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

SIAM Review, Volume 63, Issue 2, Page 405-415, January 2021. Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem. This amounts to on-the-fly construction of discrete orthogonal polynomials by Stieltjes

SIAM Review, Volume 63, Issue 2, Page 395-404, January 2021. The Rodrigues formula for the $3\times 3$ rotation matrix is hardly ever derived from first principles in a simple and intuitive way that is accessible to undergraduate students. Two different derivations of the rotation matrix are presented here. This communication is written in a simple and expository style, so that it could serve as supplementary

SIAM Review, Volume 63, Issue 2, Page 393-393, January 2021. This issue of SIAM Review presents two papers in the Education section. The first paper, “Rotations in Three Dimensions,” is written by Milton F. Maritz. Current textbooks in calculus and linear algebra typically discuss rotation on the plane and do not present rotation matrices in higher dimensions. The author provides two ways to derive

SIAM Review, Volume 63, Issue 2, Page 375-392, January 2021. Compartment models are a widely used class of models that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments. Classic examples include SIR models in epidemiology and many pharmacokinetic models used in pharmacology. These models are formulated as sets of coupled

SIAM Review, Volume 63, Issue 2, Page 373-373, January 2021. The SIGEST article in this issue is “A General Framework for Fractional Order Compartment Models,” by Christopher N. Angstmann, Austen M. Erickson, Bruce I. Henry, Anna V. McGann, John M. Murray, and James A. Nichols. The seminal compartmental publication of Kermack and McKendrick ([24] in the reference list) is approaching its 100th anniversary

SIAM Review, Volume 63, Issue 2, Page 360-372, January 2021. This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integral over others, arising in Bayesian optimization. This strong law requires that the integrand have a finite moment of order $p$ for some $p>1$.

SIAM Review, Volume 63, Issue 2, Page 317-359, January 2021. The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t={\mathcal{L}} u+f[u]$, with ${\mathcal{L}}$ a linear

SIAM Review, Volume 63, Issue 2, Page 315-315, January 2021. Under what conditions does a semilinear parabolic equation have the maximum bound principle, and do numerical approximations of it preserve the principle? These are the main questions that are addressed by authors Qiang Du, Lili Ju, Xiao Li, and Zhonghua Qiao in their paper “Maximum Bound Principles for a Class of Semilinear Parabolic Equations

SIAM Review, Volume 63, Issue 2, Page 249-313, January 2021. In 1931--1932, Erwin Schrödinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schrödinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes

SIAM Review, Volume 63, Issue 2, Page 247-247, January 2021. ``Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge,” by Yongxin Chen, Tryphon T. Georgiou, and Michele Pavon, is the Survey and Review article in this issue. There is little doubt that the authors chose to have a very intriguing title to entice SIAM Review readers into browsing through their work;

SIAM Review, Volume 63, Issue 1, Page 231-245, January 2021. The section opens with a review well written by a giant in the field of numerical solution of partial differential equations, Roland Glowinski. In this featured review, he analyzes the book Think Before You Compute: A Prelude to Computational Fluid Dynamics, by E. J. Hinch, in much detail and contributes his own thoughts on appraisal or possibilities

SIAM Review, Volume 63, Issue 1, Page 208-228, January 2021. Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm

SIAM Review, Volume 63, Issue 1, Page 181-207, January 2021. While students may find spline interpolation quite digestible based on their familiarity with the continuity of a function and its derivatives, some of its inherent value may be missed when they only see it applied to standard data interpolation exercises. In this paper, we offer alternatives in which students can qualitatively and quantitatively

SIAM Review, Volume 63, Issue 1, Page 167-180, January 2021. The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in nonperiodic cases. Commonly used improvements, such as Simpson\textquoteright s rule and the Newton--Cotes formulas, are not much (if at all) better than the even more classical quadrature

SIAM Review, Volume 63, Issue 1, Page 165-166, January 2021. This issue of SIAM Review presents three papers in the Education section. The first paper, “Improving the Accuracy of the Trapezoidal Rule,” is written by Bengt Fornberg. The trapezoidal rule is a basic technique for numerical calculation of definite one-dimensional integrals. The author provides a historical perspective of the developments

SIAM Review, Volume 63, Issue 1, Page 123-164, January 2021. Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing

SIAM Review, Volume 63, Issue 1, Page 121-121, January 2021. The SIGEST article in this issue is “What Tropical Geometry Tells Us about the Complexity of Linear Programming,” by Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig. Linear programming means optimizing a linear objective function with respect to linear equality constraints and linear inequality constraints. This

SIAM Review, Volume 63, Issue 1, Page 100-120, January 2021. Animals use stereo sampling of odor concentration to localize sources and follow odor trails. We analyze the dynamics of a bilateral model that depends on the simultaneous comparison between odor concentrations detected by left and right sensors. The general model consists of three differential equations for the positions in the plane and

SIAM Review, Volume 63, Issue 1, Page 67-99, January 2021. A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively

