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

SIAM Review, Volume 64, Issue 3, Page 623-624, August 2022. What's in a name [brand]? In the first of three Research Spotlights articles this issue, authors Joseph D. Johnson, Adam M. Redlich, and Daniel M. Abrams address this issue by developing a mathematical model for the dynamics of competition through advertising. Specifically, in “A Mathematical Model for the Origin of Name Brands and Generics

SIAM Review, Volume 64, Issue 3, Page 554-622, August 2022. In this survey we provide an overview of recent advances on scalable load balancing schemes which provide favorable delay performance and yet require minimal implementation overhead. The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. The

SIAM Review, Volume 64, Issue 3, Page 517-553, August 2022. The primary aim of this paper is to provide a review of the last few decades of research focused on delayed gradient methods for solving symmetric positive definite linear systems. Recent developments on spectral properties are stressed, as well as their implications for the formulation of new methods. Numerical experiments are conducted with

SIAM Review, Volume 64, Issue 3, Page 515-515, August 2022. This issue contains two Survey and Review papers. The first, by Qinmeng Zou and Frédéric Magoulès, is “Delayed Gradient Methods for Symmetric and Positive Definite Linear Systems.” Gradient methods are the oldest and simplest algorithms to minimize a real objective function (f(x)). The ((n+1))st approximation to the minimizer is defined as

SIAM Review, Volume 64, Issue 2, Page 503-513, May 2022. We start this section with Andreas Mang's in-depth featured review on the book Introduction to the Tools of Scientific Computing, written by Einar Smith. Seemingly the level of the book is somewhat more fundamental as it investigates the tools of the trade rather than its theoretical sophistication. We continue with aspects of scientific computing

SIAM Review, Volume 64, Issue 2, Page 485-499, May 2022. In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. We illustrate a method to generate exact

SIAM Review, Volume 64, Issue 2, Page 469-484, May 2022. In this paper, we provide assembly instructions for an easy-to-build experimental setup in order to gain practical experience with tomography. As such, this paper can be seen as a complementary work to excellent undergraduate-level mathematical textbooks concerned with the basic mathematical principles of tomography. Since the setup uses light

SIAM Review, Volume 64, Issue 2, Page 467-467, May 2022. This issue contains two papers in the Education section. The first paper, “Computed Origami Tomography,” is presented by Axel Kittenberger, Leonidas Mindrinos, and Otmar Scherzer. The paper is motivated by advances in scanner technology and the increased sophistication of the accompanying 3D image reconstruction methods. The authors have implemented

SIAM Review, Volume 64, Issue 2, Page 425-465, May 2022. Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar cubic graphs with faces of size at most 6 are Hamiltonian. We prove Barnette and Goodey's conjecture.

SIAM Review, Volume 64, Issue 2, Page 423-423, May 2022. The SIGEST article in this issue is Hamiltonicity of Cubic Planar Graphs with Bounded Face Sizes, by František Kardoš. The original version of this article appeared in the SIAM Journal on Discrete Mathematics in 2020. Barnette's Conjecture is a famous problem in graph theory which asserts that every cubic bipartite 3-connected planar graph has

SIAM Review, Volume 64, Issue 2, Page 392-421, May 2022. We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives, mainly because the state variable depends nonlinearly on the control variable and

SIAM Review, Volume 64, Issue 2, Page 360-391, May 2022. Competitive tournaments appear in sports, politics, population ecology, and animal behavior. All of these fields have developed methods for rating competitors and ranking them accordingly. A tournament is intransitive if it is not consistent with any ranking. Intransitive tournaments contain rock-paper-scissors type cycles. The discrete Helmholtz--Hodge

SIAM Review, Volume 64, Issue 2, Page 343-359, May 2022. We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a

SIAM Review, Volume 64, Issue 2, Page 341-342, May 2022. This issue features three Research Spotlight articles. The first of these is entitled “Variance and Covariance of Distributions on Graphs" and is coauthored by Karel Devriendt, Samuel Martin-Gutierrez, and Renaud Lambiotte. Given a distribution on the graph (that is, a function $p$ from the set of nodes to $[0,1]$ such that the sum of all values

