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2011

Volume 53, Issue 4, pp. 605-823

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 53, pp. 605-605 (1 page)

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The two articles in this issue's Surveys and Review section could not be more different. One looks backward, and the other, forward. The paper “John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis,” by Joseph Grcar, provides a historical account of von Neumann's and Goldstine's analysis of Gaussian elimination and argues that one can trace the roots of numerical analysis, as practiced today, to this important paper. There is a wealth of biographical material on both von Neumann and Goldstine, which is a pleasure to read. The article then dives into the contributions of these figures on error analysis of Gaussian elimination. The presentation is clear, using modern notation with which readers will be familiar, and provides convincing evidence of the importance of their contributions.
The goal of molecular dynamic simulation is to employ computational techniques to predict the properties of materials from knowledge of their molecular structure. The idea is simple enough— given a model for the forces between the atoms and a model for how they interact with their surroundings, one can use Newton's Second Law to derive the equations of motion of each atom and compute away.
To actually do the calculation is not so simple. There are plenty of computational challenges to overcome in order to be able to complete the calculation in a reasonable amount of time. For average protein molecules, the number of pairwise interactions can make a direct calculation of the interactions computationally prohibitive. Therefore, approximation schemes are necessary to make the calculation tractable on today's machines. The game is to come up with approximation schemes that trade off accuracy for computational efficiency.
The second paper, “Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations,” by Xu and Cai, is a review of approaches to developing efficient methods in molecular dynamics simulation. The focus of the article is on solvation, where the molecule is immersed in an environment with which it reacts. Solvation presents specific challenges. The paper goes over ways in which approximations lead to fast algorithms. The article provides a view into the world of biomolecular simulations and exposes where analysis has made a difference and where further developments are needed.

John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis

Joseph F. Grcar

SIAM Rev. 53, pp. 607-682 (76 pages)

Online Publication Date: November 07, 2011

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Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination—but their paper was about more than that. Von Neumann and Goldstine described what they surmized would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them.

Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations

Zhenli Xu and Wei Cai

SIAM Rev. 53, pp. 683-720 (38 pages)

Online Publication Date: November 07, 2011

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We review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications, including the Poisson–Boltzmann (PB) and the generalized Born models for electrostatic solvation energy. The focus is on analytical approaches for hybrid solvation models, especially the image charge method for a spherical cavity, and also the generalized Born theory as an approximation to the PB model. This review places much emphasis on the mathematical details behind these methods.
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Expository Research Papers

Ilse Ipsen, Section Editor

SIAM Rev. 53, pp. 721-721 (1 page)

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You may be concerned that this issue features just a single paper in the Expository Research section. But not to worry—you will get your money's worth. Based on foundations laid by the eminent Austrian-born algebraist Emil Artin (1898–1962) you will be acquainted with improved designs for bread mixers and taffy pullers. Ingredients include topology, dynamical systems, linear algebra, the golden ratio, and Lego toys.
The applications considered by Matthew Finn and Jean-Luc Thiffeault in their paper “Topological Optimization of Rod-Stirring Devices” go far beyond food preparation and extend to industrial dough production, and to glass manufacturing where inhomogeneities in molten glass are removed by stirring it with rods. The subject of this paper is the design of efficient devices that stir a fluid thoroughly.
When the fluid motion is mainly two-dimensional, as in the case of glass, for example, one can model the fluid as a two-dimensional surface. Instead of using the Stokes equations to describe the fluid motion, the authors adopt a topological approach. They view the fluid as a punctured disk (the punctures being the stirring rods), and the rod motions as mappings of this disk. The mappings associated with efficient practical stirring protocols belong to the so-called pseudo-Anosov category and produce a fluid motion that is related to chaotic dynamical systems. Implementing a pseudo-Anosov mapping requires at least three stirring rods, which is why many taffy pullers have three stirring rods.
The authors choose the topological notion of braids to describe the motion protocol of the stirring rods. Mathematical relations derived by Emil Artin identify those groups of braids that can actually be physically realized. In order to estimate how thoroughly a stirring protocol mixes up the fluid, one represents the braids as products of $2\times 2$ matrices, and then determines the spectral radius of the product. For three rods the best stirring protocol has a spectral radius equal to the golden ratio.
One must be careful, though, to balance mathematical tractability with practical engineering considerations. Stirring protocols that are optimal from a mathematical point of view are not necessarily so in practice. Part of the reason is that the mathematical approach corresponds to a sequential operation of the stirring rods, while in practice one would want them to work in parallel. The authors devise a parallel motion protocol for stirring rods and present evidence for the optimality of this parallel protocol.
This is a marvelous paper, easy to read, accessible, and most enjoyable. To top everything off, the authors built optimal stirring devices with 2 and with 4 stirring rods from Lego toy pieces. How cool is that?

