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 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 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 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 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 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 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 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