SIAM Review, Volume 66, Issue 4, Page 795-805, November 2024. If you are teaching a course (or otherwise looking for a text) in the techniques and applications of mathematical modeling, or mathematical approaches that analyze or solve those equations, you may find one of the reviews in this issue's collection interesting. Our featured review was written by Shawn Ryan, on the book Mathematical Modeling

SIAM Review, Volume 66, Issue 4, Page 778-792, November 2024. Mathematicians have traditionally been a select group of academics who produce high-impact ideas enabling substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant proportion of approximately 1% of bachelor degrees awarded in the

SIAM Review, Volume 66, Issue 4, Page 751-777, November 2024. Granular materials are everywhere, in the environment but also in our pantry. Their properties are different from those of any solid material, due to the possibility of sudden phenomena such as avalanches or landslides. Here we present a brief survey on their characteristics and on what can be found (from the past thirty years) in the recent

SIAM Review, Volume 66, Issue 4, Page 749-749, November 2024. In this issue the Education section presents two contributions. The first paper, “Sandpiles and Dunes: Mathematical Models for Granular Matter,” by Piermarco Cannarsa and Stefano Finzi Vita, presents a review of mathematical models for formation of sand piles and dunes. In nature and everyday life various materials appear as conglomerates

SIAM Review, Volume 66, Issue 4, Page 721-747, November 2024. We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous

SIAM Review, Volume 66, Issue 4, Page 719-719, November 2024. The SIGEST article in this issue, “A Bridge between Invariant Theory and Maximum Likelihood Estimation,” by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal, uncovers the deep connections between geometric invariant theory and statistical methods, specifically maximum likelihood estimation (MLE) by connecting it to norm

SIAM Review, Volume 66, Issue 4, Page 694-718, November 2024. We analyze an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. We show that the problem can be modeled as an inverse Cauchy problem for gravity-capillary waves, conduct a detailed analysis of the Cauchy problem, and give a uniqueness proof for the inverse problem. Somewhat surprisingly, this

SIAM Review, Volume 66, Issue 4, Page 683-693, November 2024. In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hp\kern .7pt$-mesh refinement for solution of \pdes, and (4) double exponential

SIAM Review, Volume 66, Issue 4, Page 681-681, November 2024. Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change

SIAM Review, Volume 66, Issue 4, Page 619-679, November 2024. There is enormous interest---both mathematically and in diverse applications---in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns

SIAM Review, Volume 66, Issue 4, Page 617-617, November 2024. Neural oscillations are periodic activities of neurons in the central nervous system of eumetazoa. In an oscillatory neural network, neurons are modeled by coupled oscillators. Oscillatory networks are employed for describing the behavior of complex systems in biology or ecology with respect to the connectivity of the network components

SIAM Review, Volume 66, Issue 3, Page 573-573, May 2024. In this issue the Education section presents “Combinatorial and Hodge Laplacians: Similarities and Differences,” by Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, and Guo-Wei Wei. Combinatorial Laplacians and their spectra are important tools in the study of molecular stability, electrical networks, neuroscience, deep learning, signal

SIAM Review, Volume 66, Issue 3, Page 535-571, May 2024. Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical

SIAM Review, Volume 66, Issue 3, Page 481-500, May 2024. It is important to choose the geographical distributions of public resources in a fair and equitable manner. However, it is complicated to quantify the equity of such a distribution; important factors include distances to resource sites, availability of transportation, and ease of travel. We use persistent homology, which is a tool from topological

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

SIAM Review, Volume 66, Issue 3, Page 575-601, May 2024. As key subjects in spectral geometry and combinatorial graph theory, respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both

SIAM Review, Volume 66, Issue 3, Page 403-477, May 2024. We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and we give concrete examples from diverse application fields such as signal and image processing, portfolio selection, and machine learning. The paper

SIAM Review, Volume 66, Issue 3, Page 501-532, May 2024. Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are

SIAM Review, Volume 66, Issue 3, Page 605-615, May 2024. The theme of this collection of book reviews is arguably about the “usefulness” of mathematics, or how we can try to understand aspects of our world by developing mathematical or data-driven models. Thus, it is fitting that our featured review is written by John Stillwell, on the book Why Does Math Work . . . If It's Not Real?, written by Dragan

SIAM Review, Volume 66, Issue 3, Page 401-401, May 2024. In “Cardinality Minimization, Constraints, and Regularization: A Survey," Andreas M. Tillmann, Daniel Bienstock, Andrea Lodi, and Alexandra Schwartz consider a class of optimization problems that involve the cardinality of variable vectors in constraints or in the objective function. Such problems have many important applications, e.g., medical

