SIAM Review, Volume 63, Issue 2, Page 315-315, January 2021.
Under what conditions does a semilinear parabolic equation have the maximum bound principle, and do numerical approximations of it preserve the principle? These are the main questions that are addressed by authors Qiang Du, Lili Ju, Xiao Li, and Zhonghua Qiao in their paper “Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes." The maximum bound principle (MBP) as described by the authors states that if the initial data and/or the boundary values are bounded pointwise in absolute value, then the absolute value of the solution is also bounded everywhere and for all time. One of the motivating examples employed is the Allen--Cahn equation where the MBP is known to hold. To generalize such results to a much broader class of problems, the authors employ an abstract form of the model evolution equations into which this motivating example and many others will fit. Using their formulation, they provide an analysis of what is required of the linear and nonlinear operators for the model equations to satisfy the MBP. The variety of problems that can be assessed according to this new framework is evident throughout. Moreover, the authors consider first- and second-order exponential time decay (ETD) schemes that will preserve the discrete MBP unconditionally. Colorful illustrations accompany the numerical demonstrations of the MBP-preserving properties of these particular ETD methods. Numerical integration is necessary in a great many applications. However, in some situations, estimates can be obtained only via approaches from the Monte Carlo family. The article “A Strong Law of Large Numbers for Scrambled Net Integration," by Art Owen and Daniel Rudolf, focuses on answering an open question regarding existence of a strong law of large numbers (SLLN) when employing a specific sampling procedure together with the randomized quasi-Monte Carlo (RQMC) for the integration. The sparsity plots in the first figure of the paper give the reader insight into the differences among the sampling in quasi-Monte Carlo (QMC) and RQMC---in QMC, one uses sample points determined (e.g., by digital nets) to cover the space more evenly, while in RQMC the QMC points are scrambled, a step that involves a random permutation in the generation of the sample points. The motivation that the authors give in the introduction for studying the existence of the SLLN in this context is from Bayesian optimization. Specifically, they note that consistent estimation of the optimal parameter can be proved assuming an SLLN for some sample values. Until now, such a tool was missing for RQMC. The authors deliver what is promised in their title, not just once, but twice: the first result is proved assuming a square integrable integrand and geometrically spaced sample sizes, while the second is shown under a relaxation of those assumptions.

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SIAM 评论，第 63 卷，第 2 期，第 315-315 页，2021 年 1 月。
在什么条件下半线性抛物线方程具有最大界原理，并且它的数值近似是否保留了该原理？这些是作者 Qqiang Du、Lili Ju、Xiao Li 和 Zhonghua Qiao 在他们的论文“Maximum Bound Principles for a Class of Semilinear Parabolic equations and Exponential Time-Differencing Schemes”中解决的主要问题。 ) 如作者所述，如果初始数据和/或边界值在绝对值中逐点有界，则解的绝对值也处处和所有时间都有界。所采用的激励示例之一是 Allen -- 已知 MBP 成立的 Cahn 方程。要将这些结果推广到更广泛的问题类别，作者采用了模型演化方程的抽象形式，这个激励示例和许多其他示例都适用于该方程。使用他们的公式，他们分析了模型方程的线性和非线性算子需要什么才能满足 MBP。可以根据这个新框架评估的各种问题自始至终都是显而易见的。此外，作者考虑了将无条件保留离散 MBP 的一阶和二阶指数时间衰减 (ETD) 方案。彩色插图伴随着这些特定 ETD 方法的 MBP 保留特性的数值演示。许多应用中都需要数值积分。然而，在某些情况下，估计只能通过蒙特卡罗家族的方法获得。具体来说，他们指出，假设某些样本值的 SLLN，可以证明对最佳参数的一致估计。直到现在，RQMC 还缺少这样的工具。作者兑现了他们标题中所承诺的，不是一次，而是两次：第一个结果被证明是假设平方可积被积函数和几何间隔的样本大小，而第二个结果是在这些假设放宽的情况下显示的。