Research Spotlights

Abstract

This issue of SIAM Review features three articles in the Research Spotlights section. All three articles use novel analyses to provide computational tools for tackling their respective applications. The first of the three articles is “Configuring Random Graph Models with Fixed Degree Sequences,” by B. K. Fosdick, D. B. Larremore, J. Nishimura, and J. Ugander. Configuration graph models---uniform distributions over a space of graphs with fixed degree sequence---are used to generate ensembles that are then used to assess the significance of empirical network properties. This paper carefully and concisely argues that certain design choices can significantly influence conclusions drawn from the respective models. The authors first establish eight different graph spaces and detail what questions should be considered in selecting a graph space model for null sampling. Readers are then invited either to skip to the case studies or to read about new space-specific MCMC routines and alternatives. The application of the work to three real-world network problems makes a convincing argument of the soundness of the approach. Code is also provided. The second paper is authored by Sanghyeon Yu and Habib Ammari and is titled “Plasmonic Interaction between Nanospheres.” In the presence of incident visible light, the electromagnetic fields on the surfaces of certain nanoscale metallic particles resonate and oscillate. Characterizing this plasmonic interaction between nearly touching nanospheres, either analytically or numerically, has remained a challenge preventing wider use in the design of nanophotonic devices, biosensing, and spectroscopy. After detailing the sources of these analytic and computational difficulties, the authors endeavor to tackle both problems in turn: they develop a fully analytic solution for two plasmonic spheres which they then use in the development of a so-called hybrid numerical scheme for computing the field in the case of multiple, nearly touching spheres. Although the details of the analyses as well as the solution techniques are quite technical, the article is structured to deliver the key elements to the reader first, with many of the details left to the appendix. Numerical results illustrate that the hybrid method is capable of efficiently capturing desired behavior. The paper “SPECTRWM: Spectral Random Walk Method for the Numerical Solution of Stochastic Partial Differential Equations,” by Nawaf Bou-Rabee, rounds out this issue's Research Spotlights section. The emphasis of this paper is on development and testing of a new stochastic partial differential equation (SPDE) numerical solver that is, according to the author, more precisely tailored to the structure of the SPDE solutions in the case of parabolic, semilinear SPDEs driven by an additive space-time noise process. SPECTRWM is a generalization of the Markov chain method to SPDEs. Two versions of the algorithm are provided, with the first intended for effective illustration of basic concepts. The second elegantly builds upon the intuition provided by the first. The utility of the algorithms is demonstrated in four interesting examples, including the heat and Burgers' SPDEs. MATLAB script which illustrates the straightforward nature of the author's proposed approach is provided in the appendix for the first algorithm applied to Burgers' SPDE.