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2006

Volume 48, Issue 4, pp. 627-821

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 48, pp. 627-627 (1 page)

Online Publication Date: November 02, 2006

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The laser, first invented in 1960, has become such a commonplace device that we rarely recall that “laser” is an acronym, standing for Light Amplification by Simulated Emission of Radiation. Lasers show up in everyday consumer electronics, such as CD players, DVD players, printers, and pointers, and are also used in such important applications as telecommunications, manufacturing, and medicine. Most of these lasers are of the continuous wave (CW) type, based on the original invention.
This issue’s Survey and Review article by J. Nathan Kutz, however, is about a different kind of laser: one designed to produce short pulses of high peak concentrated in a short burst. LIDAR (Light Detection and Ranging), an optical equivalent of radar and an important tool in atmospheric measurements, uses pulsed lasers. Pulsed lasers are expected to be a key element in the next generation of optical networks which transmits soliton pulses. Soliton‐based optical communication systems are attractive because they are capable of transmitting data across distances of over ten thousand miles without the need for repeaters.
Mode‐locked soliton lasers, described in this paper, are a type of pulsed lasers. The paper starts with a comprehensive overview of optical‐fiber–based laser models. These models are governed by nonlinear partial differential equations. The focus is on the formation of solitons, which involves the interplay between nonlinearity and dispersion. The article further demonstrates that the study of mode‐locked soliton lasers provides a rich area for mathematical research as it brings together ideas from nonlinear dynamics, multiple time scale analysis, and stability of nonlinear evolution equations.

Mode‐Locked Soliton Lasers

J. Nathan Kutz

SIAM Rev. 48, pp. 629-678 (50 pages) | Cited 47 times

Online Publication Date: November 02, 2006

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A comprehensive treatment is given for the formation of mode‐locked soliton pulses in optical fiber and solid state lasers. The pulse dynamics is dominated by the interaction of the cubic Kerr nonlinearity and chromatic dispersion with an intensity‐dependent perturbation provided by the mode‐locking element in the laser cavity. The intensity‐dependent perturbation preferentially attenuates low intensity electromagnetic radiation which makes the mode‐locked pulses attractors of the laser cavity. A review of the broad spectrum of mode‐locked laser models, both qualitative and quantitative, is considered with the basic pulse formation phenomena highlighted. The strengths and weaknesses of each model are considered with two key instabilities studied in detail: Q‐switching and harmonic mode‐locking. Although the numerous mode‐locking models are considerably different, they are unified by the fact that stable solitons are exhibited in each case due to the intensity discrimination perturbation in the laser cavity.
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Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 48, pp. 679-680 (2 pages)

Online Publication Date: November 02, 2006

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Readers may be surprised to learn that there is something on which everybody in my department agrees: the heating system is absolutely ineffective. Some offices get too hot too fast (usually inhabitated by some who like it cold), while other offices (like mine, unfortunately) take forever to become even slightly warm. The building is heated by a network of pipes through which hot steam is circulated. Question: How to design a heating system that delivers the same amount of steam to each office as fast as possible, thereby ensuring that all offices have the same temperature as fast as possible? Answer: Construct a specific matrix, called a weighted Laplacian, and choose the weights so as to maximize its second largest eigenvalue. The weights tell us how wide each pipe needs to be. Of course, this is a constrained optimization problem, because we are on a budget and can afford only a limited amount of material for the pipes.
A more general problem is discussed by Jun Sun, Stephen Boyd, Lin Xiao, and Persi Diaconis in their paper “The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem.” They express the problem of maximizing the second largest eigenvalue of the Laplacian as a semidefinite program. The dual of this program has a simple geometric interpretation: It’s the problem of positioning $n$ points in $n$‐space, so that they are as far apart as possible, but do not exceed prescribed distances between any two points. Coming back now to the heating issues in my department, we have been promised a new building. Ground breaking is to start anytime now (or so, at least, we are told). I have been thinking about giving this paper to the architects; it might inspire them to install a more effective heating system.
In the well‐written paper “Globalization Techniques for Newton–Krylov Methods and Applications to the Fully Coupled Solution of the Navier–Stokes Equations,” Roger Pawlowski, John Shadid, Joseph Simonis, and Homer Walker discuss methods for the solution of systems of nonlinear equations $F(u)=0$. Such systems arise, for instance, when one discretizes partial differential equations to solve fluid flow problems. Arguably the most popular method for solving $F(u)=0$ is Newton’s method. It starts from an initial approximation $u_0$ and produces successively better (we hope) iterates $u_{k+1}$ as updates of the previous iterate, $u_{k+1}=u_k+s_k$. The step $s_k$ is computed as the solution to the linear system $F^{\prime}(u_k)s_k=-F(u_k)$, where the Jacobian $F^{\prime}(u)$ is the matrix of derivatives. When the linear system is solved by a Krylov space method, for instance, one talks about a Newton–Krylov method.
Convincing Newton’s method to converge to the solution is not always easy, especially when the initial approximation $u_0$ is far away. A variety of strategies is available that can enhance the performance of Newton’s method. The authors discuss two. To increase robustness, one can solve the linear systems more or less accurately; this is done by terminating the linear system solution as soon as the residual norm $\|F^{\prime}(u_k)+F(u_k)s_k\|$ falls below a specified forcing term. To improve the chances for convergence, one can globalize Newton’s method by changing the length of the step $s_k$ (as opposed to its direction), or by choosing a step $s_k$ that minimizes the residual norm over a particular region. The authors prove convergence results, and perform numerical experiments on standard benchmark problems to compare different forcing terms and globalization strategies.
Are you one of those people who firmly believes that there is one and only one way to win a tennis match? And that’s by subjecting your opponent to that impossible‐to‐return 700‐horse‐power serve? Yes? Then we might have just the paper for you.
In “Monte Carlo Tennis,” Paul Newton and Kamran Aslam analyze the probability of winning in tennis, and express it in terms of the probability that a player wins a point when serving. In previous work, Paul Newton and coauthor Joe Keller had assumed that this probability is constant—throughout the whole match, and even a tournament. This amounts to assuming that points in tennis are random variables, independently and identically distributed (i.i.d.). However, this assumption fails to account for the “hot‐hand,” when everything goes just swimmingly; the “back‐to‐the‐wall” effect, when miraculous feats become possible in the face of looming loss; or simply the adjustment to new tennis balls. Do these things really make a difference? Is the i.i.d. assumption unrealistic? Paul Newton and Kamran Asham perform Monte Carlo simulations in MATLAB to answer this question. Read the paper if you want to know what they come up with.

