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

SIAM Review, Volume 65, Issue 1, Page 291-315, February 2023. We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that

SIAM Review, Volume 65, Issue 1, Page 261-290, February 2023. The chaos game representation (CGR) is an interesting method to visualize one-dimensional sequences. In this paper, we show how to construct a chaos game representation. The applications mentioned here are biological, and CGR was able to uncover patterns in DNA or proteins in them that were previously unknown. We also show how CGR might

SIAM Review, Volume 65, Issue 1, Page 259-260, February 2023. In this issue the Education section presents two contributions. The first paper is “Chaos Game Representation” (CGR), written by Eunice Y. S. Chan and Robert M. Corless. The chaos game is an algorithm which allows us to produce pictures like fractal structures associated with one-dimensional sequences of integers and, in this way, to “visualize”

SIAM Review, Volume 65, Issue 1, Page 227-257, February 2023. In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point-vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [C. Marchioro

SIAM Review, Volume 65, Issue 1, Page 225-225, February 2023. The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics

SIAM Review, Volume 65, Issue 1, Page 183-223, February 2023. We present a mathematical and computational framework for learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn, besides trajectory data, and is typically inferred from domain-specific knowledge

SIAM Review, Volume 65, Issue 1, Page 147-182, February 2023. Bump attractors are wandering localized patterns observed in in vivo experiments of spatially extended neurobiological networks. They are important for the brain's navigational system and specific memory tasks. A bump attractor is characterized by a core in which neurons fire frequently, while those away from the core do not fire. These

SIAM Review, Volume 65, Issue 1, Page 145-145, February 2023. What surprising patterns do fluid dynamics and neurobiological networks have in common? In this issue's first Research Spotlights article, “Bump Attractors and Waves in Networks of Leaky Integrate-and-Fire Neurons,” a connection is made between the waves at the onset of pipe turbulence and the localized traveling waves in spiking neural

SIAM Review, Volume 65, Issue 1, Page 59-143, February 2023. Clustering points in a vector space or nodes in a graph is a ubiquitous primitive in statistical data analysis, and it is commonly used for exploratory data analysis. In practice, it is often of interest to “refine” or “improve” a given cluster that has been obtained by some other method. In this survey, we focus on principled algorithms

SIAM Review, Volume 65, Issue 1, Page 3-58, February 2023. This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratios of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing, and machine learning. This article

SIAM Review, Volume 65, Issue 1, Page 1-1, February 2023. Marginal likelihood and Bayes factors are key components of the Bayesian analysis used in hypothesis testing and model selection. In many applications in statistics, applied mathematics, signal processing, and machine learning, the computation of normalizing constants of probability models or ratios of these constants plays a fundamental role

SIAM Review, Volume 64, Issue 4, Page 1083-1095, November 2022. The book reviews sections starts off with a featured review that is dedicated in great depth to the collected volume Control Applications for Biomedical Engineering Systems, edited by Ahmad Taher Azar. The review originated as a joint project of Anita Layton's group at the University of Waterloo and has by far the largest number of authors

SIAM Review, Volume 64, Issue 4, Page 1062-1080, November 2022. Signals cannot always be sampled at their full desired resolution. In this Education article, we explore the benefits of randomly subsampling a signal's frequency spectrum. Whereas uniform subsampling introduces structural artifacts in the time series, random subsampling introduces a type of noise whose behavior we quantify. This analysis

SIAM Review, Volume 64, Issue 4, Page 1031-1061, November 2022. A new theory of a generalized dual transform of a list of vectors is presented, emphasizing its geometry as well as its linear algebra. We review the theory for independent vectors and find a new butterfly identity. This leads us to a parallel process to compute the dual vectors. Next we take the general case of arbitrary vectors, which

SIAM Review, Volume 64, Issue 4, Page 1029-1029, November 2022. This issue of SIAM Review contains two papers in the Education section. The first, “A Generalized Dual Transform: Linear Algebra and Geometry of (Pseudo)Inverting a Matrix,” is presented by L. P. Withers, Jr. For a linear subspace $A$ of a vector space $V$, we may have a nonorthogonal basis of $A$. We could obtain an orthogonal basis (e

SIAM Review, Volume 64, Issue 4, Page 991-1028, November 2022. Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computational approach is convex optimization, which scales very efficiently to high-dimensional image models and delivers accurate point estimation results. However, in order to perform more complex

SIAM Review, Volume 64, Issue 4, Page 989-989, November 2022. The SIGEST article in this issue is “A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau,” by Alain Durmus, Éric Moulines, and Marcelo Pereyra. The authors provide new algorithms to sample from high-dimensional log-concave probability measures, where they combine Moreau--Yosida

SIAM Review, Volume 64, Issue 4, Page 954-988, November 2022. Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well-conditioned problems. In this paper, specialized variants of an interior point-proximal method of multipliers

