Research Spotlights (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 case of network-based modeling. Instead of representing a physical quantity (e.g., density) or abstract quantity (e.g., opinions), in this regime the nodal variables represent phases (points on the circle) and the network state belongs to the $n$-torus instead of $R^{n}$. A flow network is described by a commodity (e.g., electricity) that is being exchanged between adjacent nodes and is defined in terms of its flow balance equation, flow equation, and angle constraint. The authors use AC power flow equations as an illustration of a flow network and explain how the general theoretical questions posed by the authors relate to important questions about stability of power supply. An elastic network is described via an interaction energy between adjacent nodes. For insight into this type of network model on the $n$-torus, the authors describe the model of collective motion in biological and engineering networks. To frame the balance of the paper, the authors enumerate five important theoretical questions they will address to greater or lesser extent. Since they prove early on that “any result about the flow network setting is directly applicable to the elastic network setting, and vice versa," the authors are free to focus only on the flow network setting in the remainder. What follows is the development of a rigorous graph-theoretic and geometric framework to study the solutions of flow network problems on the $n$-torus. Proofs are left to the appendices to enable the reader to better follow the stages of this development. Those whose interest is piqued by this line of research will find suggestions for further consideration and application in the final section of the paper. The topic of the second Research Spotlights article in this issue deals with, to quote the author, “one of the most fundamental matrices in applied mathematics," the discrete Fourier transform (DFT) matrix. In the article “How Exponentially Ill-Conditioned Are Contiguous Submatrices of the Fourier Matrix?" author Alex Barnett considers the conditioning of its $p \times q$ submatrices as functions of their scaled submatrix shape, where shape is defined by the pair ($p/N,q/N$), with $N$ denoting the DFT matrix dimension. The topic is of importance in many applications where the amplification of noise during the process must be avoided. Such applications, described in some level of detail in the introduction, include Fourier continuation, superresolution imaging, and coherent X-ray diffraction imaging. The paper focuses on finding good lower bounds on the condition number of a $p \times q$ submatrix in terms of $p$, $q$, and $N$. The author has chosen a creative presentation format. The three main theorems are presented immediately in section 1.1. The second of these largely supersedes the first, but, as the author notes, the first is included because “its prefactor is slightly stronger and, moreover, its elementary proof (section 2) is instructive." From there, the reader is presented with a discussion of previous work and an overview of the tools and techniques needed for the proofs presented in the following sections. With the background laid, each of the next three sections is devoted to a proof of one of three theorems. The interested reader might wonder about the sharpness of the bounds in practice, and indeed the discussion in section 5.1 addresses precisely this issue. Many open questions, such as upper bounds on the condition numbers, are described in the final section, leaving the curious reader with avenues for future research in this area. Our third article addresses a technique familiar to many of our RS readers: numerical quadrature. The article “Exactness of Quadrature Formulas," by Lloyd N. Trefethen, offers a survey and new results pertaining to four well-known quadrature rules. The jumping off point for the discussion is the concept of exactness, the principle around which quadrature rules are typically designed. A quadrature rule is said to be exact for a certain class of integrands---say, polynomials of degree less than or equal to $n-1$---if the rule returns the exact value of the integral for functions in this class. One point the paper drives home is the fact that “the exactness principle is not a reliable guide to the actual accuracy" of the rule. A simple but striking example of this fact lies in the author's comparison of the accuracy of an $n$-point Newton--Cotes for fixed $n$ on two polynomial classes of degree $n$ and higher. The error for the rule on the monomials remains small as the degree is slowly increased beyond $n$, an observation the author actually explains in terms of the exactness principle, while the accuracy for Chebyshev polynomials blows up almost immediately. In the case of the Gauss--Hermite quadrature rule, that the exactness principle leads to inefficiency and suboptimality in a practical sense is illustrated artfully in section 5. Here, the author also gives a theoretical result suggesting the preferred method for evaluating integrals of type (5.1) would be to instead truncate the interval to a finite length and then apply an unweighted rule like Gauss--Legendre. In another section, modifications to Gauss quadrature are derived that have a small potential gain in the convergence rate by taking factors other than exactness into consideration. The moral of the story woven together through survey and new results is, in the parting words of the author, “if you are told that a certain formula is optimal, do not assume this is the end of the discussion."