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

Jun Sun, Stephen Boyd, Lin Xiao, and Persi Diaconis

SIAM Rev. 48, pp. 681-699 (19 pages) | Cited 15 times

Online Publication Date: November 02, 2006

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We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue $\lambda_2$ of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize $\lambda_2$ subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, , the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding” high‐dimensional data that lies on a low‐dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

Globalization Techniques for Newton–Krylov Methods and Applications to the Fully Coupled Solution of the Navier–Stokes Equations

Roger P. Pawlowski, John N. Shadid, Joseph P. Simonis, and Homer F. Walker

SIAM Rev. 48, pp. 700-721 (22 pages) | Cited 12 times

Online Publication Date: November 02, 2006

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A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually globalized, i.e., augmented with auxiliary procedures (globalizations) that improve the likelihood of convergence from a starting point that is not near a solution. In recent years, globalized Newton–Krylov methods have been used increasingly for the fully coupled solution of large‐scale problems. In this paper, we review several representative globalizations, discuss their properties, and report on a numerical study aimed at evaluating their relative merits on large‐scale two‐ and three‐dimensional problems involving the steady‐state Navier–Stokes equations.

Monte Carlo Tennis

Paul K. Newton and Kamran Aslam

SIAM Rev. 48, pp. 722-742 (21 pages) | Cited 2 times

Online Publication Date: November 02, 2006

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The probability of winning a game, set, match, or single elimination tournament in tennis is computed using Monte Carlo simulations based on each player’s probability of winning a point on serve, which can be held constant or varied from point to point, game to game, or match to match. The theory, described in Newton and Keller [Stud. Appl. Math., 114 (2005), pp. 241–269], is based on the assumption that points in tennis are independent, identically distributed (i.i.d.) random variables. This is used as a baseline to compare with the simulations, which under similar circumstances are shown to converge quickly to the analytical curves in accordance with the weak law of large numbers. The concept of the importance of a point, game, and set to winning a match is described based on conditional probabilities and is used as a starting point to model non‐i.i.d.effects, allowing each player to vary, from point to point, his or her probability of winning on serve. Several non‐i.i.d.models are investigated, including the “hot‐hand‐effect,” in which we increase each player’s probability of winning a point on serve on the next point after a point is won. The “back‐to‐the‐wall” effect is modeled by increasing each player’s probability of winning a point on serve on the next point after a point is lost. In all cases, we find that the results provided by the theoretical curves based on the i.i.d.assumption are remarkably robust and accurate, even when relatively strong non‐i.i.d.effects are introduced. We end by showing examples of tournament predictions from the 2002 men’s and women’s U.S. Open draws based on the Monte Carlo simulations. We also describe Arrow’s impossibility theorem and discuss its relevance with regard to sports ranking systems, and we argue for the development of probability‐based ranking systems as a way to soften its consequences.
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SIGEST

SIAM Rev. 48, pp. 743-743 (1 page)