SIAM Review, Volume 64, Issue 2, Page 229-340, May 2022. The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator-theoretic or probabilistic frameworks. Koopman spectral theory has emerged

SIAM Review, Volume 64, Issue 2, Page 227-227, May 2022. Steven L. Brunton, Marko Budišić, Eurika Kaiser, and J. Nathan Kutz are the authors of the Survey and Review paper in this issue, “Modern Koopman Theory for Dynamical Systems.” Koopman theory is a valuable formalism for the study of dynamical systems; it has gained popularity in recent years in connection with data-driven analysis, control theory

SIAM Review, Volume 64, Issue 1, Page 191-211, February 2022. We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the linear solve has very little negative effect on the nonlinear convergence rate. However, if one considers

SIAM Review, Volume 64, Issue 1, Page 181-190, February 2022. The reduced row echelon form ${\bf rref}(A)$ has traditionally been used for classroom examples: small matrices $A$ with integer entries and low rank $r$. This paper creates a column-row rank-revealing factorization ${A=CR}$, with the first $r$ independent columns of $A$ in $C$ and the $r$ nonzero rows of ${\bf rref}(A)$ in $R$. We want

SIAM Review, Volume 64, Issue 1, Page 153-178, February 2022. t-distributed stochastic neighborhood embedding (t-SNE), a clustering and visualization method proposed by van der Maaten and Hinton in 2008, has rapidly become a standard tool in the natural sciences. Despite its overwhelming success, it has a distinct lack of mathematical foundations and the inner workings of the algorithm are not well

SIAM Review, Volume 64, Issue 1, Page 132-150, February 2022. The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We review the subject from this point of view and show that this principle fails to predict the actual behavior in four of the best-known cases: Newton--Cotes, Clenshaw--Curtis, Gauss--Legendre

SIAM Review, Volume 64, Issue 1, Page 105-131, February 2022. Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT) matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such $p\times q$ submatrix of the $N\times N$ DFT matrix is at least $ \exp \left( \frac{\pi}{2}

SIAM Review, Volume 64, Issue 1, Page 59-104, February 2022. Networks with phase-valued nodal variables are central in modeling several important societal and physical systems, including power grids, biological systems, and coupled oscillator networks. One of the distinctive features of phase-valued networks is the existence of multiple operating conditions corresponding to critical points of an energy

SIAM Review, Volume 64, Issue 1, Page 3-56, February 2022. Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stakes issue. Vanilla Cholesky samplers imply a computational cost and memory requirements that can rapidly become prohibitive in high dimensions. To tackle these issues, multiple methods have been proposed from different communities ranging from iterative numerical

SIAM Review, Volume 64, Issue 1, Page 215-225, February 2022. By now, we have all become somewhat accustomed to COVID-19. So it is a good time to read an introduction to epidemiological dynamics from a mathematical perspective. Our first featured review examines in detail Igor Nesteruk's introductory book COVID-19 Pandemic Dynamics: Mathematical Simulations. Aspects such as the $R_0$ value, SIR model

SIAM Review, Volume 64, Issue 1, Page 179-179, February 2022. The Education section in this issue presents two contributions. The first paper, “$LU$ and $CR$ Elimination,” is written by Gilbert Strang and Cleve Moler. This is an expository paper which is helpful to instructors and students interested in linear algebra and matrix manipulations. The focus of the paper is the column-row factorization

SIAM Review, Volume 64, Issue 1, Page 151-151, February 2022. The SIAM Journal on Mathematics of Data Science (SIMODS) launched in February 2019. The SIGEST article in this issue, “Dimensionality Reduction via Dynamical Systems: The Case of t-SNE,” by George C. Linderman and Stefan Steinerberger, is therefore our first SIMODS representative. Here the authors study t-SNE, a widely adopted clustering

