SIAM Review, Volume 65, Issue 4, Page 1135-1135, November 2023.
In this issue the Education section presents three contributions. The first paper “The Reflection Method for the Numerical Solution of Linear Systems,” by Margherita Guida and Carlo Sbordone, discusses the celebrated Gianfranco Cimmino reflection algorithm for the numerical solution of linear systems $Ax=b$, where $A$ is a nonsingular $n \times n$ sparse matrix, $b \in \mathbb{R}^n$, and $n$ may be large. This innovative iterative algorithm proposed in 1938 uses the geometric reading of each equation of the system as a hyperplane to compute an average of all the symmetric reflections of an initial point $x^0$ with respect to hyperplanes. This leads to a new point $x^1$ which is closer to the solution. The iterative method constructs a sequence $x^k \in \mathbb{R}^n$ converging to the unique intersection of hyperplanes. To overcome the algorithm's efficiency issues, in 1965 Cimmino upgraded his method by introducing probabilistic arguments also discussed in this article. The method is different from widely used direct methods. Since the early 1980s, there has been increasing interest in Cimmino's method that has shown to work well in parallel computing, in particular for applications in the area of image reconstruction via X-ray tomography. Cimmino's algorithm could be an interesting subject to be deepened by students in a course on scientific computing. The second paper, “Incorporating Computational Challenges into a Multidisciplinary Course on Stochastic Processes,” is presented by Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, and Alexander J. Stewart. The authors describe their graduate-level introductory stochastic modeling course in biology for a mixed audience of mathematicians and biologists whose goal was teaching students to formulate, implement, and assess nontrivial biomathematical models and to develop research skills. This problem-based learning was addressed by proposing several computational modeling challenges based on real life applied problems; by assigning tasks to groups formed by four students, where necessarily participants had different levels of knowledge in programming, mathematics, and biology; and by creating retrospective discussion sessions. In this way the stochastic modeling was introduced using a variety of examples involving, for instance, biochemical reaction networks, gene regulatory systems, neuronal networks, models of epidemics, stochastic games, and agent-based models. As supplementary material, a detailed syllabus, homework, and the text of all computational challenges, along with code for the discussed examples, are provided. The third paper, “Hysteresis and Stability,” by Amenda N. Chow, Kirsten A. Morris, and Gina F. Rabbah, describes the phenomenon of hysteresis in some ordinary differential equations motivated by applications in a way that can be integrated into an introductory course of dynamical systems for undergraduate students. The considered ODEs involve a time dependent parameter, and looping behavior is illustrated with figures. These low-dimensional examples can be used to construct student exercises. There are many citations of related literature inviting the readers to go beyond. A discussion of possible extensions, including hysteresis in PDEs, concludes the article.

中文翻译：

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教育

《SIAM 评论》，第 65 卷，第 4 期，第 1135-1135 页，2023 年 11 月。
在本期中，教育部分提出了三项贡献。第一篇论文“线性系统数值解的反射方法”由 Margherita Guida 和 Carlo Sbordone 撰写，讨论了著名的 Gianfranco Cimmino 反射算法，用于线性系统数值解 $Ax=b$，其中 $A$ 是非奇异数$n \times n$ 稀疏矩阵，$b \in \mathbb{R}^n$，并且 $n$ 可能很大。这种于 1938 年提出的创新迭代算法使用系统每个方程的几何读数作为超平面来计算初始点 $x^0$ 相对于超平面的所有对称反射的平均值。这导致一个新点 $x^1$ 更接近解决方案。迭代方法构造一个序列 $x^k \in \mathbb{R}^n$ 收敛于超平面的唯一交集。为了克服算法的效率问题，Cimmino 在 1965 年通过引入概率论来升级他的方法，本文也对此进行了讨论。该方法不同于广泛使用的直接方法。自 20 世纪 80 年代初以来，人们对 Cimmino 的方法越来越感兴趣，该方法在并行计算中表现良好，特别是在通过 X 射线断层扫描进行图像重建领域的应用中。西米诺的算法可能是一个有趣的主题，学生可以在科学计算课程中加深理解。第二篇论文“将计算挑战纳入随机过程多学科课程”由 Mark Jayson Cortez、Alan Eric Akil、Krešimir Josić 和 Alexander J. Stewart 发表。作者描述了他们面向数学家和生物学家的研究生水平随机建模入门课程，其目标是教学生制定、实施和评估重要的生物数学模型并培养研究技能。这种基于问题的学习是通过提出一些基于现实生活应用问题的计算建模挑战来解决的；将任务分配给由四名学生组成的小组，其中参与者必然具有不同水平的编程、数学和生物学知识；并通过创建回顾性讨论会议。通过这种方式，使用各种示例引入了随机建模，例如涉及生化反应网络、基因调控系统、神经元网络、流行病模型、随机博弈和基于代理的模型。作为补充材料，提供了详细的教学大纲、作业和所有计算挑战的文本，以及所讨论示例的代码。第三篇论文“磁滞与稳定性”，由 Amenda N. Chow、Kirsten A. Morris 和 Gina F. Rabbah 撰写，描述了一些常微分方程中的磁滞现象，这些现象是由应用程序驱动的，可以集成到介绍性的文章中。本科生动力系统课程。所考虑的 ODE 涉及与时间相关的参数，并且循环行为用图来说明。这些低维示例可用于构建学生练习。书中引用了很多相关文献，欢迎读者进一步探讨。本文最后讨论了可能的扩展，包括偏微分方程中的滞后现象。