By Evelyn Sander
The latest SIAM Review's Research Spotlights section contains two papers. The first article, “Fast Randomized Iteration: Diffusion Monte Carlo through the Lens of Numerical Linear Algebra,” by Lek-Heng Lim and Jonathan Weare, introduces a family of randomized iterative methods for solving linear systems, matrix exponentiation, and eigenvalue problems. The advantage of these new methods is that they have a quite small number of operations and storage cost per iteration, and can solve problems in dimensions too high to be tractable by traditional linear algebra methods. The specific focus here is applying these methods to solving linear systems and eigenvalue problems in cases where the matrices and solution vectors are too big to manipulate or even to store. The approach uses a compression operation similar to that which is used in data assimilation. The paper gives a general overview and a useful set of references to related works, followed by stability and convergence results. It ends by presenting three concrete numerical examples, namely, computation of a partition function for the two-dimensional Ising model, a spectral gap in diffusion process with up to ten spatial dimensions, and the computation of a free energy landscape for that process. This article will be of broad interest, especially for those interested in partial differential equations and data analysis for large systems, and those familiar with diffusion Monte Carlo methods who would like to be introduced to applications beyond Markov processes.
The second article, “Fracking, Renewables, and Mean Field Games,” by Patrick Chan and Ronnie Sircar, uses mean field game models to study the competition between traditional oil producers and producers of renewable energy. It gives an overview introduction to the application, from the OPEC price setting and view of fracking to the viewpoint of competitive oligopolies. Mean field games were first introduced in the early 2000s. The idea is that by a systematic application of a continuum approximation, it is possible to get analytical solutions and straightforward computational answers that would be much more difficult in the study of the \(N\)-player system. The paper includes the details of setting the energy market competition problem in this framework and illustrates how to use this method in a concrete numerical example. For those who study systems with large numbers of players or agents, this paper is a nice walk-through of the relatively new analytical technique of mean field games.
Evelyn Sander is a professor of mathematics at George Mason University. She is the section of editor of the Research Spotlights section of SIAM Review.