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

SIAM Review, Volume 62, Issue 3, Page 685-715, January 2020. This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is simplicity, requiring only knowledge of linear algebra and graph theory. We have also isolated the algebra from

SIAM Review, Volume 62, Issue 3, Page 661-682, January 2020. Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices

SIAM Review, Volume 62, Issue 3, Page 716-726, January 2020. We present an elementary proof of a generalization of Kirchhoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which involves only the two incidence matrices that capture the graph's topology. We also point out how this result can be used

SIAM Review, Volume 62, Issue 3, Page 729-739, January 2020. These are difficult times, but some circumstances make the work in our Book Reviews section even more difficult. Due to delivery difficulties during the coronavirus pandemic, some publishers temporarily switched from the delivery of physical books for reviews to the delivery of ebooks. This is an understandable and obvious measure at the

SIAM Review, Volume 62, Issue 3, Page 683-683, January 2020. This issue of SIAM Review contains two papers in the Education section. The first paper, “Hodge Laplacians on Graphs,” is presented by Lek-Heng Lim. The classical Hodge Laplacian is a differential operator defined on any manifold equipped with a Riemannian metric. In this paper, the author provides an accessible introduction to what he calls

SIAM Review, Volume 62, Issue 3, Page 615-615, January 2020. In this issue, Research Spotlights features two exciting and topically diverse articles. The authors of the first article introduce the notion of a metaplex and use it to describe and study dynamics of complex systems having both exo- and endostructure. Unlike previous works where both the exo- and endostructures of the complex systems are

SIAM Review, Volume 62, Issue 3, Page 659-659, January 2020. The SIGEST article in this issue, “Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations,” by Jed A. Duersch and Ming Gu, concerns the use of randomization to reduce the bottleneck in a central numerical linear algebra kernel. The authors consider QR factorization, which is a key component in algorithms

SIAM Review, Volume 62, Issue 3, Page 527-527, January 2020. Andrii Mirochenko and Christophe Prieur are the authors of “Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions,” the Survey and Review paper in this issue of SIAM Review. Introduced in the late 1980s by E. Sontag, the notion of input-to-state stability (ISS) revolutionized the control theory of finite-dimensional

SIAM Review, Volume 62, Issue 2, Page 483-508, January 2020. Polynomial approximations constructed using a least-squares approach form a ubiquitous technique in numerical computation. One of the simplest ways to generate data for least-squares problems is with random sampling of a function. We discuss theory and algorithms for stability of the least-squares problem using random samples. The main lesson

SIAM Review, Volume 62, Issue 2, Page 463-482, January 2020. We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is

SIAM Review, Volume 62, Issue 2, Page 439-462, January 2020. Boundary-value problems involving the linear differential equation $\varepsilon y'' - x y' + y = 0$ have surprising properties as $\varepsilon\to 0$. We examine this equation from eight points of view, showing how it sheds light on aspects of numerical analysis (backward error analysis and ill-conditioning), asymptotics (boundary layer analysis)

SIAM Review, Volume 62, Issue 2, Page 395-436, January 2020. We consider minimization of indefinite quadratics with either trust-region (norm) constraints or cubic regularization. Despite the nonconvexity of these problems we prove that, under mild assumptions, gradient descent converges to their global solutions and give a nonasymptotic rate of convergence for the cubic variant. We also consider Krylov

SIAM Review, Volume 62, Issue 2, Page 353-391, January 2020. Using graphs to model pairwise relationships between entities is a ubiquitous framework for studying complex systems and data. Simplicial complexes extend this dyadic model of graphs to polyadic relationships and have emerged as a model for multinode relationships occurring in many complex systems. For instance, biological interactions occur

SIAM Review, Volume 62, Issue 2, Page 301-350, January 2020. The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astronomy, radar, speech recognition, quantum mechanics, and, perhaps most prominently, diffraction imaging. The mathematical study of phase retrieval problems possesses

SIAM Review, Volume 62, Issue 2, Page 511-526, January 2020. I am writing these lines in April 2020 while science is in lockdown mode globally. They will be published in about two months, hopefully when we are mostly out of the lockdown. Because of this time lag, I will avoid further remarks with relevance to the current situation and try to do business as usual. The first and featured review, of the

SIAM Review, Volume 62, Issue 2, Page 437-438, January 2020. The Education section of SIAM Review presents three papers in this issue. In the first paper Lloyd N. Trefethen reviews “Eight Perspectives on the Exponentially Ill-Conditioned Equation $\varepsilon y'' - x y' + y = 0$.” This paper illustrates how an ensemble of mathematical techniques can come together to provide interesting insights into

