Problems and Techniques
Authors: Joe Flaherty
Venue: SIAM Review (2005)

This issue's Problems and Techniques section contains three papers that treat areas that are simultaneously diverse and connected. The first paper considers a favorite topic in linear algebra---Krylov subspace methods, which made the widely cited "Top 10 Algorithms" (of the twentieth century) list published by Computing in Science and Engineering in January 2000. Thousands of papers have been written about these iterative techniques for solving large linear systems Ax = b. The issue considered by V. Simoncini and D. Szyld is the "superlinear convergence" of Krylov subspace methods, where the term means linear convergence that accelerates as the iterations proceed. The authors propose an analytic model of superlinear convergence derived from proximity of an invariant subspace to a crucial subspace in the algorithm. The paper contains a clear summary of previous work on analytic models of superlinear convergence; it then analyzes the convergence of specific methods, including GMRES, the conjugate gradient method, block Krylov subspace methods, and inexact Krylov subspace methods, accompanied in each case by illuminating numerical results. A particularly interesting result for inexact methods involves the effects of using invariant subspaces of the implicitly perturbed matrix rather than the original matrix. The second and third papers involve problems that might be seen as variations on a theme---the collision sequences generated by beads on a frictionless loop, and the stationary density for the traffic flow from random walks on a circle where passing is forbidden. (A visual similarity is evident in the first figures in both papers.) The second paper, by B. Cooley and P. K. Newton, studies beads on a ring. The authors note that the number of collisions is bounded for a finite number of particles colliding elastically on a line, and that several results are known for elastic particles moving in space. But on a ring there are usually an infinite number of collisions, and thus the emphasis shifts to the long-term properties of the dynamics, which strongly depend on assumptions about the nature of the collisions. The paper examines basic properties of the general problem in terms of matrix products, explaining in addition the relationship between a collision sequence and a billiard trajectory in a right-angle table with nonstandard reflection rules. (Figure 9 summarizes computational results that express the limiting results of this analysis.) The authors present a clear and self-contained exposition of a deceptively simple problem with extensions to questions about the connection between randomness and chaotic deterministic systems. Our third paper, by J. D. Skufca, starts by noting the similarities among the slowdown of cars in dense traffic, the enforced single-file motion of protein-controlled diffusion through a membrane, and the seemingly inevitable delays in serving oneself at a buffet dinner. The author then turns to the underlying general problem---analyzing the single-file flow of particles on a closed loop when the particles are unable to pass one another. Beginning with a Markov chain description transforming the problem to that of a generalized random walk on a bounded hyperplane, the author constructs a digraph representation that allows transition probabilities to be obtained via a shortest path determination. The ultimate goal of this analysis is to determine the efficiency of the system, measured by the expected fraction of the number of time steps when an individual worker is blocked (unable to move) because of the presence of another worker. The paper's results are derived based on a mixture of probability, statistics, graph theory, and geometry. We hope that SIAM Review readers will learn something new from the insights of these papers into several fields of applied mathematics.