Topological Optimization of Rod-Stirring Devices

Matthew D. Finn and Jean-Luc Thiffeault

SIAM Rev. 53, pp. 723-743 (21 pages)

Online Publication Date: November 07, 2011

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There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly knead a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of material lines is conveniently given by the topological entropy of the rod motion. We discuss the problem of optimizing such rod devices from a topological viewpoint. We express rod motions in terms of generators of the braid group and assign a cost based on the minimum number of generators needed to write the braid. We show that for one cost function—the topological entropy per generator—the optimal growth rate is the logarithm of the golden ratio. For a more realistic cost function, involving the topological entropy per operation where rods are allowed to move together, the optimal growth rate is the logarithm of the silver ratio, $1+\sqrt{2}$. We show how to construct devices that realize this optimal growth, which we call silver mixers.
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SIGEST

The Editors

SIAM Rev. 53, pp. 745-745 (1 page)

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The distributed consensus problem is to develop a distributed algorithm that a set of autonomous agents, for example, unmanned aircraft, can use to agree on a course of action, such as direction of motion. In the context of this issue's SIGEST paper, a distributed consensus algorithm is an iterative method in which, at every step, each agent updates its local data by taking a weighted average of the local data from some of the other agents. The mathematical task is to understand the convergence properties of the algorithms. You can see consensus algorithms in nature as well, for example, in flocks of birds.
“Convergence Speed in Distributed Consensus and Averaging” by Alex Olshevsky and John N. Tsitsiklis, which appeared in the SIAM Journal on Control and Optimization, considers a special case of the distributed averaging problem. In this problem each agent owns local scalar data, and the objective is to generate a global scalar which is the average of the local values.
An example from the paper, one of several in the very well written introduction, is a wireless multihop network. In these networks the nodes may be in different countries and are not under any centralized control. Some nodes may attempt selfish behavior, for example, not forwarding traffic meant for other nodes. Distributed averaging is one way to enforce cooperation by using reputation management to detect selfish behavior. Suppose each node observes its neighbors and forms a local opinion. We can then use distributed averaging to merge the local opinions into a single reputation measure for each node.
The authors develop several new algorithms and derive worst-case bounds for the convergence times of the new algorithms. They also describe a new polynomial time algorithm ($O(n^3)$ operations, where $n$ is the number of nodes in the network) for time-varying network topologies.
The paper is very accessible to anyone who has some knowledge of graph theory and Markov chains.

Convergence Speed in Distributed Consensus and Averaging

Alex Olshevsky and John N. Tsitsiklis

SIAM Rev. 53, pp. 747-772 (26 pages)

Online Publication Date: November 07, 2011

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We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.
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Education

Louis F. Rossi, Section Editor

SIAM Rev. 53, pp. 773-773 (1 page)