SIAM Review, Volume 62, Issue 2, Page 393-393, January 2020. The SIGEST article in this issue, “First-Order Methods for Nonconvex Quadratic Minimization,” by Yair Carmon and John C. Duchi, looks at the minimization of quadratic functions that are (a) potentially indefinite and (b) have been regularized either via a trust-region constraint or a cubic penalty term. This task forms an important component

SIAM Review, Volume 62, Issue 2, Page 351-351, January 2020. The current issue of SIAM Review features the Research Spotlights article “Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian," coauthored by Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaie. Simplicial complexes (SCs) allow one to model multimode relationships in a way that standard

SIAM Review, Volume 62, Issue 2, Page 299-299, January 2020. In a previous issue of SIAM Review, I started my introduction to the Survey and Review section by saying that it is safe to bet that most readers use matrices in their work. I would also safely bet that most readers have used the (discrete or continuous) Fourier transform at some point, since, for reasons I have not yet completely understood

SIAM Review, Volume 62, Issue 1, Page 283-297, January 2020. The frame for the Book Reviews section this time is formed by the books of two very well known and successful SIAM Fellows. In his featured review Akhtar Khan introduces the new book by Boris Mordukhovich, Variational Analysis and Applications. He praises the very extensive work as being “not only of great benefit to the research community

SIAM Review, Volume 62, Issue 1, Page 229-230, January 2020. The Education section of SIAM Review presents three papers in this issue. In the first paper Robert M. Corless and Leili Rafiee Sevyeri discuss “The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding.” The authors use several classical examples in numerical analysis to discuss propagation of error and the role of sensitivity

SIAM Review, Volume 62, Issue 1, Page 197-197, January 2020. The SIGEST article in this issue, “Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models,” by Xiaochuan Tian and Qiang Du, concerns differential equation models that (a) involve a physical parameter, and (b) change their nature when this parameter reaches an asymptotic limit

SIAM Review, Volume 62, Issue 1, Page 131-131, January 2020. Tensors, or multiway arrays, are often used for storage of high-dimensional data. In order to have compression, completion, or interpretation of such data, the data tensor is factored according to a proposed model consistent with some belief about the data. The first Research Spotlights article, “Generalized Canonical Polyadic Tensor Decomposition

SIAM Review, Volume 62, Issue 1, Page 1-1, January 2020. Bernard Brogliato and Aneel Tanwani are the authors of “Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability,” the Survey and Review paper in this issue. To get a feeling for the problems being dealt with, the reader may have a look at the electrical circuit in Figure 4.1(c) (also

SIAM Review, Volume 62, Issue 1, Page 264-280, January 2020. In this work we introduce a new operator based approach to matrix analysis. Our main technical tool comprises an extension of a tool introduced long ago by MacMahon to analyze the partitions of natural numbers: the Omega operator calculus. More precisely, we construct an operator acting linearly on absolutely convergent matrix valued expansions

SIAM Review, Volume 62, Issue 1, Page 244-263, January 2020. Most undergraduates have limited experience with mathematical modeling. In an effort to respond to various initiatives, such as the recommendations outlined in [S. Garfunkel and M. Montgomery, eds., GAIMME: Guidelines for Assessment & Instruction in Mathematical Modeling Education, SIAM, 2016], this paper describes a course on the mathematical

SIAM Review, Volume 62, Issue 1, Page 231-243, January 2020. We look at two classical examples in the theory of numerical analysis, namely, the Runge example for interpolation and Wilkinson's example (actually two examples) for rootfinding. We use the modern theory of backward error analysis and conditioning, as instigated and popularized by Wilkinson but refined by Farouki and Rajan. By this means

SIAM Review, Volume 62, Issue 1, Page 199-227, January 2020. Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied

SIAM Review, Volume 62, Issue 1, Page 164-195, January 2020. Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications in fields such as biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of

SIAM Review, Volume 62, Issue 1, Page 133-163, January 2020. Tensor decomposition is a fundamental unsupervised machine learning method in data science, with applications including network analysis and sensor data processing. This work develops a generalized canonical polyadic (GCP) low-rank tensor decomposition that allows other loss functions besides squared error. For instance, we can use logistic

SIAM Review, Volume 62, Issue 1, Page 3-129, January 2020. This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations is coupled with a static or time-varying set-valued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems, including sweeping processes, differential inclusions with maximal