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1999

Volume 41, Issue 4, pp. 635-851


Survey and Review

Nick Trefethen, Section Editor

SIAM Rev. 41, pp. 635-635 (1 page)

Online Publication Date: August 02, 2006

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Do mouthbreeder fish (Tilapia mossambica) know about centroidal Voronoi tesselations? They act as if they do, according to Figure 2.2 of the first Survey and Review article of this issue, by Du, Faber, and Gunzburger (the photo is due to G. W. Barlow). The definitions are simple: a Voronoi tesselation is a division of space into polyhedral regions of points closest to a fixed set of reference points, and a centroidal Voronoi tesselation is one with the special property that each reference point lies at the centroid of its polyhedron. Du, Faber, and Gunzburger show that this simple idea has applications in many areas of science and engineering, both actual and prospective, and they discuss algorithms for computing these tesselations. (The algorithm used by the fish proves to be slow.) Readers may get a quick impression of the compelling geometry of this subject by looking at Figures 7.1--7.6.
And do populations of sharks and fish, or of hawks and doves, oscillate in time because of spatial effects? Questions like this from ecology and epidemiology are the starting point for our second article, by Durrett. Like the first article, this one is concerned with spatial structure and with applications partly in biology. The substance is entirely different, however, for here the essential matter is nonlinear dynamics. The fundamental question that concerns Durrett is, What is the role played by spatial variation in large interacting systems? The answer varies from application to application, depending subtly on the rates and detailed nature of spatial interactions and mixing. Along the way, further questions come up. What is the significance of discrete vs. continuous space? of discrete vs. continuous time? of stochastic vs. deterministic dynamics? These are universal issues of applied mathematics, which will give us plenty to think about---dare we say it?---in the millennium around the corner.

Centroidal Voronoi Tessellations: Applications and Algorithms

Qiang Du, Vance Faber, and Max Gunzburger

SIAM Rev. 41, pp. 637-676 (40 pages) | Cited 153 times

Online Publication Date: August 02, 2006

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A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methodsfor computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.

Stochastic Spatial Models

Rick Durrett

SIAM Rev. 41, pp. 677-718 (42 pages) | Cited 47 times

Online Publication Date: August 02, 2006

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In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and Levin (1994): the behavior of these models can be predicted from the properties of the mean field ODE, i.e., the equations for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genetics, population biology, epidemiology, and ecology. Some of our findings are only conjectures based on simulation, but in a number of cases we are able to prove results for systems with "fast stirring" by exploiting connections between the spatial model and an associated reaction diffusion equation.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 41, pp. 719-719 (1 page)

Online Publication Date: August 02, 2006

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We offer three articles in this edition of Problems and Techniques, describing optimal radiation therapy strategies, calculating Green's functions in the complex plane, and finding the zeros of complex functions by inverting their asymptotic approximations.
Radiation therapy is a long-standing tool for treating cancer patients. The goal is to kill tumor cells by bombarding them with beams of radiation while minimizing damage to healthy cells that are in the beam's path. The authors of our first article, David Shepard, Michael Ferris, Gustavo Olivera, and Thomas Mackie, examine a new technique called ``tomotherapy,' which delivers beams of radiation from points normal to a circular arc around the patient's body in much the same fashion that computed tomography (CT) constructs images. The radiation beam is delivered in narrow slits that affect only a small slice of the patient. The patient lies on a couch that, like a CT apparatus, can be moved into and out of the radiation field. The mathematical challenge is to find an optimal delivery strategy that, for example, minimizes the total radiation to the patient while delivering a lethal dose to the tumor. The authors present several optimization problems and solution techniques. If you would like to have a go at figuring dosages, their software and data are available on a web site.
Our second article, by Mark Embree and Nick Trefethen, involves computing the Green's function in the complex plane in a domain exterior to a set of symmetric regions relative to and on the real axis. The symmetric configuration leads to a simple solution for the Green's function through a sequence of Schwarz--Christoffel conformal maps. Several nice examples illustrate both the process and the results. Applications are presented in potential theory and polynomial approximation, with the latter involving several interesting minimax problems that can be solved by exploiting their relationship to the Green's function. The polynomial approximations may be used in, e.g., signal processing and Krylov iteration.
Our third article, by Bruce Fabijonas and Frank Olver, is another feature for fans of complex analysis and approximation theory. The goal here is a technique for finding the zeros of complex functions that are known near singularities through asymptotic approximations. After reviewing the relevant theory, the authors show how to construct asymptotic expansions of functions and their zeros. Airy functions are used to test the theory and expansions are generated for them, their first derivatives, their zeros, and their stationary points. Software for symbolic computation is available for use by those who want to try their hand. Bounds on the accuracy of the results are discussed, and the authors suggest a conjecture for us to prove.
Although the subject matter in this installment of Problems and Techniques is pretty far removed from my domain of influence, I thoroughly enjoyed all of the articles. I hope that you will too. Please join me in thanking all of our authors for their fine contributions.