SIAM Review, Volume 64, Issue 4, Page 921-953, November 2022. We consider filtering in high-dimensional non-Gaussian state-space models with intractable transition kernels, nonlinear and possibly chaotic dynamics, and sparse observations in space and time. We propose a novel filtering methodology that harnesses transportation of measures, convex optimization, and ideas from probabilistic graphical

SIAM Review, Volume 64, Issue 4, Page 919-919, November 2022. The first Research Spotlights article in this issue is concerned with filtering, a task of paramount importance in a great many applications such as numerical weather prediction and geophysical data assimilation. Authors Alessio Spantini, Ricardo Baptista, and Youssef M. Marzouk, in their article “Coupling Techniques for Nonlinear Ensemble

SIAM Review, Volume 64, Issue 4, Page 866-918, November 2022. This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic

SIAM Review, Volume 64, Issue 4, Page 831-865, November 2022. This paper is concerned mainly with the deceptively simple integral equation $$ \hskip2cm u(x) \pm \frac{1}{\pi}\int_{-1}^{1} \frac{\alpha u(y)}{\alpha^2+(x-y)^2}\, {\rm d} y =1,\qquad -1\leq x\leq 1, $$ where $\alpha$ is a real nonzero parameter and $u$ is the unknown function. This equation is classified as a Fredholm integral equation

SIAM Review, Volume 64, Issue 4, Page 763-830, November 2022. Many dynamical systems can be successfully analyzed by representing them as networks. Empirically measured networks and dynamic processes that take place in these situations show heterogeneous, non-Markovian, and intrinsically correlated topologies and dynamics. This makes their analysis particularly challenging. Randomized reference models

SIAM Review, Volume 64, Issue 4, Page 761-761, November 2022. This issue of SIAM Review contains three Survey and Review papers. When taken together, they show the enormous variety of mathematical ideas and techniques currently being used by applied mathematicians and also the wide spectrum of application fields now being tackled, ranging from very classic areas like electromagnetism to the investigation

SIAM Review, Volume 64, Issue 3, Page 751-759, August 2022. The section starts with two hot topics combined into one book: data-driven science and COVID-19. In our featured review, Anita Layton reviews the book Computational Epidemiology: Data-Driven Modeling of COVID-19, by Ellen Kuhl. She concludes her detailed review with the assessment: “This is a timely and truly wonderful book.” Dynamics continues

SIAM Review, Volume 64, Issue 3, Page 728-747, August 2022. The Gray--Scott model is a widely studied autocatalytic model that exhibits a range of interesting pattern formation behavior, as well as a rich structure of dynamics that includes many ideas from a typical undergraduate dynamical systems course, and some from beyond. Gaining an understanding of the solutions to this model is arguably most

SIAM Review, Volume 64, Issue 3, Page 713-727, August 2022. We consider, and study with elementary calculus, the polyhedral norms $||x||_{(k)}=$ sum of the $\mathit{k}$ largest among the $|x_{i}|$'s. Besides their basic properties, we provide various expressions of the unit balls associated with them and determine all the facets and vertices of these balls. We do the same with the dual norm $||.||_{(k)}^{\ast

SIAM Review, Volume 64, Issue 3, Page 711-712, August 2022. This issue of SIAM Review presents two papers in the Education section. The first paper, “Deforming $\|\cdot\|_1$ into $\|\cdot\|_\infty$ via Polyhedral Norms: A Pedestrian Approach,” is contributed by Manlio Gaudioso and Jean-Baptiste Hiriart-Urruty. The paper starts from the well-known properties of the $\ell_p$-norms in the Euclidean space

SIAM Review, Volume 64, Issue 3, Page 689-709, August 2022. Federal student loans are fixed-rate debt contracts with three main special features: (i) borrowers can use income-driven schemes to make payments proportional to their income above subsistence; (ii) after several years of being in good standing, the remaining balance is forgiven but taxed as ordinary income; and (iii) accrued interest is

SIAM Review, Volume 64, Issue 3, Page 687-687, August 2022. In this section we present “Minimizing the Repayment Cost of Federal Student Loans,” by Paolo Guasoni and Yu-Jui Huang. This is the highlighted SIGEST version of an article that first appeared in the SIAM Journal on Financial Mathematics (SIFIN) in 2021. The article studies optimal strategies for repayment of federal student loans in the USA

SIAM Review, Volume 64, Issue 3, Page 650-685, August 2022. The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility

SIAM Review, Volume 64, Issue 3, Page 640-649, August 2022. The solution of systems of linear(ized) equations lies at the heart of many problems in scientific computing. In particular, for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the conjugate gradient method (for definite problems) or MINRES

SIAM Review, Volume 64, Issue 3, Page 625-639, August 2022. Firms in the U.S. spend over $200 billion each year advertising their products to consumers, around one percent of the country's gross domestic product. It is of great interest to understand how that aggregate expenditure affects prices, market efficiency, and overall welfare. Here, we present a mathematical model for the dynamics of competition