Research Spotlights

Abstract

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 measure is necessary in their applications of interest to give a more complete description of the operator and associated dynamics. Their article, “Computing Spectral Measures of Self-Adjoint Operators," begins with the formal mathematical description of the spectral measure and associated assumptions. Several applications that rely on the computation of spectral measure are highlighted throughout the article: for example, applications in particle and condensed matter physics are discussed in the context of a survey of previous work in estimation of spectral measure. The crux of the proposed approach, first formulated in section 4 and generalized to higher order kernels for improved accuracy in section 5, is to evaluate a smoothed approximation (i.e., defined through convolution with appropriate kernel) to the spectral measure by evaluating the resolvent, with the latter calculation akin to solving shifted systems of linear equations. The authors detail carefully and demonstrate graphically the challenges associated with designing a tractable and robust scheme. The discussion of algorithmic issues in section 6, including the ability of their approach to dynamically adjust to reach desired accuracy, also features use cases of their associated publicly available MATLAB code. Within the body of the article, the versatility of their proposed framework is well illustrated on differential, integral, and lattice operators. Readers may be interested in the suggestions in section 8 on possible further use cases for the new framework, such as in “understanding the behavior of large real-world networks and new random graph models." Our second article, “Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps," is coauthored by Sarafa A. Iyaniwura, Tony Wong, Colin B. Macdonald, and Michael J. Ward. Narrow escape or capture problems are those portrayed in the introduction as first passage time problems that describe the expected time for a Brownian particle to reach some absorbing set with small measure. Two of the applications in which such problems arise include the time it takes for a diffusing surface-bound molecule (the “particle” in this case) to reach a localized signaling region on the cell membrane and the time it takes for a predator to locate its prey. The authors define the “average MFPT" for a diffusion process to be the expected time for capture given a uniform distribution of starting points for the random walk. The optimal trap configurations for the average MFPT in geometries other than the disk had been unsolved and provided the impetus for the authors to investigate the question in the context of near-disk and elliptical domains. Through a combination of asymptotic analysis and numerical techniques (e.g., use of numerical quadrature and numerical time stepping for solving equations (3.4) and (4.3)), the authors design “hybrid asymptotic numerical" approaches to predicting optimal configurations of small stationary circular absorbing traps that minimize the average MFPT in these new domains. Though much of the paper is devoted to detailed derivations that will take some time for the reader to absorb, one can get some immediate appreciation for the results from the graphical illustrations in which the new results are compared against numerical PDE generated solutions. Extensions of the approach and remaining open problems are included in the last section for consideration by the reader.