SIAM Review, Volume 64, Issue 1, Page 57-58, February 2022. We are fortunate to have three Research Spotlights articles, on a broad range of topics, in the current issue of SIAM Review. The first of these articles is “Flow and Elastic Networks on the $n$-Torus: Geometry, Analysis, and Computation." In it, authors Saber Jafarpour, Elizabeth Y. Huang, Kevin D. Smith, and Francesco Bullo explore a special

SIAM Review, Volume 64, Issue 1, Page 1-1, February 2022. Gaussian (also known as normal) probability distributions in (\mathbb R^d\) play a central role in statistics and are important in many branches of pure and applied mathematics, statistical physics, and several other fields. In the univariate (d=1) case, the corresponding bell curve is known even to many people outside the sciences. At present

SIAM Review, Volume 63, Issue 4, Page 867-875, January 2021. The first half of this section is devoted to books related to data science. The featured review on the book Insights from Data with R. An Introduction for the Life and Environmental Sciences, authored by Owen L. Petchey, Andrew P. Beckerman, Natalie Cooper, and Dylan Z. Childs, praises the didactic approach of the book, which starts with

SIAM Review, Volume 63, Issue 4, Page 854-864, January 2021. The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them

SIAM Review, Volume 63, Issue 4, Page 825-853, January 2021. Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from the high computational cost and excessive memory requirements associated with the high-dimensionality and heterogeneous characteristics of industrial size networks. Graph embedding techniques can be effective

SIAM Review, Volume 63, Issue 4, Page 823-823, January 2021. The Education section in this issue presents two papers. The first paper, “Understanding Graph Embedding Methods and Their Applications,” is written by Mengjia Xu. Graph theory is part of discrete mathematics, whose origin is attributed to the paper of Leonhard Euler, “Seven Bridges of Königsberg,” published in 1736. With the rise of data

SIAM Review, Volume 63, Issue 4, Page 783-821, January 2021. We consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set $\mathcal{M}\subset\mathbb{R}^m$. Such hybrid discrete-continuous problems occur in, e.g., topology optimization or medical imaging and are challenging due to

SIAM Review, Volume 63, Issue 4, Page 781-781, January 2021. The SIGEST article in this issue, which comes from the SIAM Journal on Control and Optimization (SICON), is “Convex Relaxation of Discrete Vector-Valued Optimization Problems,” by Christian Clason, Carla Tameling, and Benedikt Wirth. The authors tackle a new class of problems where a vector-valued control must take pointwise values in a given

SIAM Review, Volume 63, Issue 4, Page 756-779, January 2021. It has been found advantageous for finite-volume discretizations of flow equations to possess additional (secondary) invariants, besides the (primary) invariants from the constituting conservation laws. This paper presents general (necessary and sufficient) requirements for a method to convectively preserve discrete kinetic energy. The key

SIAM Review, Volume 63, Issue 4, Page 723-755, January 2021. We present a successful deployment of high-fidelity large-eddy simulation (LES) technologies based on spectral/$hp$ element methods to industrial flow problems, which are characterized by high Reynolds numbers and complex geometries. In particular, we describe the numerical methods, software development, and steps that were required to perform

SIAM Review, Volume 63, Issue 4, Page 721-721, January 2021. The car-themed cover art on the current issue of SIAM Review is courtesy of the first of two Research Spotlights articles this issue. Indeed, the paper “Industry-Relevant Implicit Large-Eddy Simulation of a High-Performance Road Car via Spectral/$hp$ Element Methods" delivers what the title implies: using a high-fidelity, large-eddy simulation

SIAM Review, Volume 63, Issue 4, Page 653-720, January 2021. Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today; our

SIAM Review, Volume 63, Issue 4, Page 651-651, January 2021. The Survey and Review paper in this issue, “Computation and Applications of Mathieu Functions: A Historical Perspective,” by Chris Brimacombe, Robert M. Corless, and Mair Zamir, deals with a well-known family of special functions. Special functions are of course an important element in the toolkit of many applied mathematicians. For instance