SIAM Review, Volume 66, Issue 3, Page 533-533, May 2024. The SIGEST article in this issue is “Operator Learning Using Random Features: A Tool for Scientific Computing,” by Nicholas H. Nelsen and Andrew M. Stuart. This work considers the problem of operator learning in infinite-dimensional Banach spaces through the use of random features. The driving application is the approximation of solution operators

SIAM Review, Volume 66, Issue 2, Page 391-399, May 2024. As I sat down to write this introduction, I became curious how the books chosen for review have changed over the past decades. So I scanned through a few SIREV Book Review section introductions written 10, 20 or more years ago by former section editors. That act of procrastination allows me to put the current collection of reviews in “historical

SIAM Review, Volume 66, Issue 2, Page 368-387, May 2024. This tutorial describes several basic and much-studied types of interactions with incomplete information, analyzing them by means of evolutionary game dynamics. The games include sender-receiver games, owner-challenger contests, costly advertising, and calls for help. We model the evolution of populations of players reacting to each other and

SIAM Review, Volume 66, Issue 2, Page 355-367, May 2024. We treat the Poincaré metric on the disc. In particular we emphasize the fact that it is the canonical holomorphically invariant metric on the unit disc. Then we generalize these ideas to the Bergman metric on a domain in complex space. Along the way we treat the Bergman kernel and study its invariance and uniqueness properties.

SIAM Review, Volume 66, Issue 2, Page 353-353, May 2024. In this issue the Education section presents two contributions. The first paper, “The Poincaré Metric and the Bergman Theory,” by Steven G. Krantz, discusses the Poincaré metric on the unit disc in the complex space and the Bergman metric on an arbitrary domain in any dimensional complex space. To define the Bergman metric the notion of Bergman

SIAM Review, Volume 66, Issue 2, Page 319-352, May 2024. We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this class of problems converge slowly in practice, involve subproblems that can be as difficult as the original problem, or lack rigorous convergence guarantees. In

SIAM Review, Volume 66, Issue 2, Page 317-317, May 2024. The SIGEST article in this issue is “Nonsmooth Optimization over the Stiefel Manifold and Beyond: Proximal Gradient Method and Recent Variants,” by Shixiang Chen, Shiqian Ma, Anthony Man-Cho So, and Tong Zhang. This work considers nonsmooth optimization on the Stiefel manifold, the manifold of orthonormal $k$-frames in $\mathbb{R}^n$. The authors

SIAM Review, Volume 66, Issue 2, Page 287-315, May 2024. We present a new version of the fast Gauss transform (FGT) for discrete and continuous sources. Classical Hermite expansions are avoided entirely, making use only of the plane-wave representation of the Gaussian kernel and a new hierarchical merging scheme. For continuous source distributions sampled on adaptive tensor product grids, we exploit

SIAM Review, Volume 66, Issue 2, Page 285-285, May 2024. The Gauss transform---convolution with a Gaussian in the continuous case and the sum of $N$ Gaussians at $M$ points in the discrete case---is ubiquitous in applied mathematics, from solving ordinary and partial differential equations to probability density estimation to science applications in astrophysics, image processing, quantum mechanics

SIAM Review, Volume 66, Issue 2, Page 205-284, May 2024. This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent

SIAM Review, Volume 66, Issue 2, Page 203-203, May 2024. Inverse problems arise in various applications---for instance, in geoscience, biomedical science, or mining engineering, to mention just a few. The purpose is to recover an object or phenomenon from measured data which is typically subject to noise. The article “Computational Methods for Large-Scale Inverse Problems: A Survey on Hybrid Projection

SIAM Review, Volume 66, Issue 1, Page 193-201, February 2024. If you are keen to understand the world around us by developing mathematical or data-driven models, or if you are interested in the methodologies that can be used to analyze those models, this collection of reviews may help you identify a useful book or two. Our featured review was written by Tim Hoheisel, on the book Convex Optimization:

SIAM Review, Volume 66, Issue 1, Page 161-190, February 2024. Uncertainty quantification (UQ) in machine learning is currently drawing increasing research interest, driven by the rapid deployment of deep neural networks across different fields, such as computer vision and natural language processing, and by the need for reliable tools in risk-sensitive applications. Recently, various machine learning

SIAM Review, Volume 66, Issue 1, Page 149-160, February 2024. In a recent article in this journal, Gouveia and Stone [``Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods,” SIAM Rev., 64 (2022), pp. 485--499] described a method for finding exact solutions to resonantly forced linear ordinary differential equations, and for finding the general

SIAM Review, Volume 66, Issue 1, Page 147-147, February 2024. In this issue the Education section presents two contributions. The first paper, “Resonantly Forced ODEs and Repeated Roots,” is written by Allan R. Willms. The resonant forcing problem is as follows: find $y(\cdot)$ such that $L[y(x)]=u(x)$, where $L[u(x)]=0$ and $L=a_0(x) + \sum_{j=1}^n a_j(x) \frac{d^j}{dx^j}$. The repeated roots problem

