由 Ludovic Chamoin 和 Frédéric Legoll 撰写，是“有限元计算中后验误差估计的介绍性评论”。它致力于验证方法的基本概念和工具，提供可计算和数学认证的误差范围，并解决空间域中误差定位的问题。本综述的重点是一个特定的方法和问题，即线性椭圆扩散问题的一致有限元方法。对偶分析工具和平衡概念使得不同的后验误差估计方法能够得到统一的视角，例如通量恢复方法、残差方法和基于对偶性的本构关系误差方法。考虑的其他主题包括面向目标的误差估计、计算成本以及对其他有限元方案和其他数学问题的扩展。虽然演示文稿是独立的，但假设读者熟悉有限元方法。本书以跨学科风格编写，旨在对应用数学家和工程师有用。