SIAM Review, Volume 63, Issue 3, Page 487-487, January 2021.
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.

中文翻译：

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研究焦点

SIAM 评论，第 63 卷，第 3 期，第 487-487 页，2021 年 1 月。
本期两篇 Research Spotlights 文章中的第一篇给出了一个通用框架和相应算法，用于计算自伴随算子的谱测度的精确近似值，其关键是算子的求解。正如作者 Matthew Colbrook、Andrew Horning 和 Alex Townsend 详细说明的那样，频谱测量对于他们感兴趣的应用程序是必要的，以便更完整地描述算子和相关动力学。他们的文章“计算自伴随算子的谱度量”以谱度量和相关假设的正式数学描述开始。文章中强调了几个依赖于谱度量计算的应用程序：例如，粒子和凝聚态物理中的应用在对先前光谱测量估计工作的调查的背景下进行了讨论。所提出的方法的关键，首先在第 4 节中阐述并在第 5 节中推广到更高阶内核以提高精度，是通过评估解析度来评估频谱度量的平滑近似（即，通过与适当内核卷积定义），后者的计算类似于求解线性方程的移位系统。作者仔细地详细说明并以图形方式展示了与设计一个易于处理且稳健的方案相关的挑战。第 6 节中对算法问题的讨论，包括他们的方法动态调整以达到所需精度的能力，还提供了相关的公开可用 MATLAB 代码的用例。在文章的正文中，他们提出的框架的多功能性在微分、积分和格算子上得到了很好的说明。读者可能对第 8 节中关于新框架可能的进一步用例的建议感兴趣，例如“理解大型现实世界网络的行为和新的随机图模型。”我们的第二篇文章，“均值优先的优化”具有小型吸收陷阱的二维近盘和椭圆域中的通道时间”，由 Sarafa A. Iyaniwura、Tony Wong、Colin B. Macdonald 和 Michael J. Ward 合着。窄逃逸或捕获问题在介绍中被描述为首次通过时间问题，描述了布朗粒子以小度量到达某个吸收集的预期时间。出现此类问题的两个应用包括扩散的表面结合分子（在这种情况下为“粒子”）到达细胞膜上的局部信号区域所需的时间以及捕食者定位所需的时间它的猎物。作者将扩散过程的“平均 MFPT”定义为在随机游走起点均匀分布的情况下捕获的预期时间。圆盘以外几何形状中平均 MFPT 的最佳陷阱配置尚未解决，这为作者在近圆盘和椭圆域的背景下研究该问题提供了动力。通过渐近分析和数值技术的结合（例如，使用数值正交和数值时间步长求解方程 (3.4) 和 (4.3)），作者设计了“混合渐近数值”方法来预测小型静止圆形吸收器的最佳配置在这些新领域中最小化平均 MFPT 的陷阱。尽管本文的大部分内容都致力于详细的推导，读者需要一些时间来吸收，通过将新结果与 PDE 生成的数值解进行比较的图形说明，人们可以立即对结果有所了解。该方法的扩展和剩余的未解决问题包含在最后一节中供读者考虑。