Survey and Review

Abstract

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.