SIAM Review, Volume 63, Issue 3, Page 641-650, January 2021. The majority of books in this issue are devoted to modeling in one form or another. The featured review by Thomas Wick discusses a new book by an internationally renowned expert in the field of numerical modeling. It is the book A Primer on Mathematical Modelling, by Alfio Quarteroni and Paola Gervasio. Wick talks enthusiastically about the

SIAM Review, Volume 63, Issue 3, Page 625-637, January 2021. For students in applied mathematics courses, the phenomenon of delay-induced stability and instability offers exciting educational opportunities. Exploration of the onset of instability in delay differential equations (DDEs) invites a blend of analysis (real, complex, and functional), algebra, and computational methods. Moreover, stabilization

SIAM Review, Volume 63, Issue 3, Page 585-624, January 2021. The high speed of $x_{k}\rightarrow x^\ast\in{\mathbb R}$ is usually measured using the $C$-, $Q$-, or $R$-orders: \[ łim \tfrac |x^\ast - x_k+1||x^\ast - x_k|^p_0\in(0,+\infty), łim \tfrac łn |x^\ast - x_k+1|łn |x^\ast - x_k| =q_0, or łim \bigěrtłn |x^\ast - x_k| \big ěrt ^\frac1k =r_0. \] By connecting them to the natural, term-by-term

SIAM Review, Volume 63, Issue 3, Page 583-583, January 2021. The Education section in this issue contains two papers. The first paper, “How Many Steps Still Left to ${x}^*$?,” is written by Emil Cătinaş. This is an important question for everyone who works with numerical methods. One could compare two convergent sequences and analyze which one would get closer to its limit in a given number of steps

SIAM Review, Volume 63, Issue 3, Page 559-582, January 2021. We present a randomized method for computing the min-plus product (a.k.a. tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. In the real random-access machine model, where additions and comparisons

SIAM Review, Volume 63, Issue 3, Page 557-557, January 2021. The SIGEST article in this issue is “From Circuit Complexity to Faster All-Pairs Shortest Paths,” by R. Ryan Williams. The original paper appeared in SIAM Journal on Computing (SICOMP) in 2018 and, as indicated by the high citation count, ideas from this work have inspired a broad range of future activity. For a given $n$-node graph, the

SIAM Review, Volume 63, Issue 3, Page 525-555, January 2021. The determination of the mean first passage time (MFPT) for a Brownian particle in a bounded 2-D domain containing small absorbing traps is a fundamental problem with biophysical applications. The average MFPT is the expected capture time assuming a uniform distribution of starting points for the random walk. We develop a hybrid asymptotic-numerical

SIAM Review, Volume 63, Issue 3, Page 489-524, January 2021. Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators

SIAM Review, Volume 63, Issue 3, Page 487-487, January 2021. The first of our two Research Spotlights articles in this issue gives a general framework and corresponding algorithm for computing accurate approximations to the spectral measures of self-adjoint operators, the key to which is the resolvent of the operator. As authors Matthew Colbrook, Andrew Horning, and Alex Townsend detail, the spectral

SIAM Review, Volume 63, Issue 3, Page 435-485, January 2021. Complex systems, composed at the most basic level of units and their interactions, describe phenomena in a wide variety of domains, from neuroscience to computer science and economics. The wide variety of applications has resulted in two key challenges: the generation of many domain-specific strategies for complex systems analyses that are

SIAM Review, Volume 63, Issue 3, Page 433-433, January 2021. Our Survey and Review paper, “The Why, How, and When of Representations for Complex Systems,” by Leo Torres, Ann S. Blevins, Danielle Bassett, and Tina Eliassi-Rad, lists 233 references. Some of them were published in applied mathematics journals, like SIAM Review or SIAM Journal on Applied Algebra and Geometry. Others appeared in well-known

SIAM Review, Volume 63, Issue 2, Page 419-431, January 2021. The April issue of SIAM News sadly reported on the death of Bob O'Malley (https://sinews.siam.org/Details-Page/obituary-robert-e-omalley-jr). He led the Book Reviews section actively and dynamically for many years. This section and its editors together with the SIAM community owe him a great debt of gratitude. This time, our section has 3