Education

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.