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The versatility of mathematics makes it indispensable when attempting to understand what cannot be directly measured. In this issue's article, “Lin & Segel's Standing Gradient Problem Revisited: A Lesson in Mathematical Modeling and Asymptotics,” author Stephen O'Brien gives us a fresh look at the standing gradient problem as originally discussed in Lin and Segel's classic Mathematics Applied to Deterministic Problems in the Natural Sciences. In physiology, a standing gradient is a condition where the solute concentration in a solution surrounding a secreting tissue is greater than the concentration within the tissue itself. The experimental observation is simple enough, but after ruling out chemical pumping action, the physical processes that create the standing gradient remain elusive. At the time, one hypothesis was that infoldings in the tissue, essentially small closed tubes that extend into the bathing fluid, might facilitate the standing gradient. However, the precise interactions between convection, diffusion, and osmosis in and around the tubes are complex. Only mathematical modeling and analysis would demonstrate that the infoldings could create and sustain the standing gradient. This article lays out the basic mathematical model for these processes and systematically carries the reader through the classic stages of modeling: formulation, nondimensionalization, simplification, and finally analysis of solutions. In this case, the latter is accomplished through matched asymptotic expansions.
In doing so, the article highlights the importance of assumptions in modeling activities. Arguably there are two categories of assumption of central interest. First, there are those that are necessary because they lead to a well-posed problem. Typically, these assumptions involve a precise description of the mechanisms in the process and will lead to a quantification of the problem. Second, there are those that are convenient because they transform the problem into one that can be solved using available techniques. These types of assumptions creep into the final stages either for sound physical reasons or as an act of desperation to squeeze something out of a hard problem. For the student, these assumptions are often introduced early in the discussion as what one of my graduate professors called “hind-foresight.” In this case, the author takes a fresh look at the standing gradient problem by relaxing one of Lin and Segel's original assumptions that would fall into this second category. In their classic treatment, Lin and Segel begin with two small parameters to generate a leading order expression. One of these is the ratio of the thickness of the tip of the tube to the length of the tube, $\gamma$, which Lin and Segel assume is always small. This article relaxes that assumption to generate two uniformly valid solutions for the cases when $\gamma = O(1)$ and $\gamma \ll 1$.
The paper is a pleasure to read, both for the mathematical discussion and historical perspective it provides. O'Brien's thorough treatment of the standing gradient problem is self-contained and would work well as a module in either a modeling course or an asymptotics course offered for advanced undergraduates or graduate students. The model problem tucked into the appendix would serve as a nice exercise for further exploration. As a bonus, the author paints a vivid portrait of the response to Lin and Segel's book and its impact on the applied mathematics community. O'Brien draws connections between the publication of the book, the growth and proliferation of Oxford-style industry study groups, and the infiltration of sophisticated mathematics into diverse fields such as heat transfer, aerodynamics, and, more recently, finance and emerging areas in the life sciences. Even outside its instructional uses, the article draws us in to reflect upon an interesting problem and the evolution of modeling as a discipline.
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Lin & Segel's Standing Gradient Problem Revisited: A Lesson in Mathematical Modeling and Asymptotics

S. B. G. O'Brien

SIAM Rev. 53, pp. 775-796 (22 pages)

Online Publication Date: November 07, 2011

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We revisit a physiological standing gradient problem of Lin and Segel from their landmark text on mathematical modeling [C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988] with a view to giving it an up-to-date perspective. In particular, via an alternative nondimensionalization, we show that the problem can be analyzed using the tools of singular perturbation theory and matched asymptotic expansions. In the spirit of the aforementioned authors, the development is didactic in style. Solving the problem requires many of the necessary skills of continuous modern mathematical modeling: formulation from a physical description of the process, scaling and asymptotic simplification, and solution using advanced perturbation (boundary layer) techniques.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 53, pp. 799-823 (25 pages)

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Readers will be lucky to find two featured reviews in this issue, praising their subjects. One, by Rabi Bhattacharya, is about Fischer's History of the Central Limit Theorem. The other, by Alfio Quarteroni, reviews Huang and Russell's Adaptive Moving Mesh Methods. The history of the CLT involves lots of prominent mathematicians over a period of two hundred years, mostly before asymptotics became ordinary. The other is central to current efforts to computationally solve practical problems involving partial differential equations using adaptive meshes.
Other reviews are about computer arithmetic, infinite-dimensional dynamical systems, chaos, population dynamics, and other topics. They illustrate the continuing breadth and vitality of current publications in applied math, in our increasingly electronic environment. We hope you enjoy reading them.
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