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

SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023. In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim$$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k→∞$, the total number of degrees of freedom needed to

SIAM Review, Volume 65, Issue 3, Page 774-805, August 2023. The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment

SIAM Review, Volume 65, Issue 3, Page 735-773, August 2023. We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of

SIAM Review, Volume 65, Issue 3, Page 733-733, August 2023. The three articles in this issue's Research Spotlight section highlight the breadth of problems and approaches that have differential equations as a central component. In the first article, “Neural ODE Control for Classification, Approximation, and Transport,” authors Domènec Ruiz-Balet and Enrique Zuazua seek to expand understanding of some

SIAM Review, Volume 65, Issue 3, Page 732-732, August 2023. This erratum corrects an error in the coefficients of equation (6.23) in the original paper [H. Miao, X. Xia, A. S. Perelson, and H. Wu, SIAM Rev., 53 (2011), pp. 3--39].

SIAM Review, Volume 65, Issue 3, Page 686-731, August 2023. Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships

SIAM Review, Volume 65, Issue 3, Page 601-685, August 2023. Over the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin--Osher--Fatemi (ROF) model

SIAM Review, Volume 65, Issue 3, Page 599-599, August 2023. Apart from a short erratum, which concerns the correction of some coefficients in a differential equation in the original paper, this issue contains two Survey and Review articles. “On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance,” authored by Monica Pragliola, Luca Calatroni, Alessandro Lanza, and Fiorella

SIAM Review, Volume 65, Issue 2, Page 591-598, May 2023. We begin the section with Alexander Mamonov's review on the book An Introduction to the Mathematical Theory of Inverse Problems, written by Andreas Kirsch. Our reviewer describes it as a classic in the field of inverse problems and recommends it to all readers inclined to the theory and solution of inverse problems. The next review discusses

SIAM Review, Volume 65, Issue 2, Page 563-588, May 2023. A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation

SIAM Review, Volume 65, Issue 2, Page 539-562, May 2023. While Nesterov's algorithm for computing the minimum of a convex function is now over forty years old, it is rarely presented in texts for a first course in optimization. This is unfortunate since for many problems this algorithm is superior to the ubiquitous steepest descent algorithm, and it is equally simple to implement. This article presents

SIAM Review, Volume 65, Issue 2, Page 537-537, May 2023. The Education section in this issue presents two contributions. In `"Nesterov's Method for Convex Optimization," Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized

SIAM Review, Volume 65, Issue 2, Page 495-536, May 2023. We present a unifying Perron--Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad set of problems involving

SIAM Review, Volume 65, Issue 2, Page 493-493, May 2023. The SIGEST article in this issue is “Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors,” by Antoine Gautier, Francesco Tudisco, and Matthias Hein. Most computational and applied mathematicians will be aware of the results that Perron published in 1907 about the eigensystems of positive matrices, which were then extended by Frobenius

SIAM Review, Volume 65, Issue 2, Page 471-492, May 2023. Most models of epidemic spread, including many designed specifically for COVID-19, implicitly assume mass-action contact patterns and undirected contact networks, meaning that the individuals most likely to spread the disease are also the most at risk of contracting it from others. Here, we review results from the theory of random directed graphs

SIAM Review, Volume 65, Issue 2, Page 439-470, May 2023. Contour integral methods for eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization and rational interpolation techniques in control theory. This connection casts contour integral methods for linear and

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

SIAM Review, Volume 65, Issue 2, Page 375-435, May 2023. Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview,

SIAM Review, Volume 65, Issue 2, Page 331-374, May 2023. Hawkes processes are a type of point process that models self-excitement among time events. They have been used in a myriad of applications, ranging from finance and earthquakes to crime rates and social network activity analysis. Recently, a variety of different tools and algorithms have been presented at top-tier machine learning conferences

SIAM Review, Volume 65, Issue 2, Page 329-329, May 2023. A point process is called self-exciting if the arrival of an event increases the probability of similar events for some period of time. Typical examples include earthquakes, which frequently cause aftershocks due to increased geological tension in their region; raised intrusion rates in the vicinity of a burglary; retweets in social media incited

SIAM Review, Volume 65, Issue 1, Page 319-328, February 2023. Our section starts with Rob Kirby's featured review on the book Numerical Methods for Elliptic and Parabolic Partial Differential Equations, written by Peter Knabner and Lutz Angermann. Our reviewer finds several aspects in this book not similarly covered in other books and recommends that the reader of the review fill a bit of open bookshelf