SIAM Review, Volume 66, Issue 1, Page 125-146, February 2024. We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$ subject to homogeneous Dirichlet or Neumann boundary conditions, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q \Lambda Q^T$, where $Q=Q(x

SIAM Review, Volume 66, Issue 1, Page 123-123, February 2024. The SIGEST article in this issue is “A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators,” by Bjørn Fredrik Nielsen and Zdeněk Strakoš. This paper studies the eigenvalues of second-order self-adjoint differential operators in the continuum and discrete settings. In particular, they investigate

SIAM Review, Volume 66, Issue 1, Page 91-122, February 2024. How can we quantify uncertainty if our favorite computational tool---be it a numerical, statistical, or machine learning approach, or just any computer model---provides single-valued output only? In this article, we introduce the Easy Uncertainty Quantification (EasyUQ) technique, which transforms real-valued model output into calibrated

SIAM Review, Volume 66, Issue 1, Page 89-89, February 2024. As modeling, simulation, and data-driven capabilities continue to advance and be adopted for an ever expanding set of applications and downstream tasks, there has been an increased need for quantifying the uncertainty in the resulting predictions. In “Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-Valued

SIAM Review, Volume 66, Issue 1, Page 3-88, February 2024. Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of

SIAM Review, Volume 66, Issue 1, Page 1-1, February 2024. Numerical methods for partial differential equations can only be successful if their numerical solutions reflect fundamental properties of the physical solution of the respective PDE. For convection-diffusion equations, the conservation of some specific scalar quantities is crucial. When physical solutions satisfy maximum principles representing

SIAM Review, Volume 65, Issue 4, Page 1187-1197, November 2023. Our section starts with the featured review of Glenn Ledder's book Mathematical Modeling for Epidemiology and Ecology. This review is a joint work of 10 authors from Anita Layton's group. This shows that one can efficiently combine a reading course with an introduction to scientific work and the writing of a review. All reviewers are enthusiastic

SIAM Review, Volume 65, Issue 4, Page 1171-1184, November 2023. A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate

SIAM Review, Volume 65, Issue 4, Page 1152-1170, November 2023. Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be challenging. Students often arrive with different mathematical backgrounds, different interests

SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023. We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $\mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the

SIAM Review, Volume 65, Issue 4, Page 1135-1135, November 2023. In this issue the Education section presents three contributions. The first paper “The Reflection Method for the Numerical Solution of Linear Systems,” by Margherita Guida and Carlo Sbordone, discusses the celebrated Gianfranco Cimmino reflection algorithm for the numerical solution of linear systems $Ax=b$, where $A$ is a nonsingular

SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023. Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent

SIAM Review, Volume 65, Issue 4, Page 1107-1107, November 2023. The SIGEST article in this issue is “Are Adaptive Galerkin Schemes Dissipative?” by Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, and Marie Farge. “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” With this quote from Bertrand Russell from 1931 commences this issue's SIGEST article

SIAM Review, Volume 65, Issue 4, Page 1074-1105, November 2023. Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to parameterize these coefficients is large, and oobtaining meaningful

SIAM Review, Volume 65, Issue 4, Page 1031-1073, November 2023. Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical

SIAM Review, Volume 65, Issue 4, Page 1029-1029, November 2023. This issue's two Research Spotlights highlight techniques for obtaining ever more realistic solutions to challenging systems of partial differential equations (PDEs). Although borne from different fields of applied mathematics, both papers aim to leverage prior information to improve the fidelity and practical solution of PDEs. How predictive

SIAM Review, Volume 65, Issue 4, Page 963-1028, November 2023. This article is a review of basic concepts and tools devoted to a posteriori error estimation for problems solved with the finite element method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The main goal of this review is to present

SIAM Review, Volume 65, Issue 4, Page 919-962, November 2023. The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey

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

SIAM Review, Volume 65, Issue 3, Page 905-915, August 2023. This collection of reviews encompasses a wide range of topics. We kick off with an insightful featured review by Chris Oats on the book Probabilistic Numerics, written by Philipp Hennig, Michael A. Osborne, and Hans P. Kersting. Oats expresses his own fascination with the topic and highly recommends delving into the substantial tome. Continuing

SIAM Review, Volume 65, Issue 3, Page 887-902, August 2023. Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system

SIAM Review, Volume 65, Issue 3, Page 869-886, August 2023. Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because

SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023. In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical

SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023. Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse

SIAM Review, Volume 65, Issue 3, Page 829-829, August 2023. The SIGEST article in this issue, which comes from the SIAM/ASA Journal on Uncertainty Quantification, is “Bayesian Inverse Problems Are Usually Well-Posed,” by Jonas Latz. The author investigates the well-posedness of Bayesian approaches to inverse problems, generalizing the framework of well-posedness introduced by Andrew Stuart to a set