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2005

Volume 47, Issue 4, pp. 627-858


Survey and Review

Michael Overton, Associate Editor

SIAM Rev. 47, pp. 627-627 (1 page)

Online Publication Date: August 04, 2006

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Graph coloring is an old idea, bringing to mind such celebrated results as the four-color theorem and NP-hardness of the chromatic number of a graph, but not, for most people, the efficient approximation of derivatives. Yet during recent years graph coloring has emerged as a highly effective tool for approximating derivatives, whether Jacobians (matrices of first derivatives of a vector-valued function) or Hessians (matrices of second derivatives of a scalar-valued function). The main issue is how best to exploit the sparsity and symmetry in these matrices to compute them with the least work. Remarkably, the techniques for exploiting sparsity are essentially the same whether the derivatives are approximated by the very old technique of finite differences or computed exactly by the comparatively recent method of automatic differentiation. In both cases, the columns or rows (or both) of a derivative matrix must be partitioned into a small number of groups in order to reduce the number of function evaluations required. It is in formulating, analyzing, and designing algorithms for such problems that graph coloring has proved to be a powerful tool. Indeed, modern software for computing large, sparse Jacobians and Hessians relies on graph coloring algorithms to make the computations feasible.
In "What Color Is Your Jacobian?~Graph Coloring for Computing Derivatives," Gebremedhin, Manne, and Pothen give a comprehensive survey of the work in this area during the past 25 years, presenting a unifying framework, several new algorithms, substantial computational results, and a survey of relevant theoretical results.

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Assefaw Hadish Gebremedhin, Fredrik Manne, and Alex Pothen

SIAM Rev. 47, pp. 629-705 (77 pages) | Cited 25 times

Online Publication Date: August 04, 2006

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Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrix estimation problems. The framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance-$2$ coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrix as an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.

Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 47, pp. 707-707 (1 page)

Online Publication Date: August 04, 2006

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Suppose you are a bicycle messenger in the busy business district of a large city. Here is what your day looks like: You are to deliver incoming mail to building i, and you are to pick up outgoing mail destined only for building i + 1. All in all you are responsible for m buildings. Your bike, however, is not of the highest quality: It can accommodate only mail for a single delivery. Riding from building i to building j requires tij minutes. Once you arrive at a building, you need d minutes to get off your bike, run into the building, and get onto your bike again. Moreover, once incoming mail has arrived at building i, it takes pi minutes for the outgoing mail to be ready (if this is too long, you may want to make another delivery and pickup in the meantime). You are to visit each building k times.
Question: In which order should you visit the buildings so that you finish your work as fast as possible?
The above description is a simplified version of the problem discussed in the paper by Milind W. Dawande, H. Neil Geismar, and Suresh P. Sethi. There the messenger is a robot, the buildings are machines, and the mail represents parts to be processed by the machines. The authors show that there exists a cyclic schedule that maximizes long-term throughput. Cyclic schedules are preferred in industrial environments, because they are easy to implement and control. Since the literature on robotic cell scheduling is full of different models for different kinds of industrial applications, it is important to know that all optimal schedules can be reduced to cyclic schedules. The authors end their paper by describing several challenging open problems.
The paper by Miguel Torres-Torriti and Hannah Michalska describes a software package (LTP), implemented in Maple, for the symbolic manipulation of expressions that occur in the context of Lie algebra theory. This theory has found applications in classical and quantum mechanics, analysis of dynamical systems, construction of nonlinear filters, and the design of feedback control laws for nonlinear systems. Since the symbolic computations are often complex and tedious, the development of software for applications of Lie algebra theory is crucial. The LTP software package is targeted at applications such as solution of differential equations evolving on Lie groups, and structure analysis of general dynamical systems.

Dominance of Cyclic Solutions and Challenges in the Scheduling of Robotic Cells

H. Neil Geismar, Milind W. Dawande, and Suresh P. Sethi

SIAM Rev. 47, pp. 709-721 (13 pages) | Cited 10 times

Online Publication Date: August 04, 2006

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We consider the problem of scheduling operations in bufferless robotic cells that produce identical parts. Maximizing the long-term average throughput of parts is an important problem in both theory and practice. We define an appropriate state space required to analyze this problem and show that cyclic schedules which repeat a fixed sequence of robot moves indefinitely are the only ones that need to be considered. For the different classes of robotic cells studied in the literature, we discuss the current state of knowledge with respect to cyclic schedules. Finally, we discuss the importance of two fundamental open problems concerning optimal cyclic schedules, special cases for which these problems have been solved, and attempts to solve the general case.

A Software Package for Lie Algebraic Computations

Miguel Torres-Torriti and Hannah Michalska

SIAM Rev. 47, pp. 722-745 (24 pages) | Cited 2 times

Online Publication Date: August 04, 2006

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The paper presents a computer algebra package that facilitates Lie algebraic symbolic computations required in the solution of a variety of problems, such as the solution of right-invariant differential equations evolving on Lie groups. Lie theory is a powerful tool, helpful in the analysis and design of modern nonlinear control laws, nonlinear filters, and the study of particle dynamics. The practical application of Lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool. The aim of the package is to facilitate and encourage further research relying on Lie algebraic computations.