SIAM Review, Volume 63, Issue 1, Page 65-65, January 2021. We have two Research Spotlights papers in the lineup this issue. The first of these, “Persistent Homology of Geospatial Data: A Case Study with Voting," written by Michelle Feng and Mason Porter, is quite timely. The article begins with a gentle introduction into the what and why of persistent homology (PH): after converting the point cloud

SIAM Review, Volume 63, Issue 1, Page 3-64, January 2021. Computing the stationary distributions of a continuous-time Markov chain (CTMC) involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and the equations cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space

SIAM Review, Volume 63, Issue 1, Page 1-1, January 2021. The use of probabilistic ideas by applied mathematicians has seen a continued increase in recent decades. Probability now appears frequently at the modeling stage. There is widespread interest in investigating the effects of noise and uncertainty. Probabilistic algorithms are routinely applied with much success to the solution of deterministic

SIAM Review, Volume 62, Issue 4, Page 985-994, January 2020. The first and featured review is on the book Introduction to Numerical Methods for Variational Problems, by Hans Petter Langtangen and Kent-Andre Mardal. It is one of the last books co-authored by Hans Petter Langtangen, who was a very influential, enthusiastic researcher and teacher, and nevertheless a very kind person. I met him several

SIAM Review, Volume 62, Issue 4, Page 936-981, January 2020. This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's

SIAM Review, Volume 62, Issue 4, Page 901-935, January 2020. We study the time correlation functions of coupled linear Langevin dynamics with and without inertial effects, both analytically and numerically. The model equation represents the physical behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and

SIAM Review, Volume 62, Issue 4, Page 899-900, January 2020. This issue of SIAM Review presents two papers in the Education section. The first paper, “Time Correlation Functions of Equilibrium and Nonequilibrium Langevin Dynamics: Derivations and Numerics Using Random Numbers,” is coauthored by Xiaocheng Shang and Martin Krog̈er. Langevin dynamics is a term describing a mathematical model of the dynamics

SIAM Review, Volume 62, Issue 4, Page 869-897, January 2020. In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb

SIAM Review, Volume 62, Issue 4, Page 867-867, January 2020. In this section we present “A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularizing Using Geometric Singular Perturbation Theory,” by Elena Bossolini, Morten Brøns, and Kristian Uldall Kristiansen. This is the highlighted SIGEST version of an article that first appeared in the SIAM Journal

SIAM Review, Volume 62, Issue 4, Page 837-865, January 2020. Forecasting elections---a challenging, high-stakes problem---is the subject of much uncertainty, subjectivity, and media scrutiny. To shed light on this process, we develop a method for forecasting elections from the perspective of dynamical systems. Our model borrows ideas from epidemiology, and we use polling data from United States elections

SIAM Review, Volume 62, Issue 4, Page 817-836, January 2020. The field of network synchronization has seen tremendous growth following the introduction of the master stability function (MSF) formalism, which enables the efficient stability analysis of synchronization in large oscillator networks. However, to make further progress we must overcome the limitations of this celebrated formalism, which

SIAM Review, Volume 62, Issue 4, Page 781-816, January 2020. Uncertainties in velocity data are often ignored when computing Lagrangian particle trajectories of fluids. Modeling these as noise in the velocity field leads to a random deviation from each trajectory. This deviation is examined within the context of small (multiplicative) stochasticity applying to a two-dimensional unsteady flow operating

SIAM Review, Volume 62, Issue 4, Page 779-780, January 2020. We are fortunate to have three topically diverse papers featured in Research Spotlights in the current issue. The first of these, “Stochastic Sensitivity: A Computable Lagrangian Uncertainty Measure for Unsteady Flows," deals with modeling uncertainties in velocity data for the purpose of quantifying their influence when computing Lagrangian

SIAM Review, Volume 62, Issue 4, Page 743-777, January 2020. The prevalence of convolution in applications within signal processing, deep neural networks, and numerical solvers has motivated the development of numerous fast convolution algorithms. In many of these problems, convolution is performed on terabytes or petabytes of data, so even constant factors of improvement can significantly reduce the

SIAM Review, Volume 62, Issue 4, Page 741-741, January 2020. Finding the convolution of two vectors is a ubiquitous task in applied mathematics. Signal processing, image processing, deep neural networks, the numerical solution of partial differential equations, and other current applications require the computation of convolutions, often on terabytes or petabytes of data. Convolution was incognito

SIAM Review, Volume 62, Issue 3, Page 617-645, January 2020. In a complex system, the interplay between the internal structure of its entities and their interconnection may play a fundamental role in the global functioning of the system. Here, we define the concept of metaplex, which describes such a trade-off between the internal structure of entities and their interconnections. We then define a dynamical

SIAM Review, Volume 62, Issue 3, Page 646-657, January 2020. Since the 1960s, Democrats and Republicans in the U.S. Congress have taken increasingly polarized positions, while the public's policy positions have remained centrist and moderate. We explain this apparent contradiction by developing a dynamical model that predicts ideological positions of political parties. Our approach tackles the challenge

SIAM Review, Volume 62, Issue 3, Page 529-614, January 2020. In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows for the estimation of the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output