Online Publication Date: November 02, 2006

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What do phenomena as distinct as the formation of clouds and smog, the kinetics of polymerization, and the formation of schools of fish have in common? They all involve the growth of clusters, and in these cases and many others this process has been modeled mathematically by a mean‐field model of clustering dynamics referred to as Smoluchowski’s coagulation equation.
The SIGEST selection in this issue, from the SIAM Journal on Mathematical Analysis, makes an important and deep contribution to the study of these types of dynamical systems, in the case when the solutions are “self‐similar.” Historically, dynamical systems developed tools to study the existence and stability of equilibrium (steady‐state) solutions to the evolution equations governing physical systems. However, some systems do not evolve towards steady states. One alternate possibility is that systems become chaotic, forever wandering in a disordered fashion through the phase space. Another possibility is that they approach a self‐similar solution, where the shape of the solution is universal, but the size of the solution increases or decreases (usually geometrically) in time.
Once we began looking for self‐similar solutions in nature, we began to find them everywhere—in the growth of an icicle, the spreading of a liquid drop, the gravitational collapse of a gas cloud, or the coarsening of particles in a solidifying alloy. Perhaps the most familiar example is associated with the central limit theorem; if we repeat a random process, such as flipping a coin, and then look at an additive measure associated with the process, such as counting the number of heads that appear, we see that the distribution approaches a universal normal distribution; however, the width of the distribution grows as the square‐root of the number of coin tosses increases. The central limit theorem not only identifies this asymptotic state but also assures us that it is stable.
The paper “Dynamical Scaling in Smoluchowski’s Coagulation Equations: Uniform Convergence” by Govind Menon and Robert L. Pego studies a dynamical process with this self‐similar character, namely, a general model of coagulation. An example is an ocean filled with schools of fish of many different sizes; if two groups of fish cross paths, they merge to make an even larger school. Like the normal distribution in the central limit theorem, models of this process yield a universal distribution of school sizes. Depending on the nature of the interaction, sometimes this coarsening process will continue for all time and sometime after a finite amount of time the coarsening will yield “blow‐up,” which can be thought of as coagulation into one cluster of infinite size (corresponding to all the fish ending up in one school in a finite time), in which case the continuum approximation that is used ceases to be valid.
The contribution of this paper is to show that all initial conditions satisfying some simple restrictions approach these self‐similar distributions uniformly when measured in an appropriate norm. In so doing, the paper establishes rigorously a phenomenon that had been observed and believed previously, but without rigorous mathematical justification. We hope that the SIAM readership will appreciate and enjoy this important contribution.

Dynamical Scaling in Smoluchowski’s Coagulation Equations: Uniform Convergence

Govind Menon and Robert L. Pego

SIAM Rev. 48, pp. 745-768 (24 pages) | Cited 4 times

Online Publication Date: November 02, 2006

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Smoluchowski’s coagulation equation is a fundamental mean‐field model of clustering dynamics. We consider the approach to self‐similarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels $K(x,y)=2,x+y$, and $xy$. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a self‐similar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.
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Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 48, pp. 769-769 (1 page)

Online Publication Date: November 02, 2006

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Violin strings and rubber bands, jump ropes and suspensions bridges, whips and chains; amazingly all these objects can be approximated as one‐dimensional elastic materials. However, the unity of the underlying physical model is lost when students are introduced to the subject via the wave equation as a small amplitude model of a vibrating string. While this classical derivation is essentially correct, it obscures the fact that the approximation being made is really that the string is perfectly elastic (i.e., its tension is proportional to its length), not that the oscillations are of small amplitude.
Fortunately, the article in this issue’s Education section, “Strings, Chains, and Ropes” by Darryl Yong, provides a compact and accessible derivation of the equations for an elastic medium. An honest derivation of the wave equation follows easily, and we also get a menagerie of other examples. For example, the oscillation modes of a swinging chain are related to an eigenfunction problem with a solution in Bessel functions. The resulting classroom demonstration is engaging and easily executed.
For those of you teaching an introductory PDE course, I urge you to consider incorporating this derivation as an introduction to the wave equation. You can also demonstrate that Bessel functions have applications beyond cylindrically coordinates (where students usually first encounter them). The article is written at a level accessible to undergraduates and suggests a number of starting points for projects both theoretical and experimental.
And, as an added bonus, you now have an excuse to bring your whips and chains into the classroom....

Strings, Chains, and Ropes

Darryl Yong

SIAM Rev. 48, pp. 771-781 (11 pages) | Cited 4 times

Online Publication Date: November 02, 2006

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Following Antman [Amer. Math. Mon., 87 (1980), pp. 359–370], we advocate a more physically realistic and systematic derivation of the wave equation suitable for a typical undergraduate course in partial differential equations. To demonstrate the utility of this derivation, three applications that follow naturally are described: strings, hanging chains, and jump ropes.
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Book Reviews

Bob O’Malley, Section Editor

SIAM Rev. 48, pp. 785-821 (37 pages)

Online Publication Date: November 02, 2006

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This issue’s featured review by Anne Greenbaum is about Trefethen and Embree’s Princeton monograph Spectra and Pseudospectra. It’ll convince you that your old dose of spectral theory is insufficient. You’ll have to learn more to understand transient effects and nonnormal dynamics. The issue also includes well‐argued opinions about twenty‐two other books, covering a variety of mathematical topics.
As usual, we thank our reviewers for giving us the benefit of their expertise.
I also want to thank Frances Chen, who has been efficiently and patiently organizing these reviews since (late in) the last century! Crystal Peterson, who has been soliciting reviews and mailing review copies for over a year, is now taking over Frances’ duties as well. If you look forward to reading these reviews, like I do, you’ve got Frances and Crystal to thank.
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