SIGEST

SIAM Rev. 47, pp. 746-746 (1 page)

Online Publication Date: August 04, 2006

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How do fish in a mountain stream, or bacteria in our intestines, continue to survive upstream, against a current? Why aren’t they washed out, eventually leading to their extinction? This "drift paradox"—the persistence of organisms which cannot "swim" against the current, yet live upstream—is part of "spatial ecology," the study of how populations of living organisms interact with the geometry and topography of their physical environment. The SIGEST selection in this issue, from the SIAM Journal on Applied Mathematics, makes groundbreaking contributions in the area of spatial ecology, including an enhanced explanation of the drift paradox.
The selected paper is "The Effect of Dispersal Patterns on Stream Populations," by Frithjof Lutscher, Elizaveta Pachepsky, and Mark Lewis. As the title suggests, the paper analyzes the movement and persistence of populations in streams that can include single directional flows. This category covers a wide variety of applications, ranging from flora and fauna in rivers, to bacteria in the human digestive system, to fish in streams, just to name a few. In some cases, ecologists may be aiming to understand how to enhance survival of the population; in other cases, such as invading species, they may be aiming to understand how to prevent the population from surviving.
The contribution of this paper is the modeling and analysis of the dispersal of populations in streams under a considerably more complex and realistic set of conditions than had previously been analyzed. The conditions covered by the analysis include both population growth (e.g., through reproduction) and movement of the existing population by two different methods, gradual flow and long-distance changes. The incorporation of the long-distance changes is particularly important and leads to different conclusions than had previously been reached without including this condition. In particular, the authors address and explain the drift paradox, showing that populations always can persist under high flow rates, provided the frequency of rare long-distance dispersal events is sufficient.
The coverage of this range of conditions requires the authors to formulate and analyze an integrodifferential equations model, as opposed to the purely partial differential equation model that had been used in previous analysis under simpler conditions. The theory required to analyze this type of model is more difficult and is developed for the first time in this paper. The issues that are successfully investigated, under a variety of different conditions, include the minimal stream size necessary for the population to sustain itself, and the maximum stream flow under which the population can survive.
The paper is very nicely written, with explanations embedded throughout it that help make it understandable to the nonexpert. It truly is an example of applied mathematics at its best: important new mathematics, leading to understanding of important applications, conducted by a team of applied mathematicians and application scientists.

The Effect of Dispersal Patterns on Stream Populations

Elizaveta Pachepsky, Frithjof Lutscher, and Mark A. Lewis

SIAM Rev. 47, pp. 749-772 (24 pages) | Cited 15 times

Online Publication Date: August 04, 2006

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Individuals in streams are constantly subject to predominantly unidirectional flow. The question of how these populations can persist in upper stream reaches is known as the "drift paradox." We employ a general mechanistic movement-model framework and derive dispersal kernels for this situation. We derive thin- as well as fat-tailed kernels. We then introduce population dynamics and analyze the resulting integrodifferential equation. In particular, we study how the critical domain size and the invasion speed depend on the velocity of the stream flow. We give exact conditions under which a population can persist in a finite domain in the presence of stream flow, as well as conditions under which a population can spread against the direction of the flow. We find a critical stream velocity above which a population cannot persist in an arbitrarily large domain. At exactly the same stream velocity, the invasion speed against the flow becomes zero; for larger velocities, the population retreats with the flow.

Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 47, pp. 773-773 (1 page)

Online Publication Date: August 04, 2006

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As a tall, spindly adolescent I was constantly asked if I played basketball, a query I abhorred, as I much preferred be buried in a mathematics book with my true love, calculus. My appreciation of the sport came much later, and then as a spectator at the University of Arizona (Go Wildcats!). Perhaps I would have felt differently if somebody had given me the article "Modeling Basketball Free Throws," by Jeorg Gablonsky and Andrew Lang, featured in this issue's Education section.
The authors should be applauded for producing a manuscript that is accessible to introductory calculus students (a novelty for this section) and, frankly, just plain fun to read. The article explores the best way to shoot a basketball free throw by successively considering release angle, release velocity, air resistance, and the shooter's height. They start with Shaquille O'Neal, provide a classic example of the iterative process of mathematical model building, work their way through some differential equations and numerical methods, introduce the basic idea of multiobjective optimization, and end with some practical advice for the next time one is shooting a free throw.
Amazingly, this is all presented using only basic calculus and written in a manner that students should find friendly and compelling. The paper illustrates how mathematics can help us understand the world around us, and suggests a number of projects and extensions suitable for those who are just beginning college mathematics.
In short, this paper is a slam-dunk of an introduction to the mathematics of sport. I encourage you to give it to your students---who knows, one of them may turn out to be the next Joe Keller or, perhaps, the next Shaquille O'Neal?

Modeling Basketball Free Throws

Joerg M. Gablonsky and Andrew S. I. D. Lang

SIAM Rev. 47, pp. 775-798 (24 pages)

Online Publication Date: October 31, 2005

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This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 47, pp. 799-858 (60 pages)

Online Publication Date: August 04, 2006

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Willy Hereman has written a feature review about the first two of four volumes of a Mathematica GuideBook by Michael Trott. Surveying 2,400 pages of text and an accompanying DVD is clearly a herculean task, which he has (fortunately) carried out with enthusiasm. We hope his effort and Trott's will help you use Mathematica more effectively.
Altogether, thirty-five books on a diverse number of subjects are reviewed in this issue. Overall, our reviewers are quite enthusiastic about them. Increasingly, however, potential reviewers are returning negative comments about certain new publications that they claim don't deserve the attention a review would provide. This suggests that we should encourage our friends in publishing to raise their standards. Poorly conceived and edited manuscripts should be rejected until their value becomes evident.
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