Optimizing the Delivery of Radiation Therapy to Cancer Patients

David M. Shepard, Michael C. Ferris, Gustavo H. Olivera, and T. Rockwell Mackie

SIAM Rev. 41, pp. 721-744 (24 pages) | Cited 55 times

Online Publication Date: August 02, 2006

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In the field of radiation therapy, much of the research is aimed at developing new and innovative techniques for treating cancer patients with radiation. In recent years, new treatment machines have been developed that provide a much greater degree of computer control than was available with the machines of previous generations. One innovation has been the development of an approach called "tomotherapy.' Tomotherapy can be defined as computer-controlled rotational radiotherapy delivered using an intensity-modulated fan beam of radiation.
The successful implementation of the new delivery techniques requires the development of a suitable approach for optimizing each patient's treatment plan. One of the challenges is to quantify optimality in radiation therapy. We have tested a variety of objective functions and constraints in pursuit of a formulation that performs well for a wide variety of disease sites. An additional challenge stems from the sizable amount of data and the large number of variables that are involved in each optimization. This paper presents several approaches to optimizing treatment plans in radiation therapy, and the advantages and disadvantages of a number of formulations are explored.

Green's Functions for Multiply Connected Domains via Conformal Mapping

Mark Embree and Lloyd N. Trefethen

SIAM Rev. 41, pp. 745-761 (17 pages) | Cited 12 times

Online Publication Date: August 02, 2006

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A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iterations. By making the end of the strip jagged, the method can be generalized to weighted Green's functions and weighted approximations.

On the Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions

Bruce R. Fabijonas and F. W. J. Olver

SIAM Rev. 41, pp. 762-773 (12 pages) | Cited 3 times

Online Publication Date: August 02, 2006

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The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available.
The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.

SIGEST

SIAM Rev. 41, pp. 775-775 (1 page)

Online Publication Date: August 02, 2006

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This issue's SIGEST paper, "The ring loading problem,' by Alexander Schrijver, Paul Seymour, and Peter Winkler, appeared originally in volume 11 of SIAM Journal on Discrete Mathematics, February 1998. Beginning with an application of great and growing practical importance to modern communication networks, the authors show that the associated mathematical formulation, a highly special integer multicommodity flow problem, is NP-complete. They then describe their (ultimately successful) search for a fast algorithm that provably produces a solution within 5% of the optimum load.
The paper is a lively, appealing mix of algorithmic theoretical computer science, operations research, and graph theory and has already inspired several other papers in the area. The exposition is outstanding, allowing even novices to understand both theoretical and practical aspects of the problem. The continuing, fruitful interactions between mathematics and numerical experiments are exceptionally interesting. For the paper's appearance in SIGEST, the authors have added a new approximation algorithm and comments about a further application area. We are grateful to them for their contribution to SIAM Review.

The Ring Loading Problem

Alexander Schrijver, Paul Seymour, and Peter Winkler

SIAM Rev. 41, pp. 777-791 (15 pages) | Cited 1 time

Online Publication Date: August 02, 2006

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The following problem arose in the planning of optical communications networks which use bidirectional SONET rings. Traffic demands di,j are given for each pair of nodes in an $n$-node ring; each demand must be routed one of the two possible ways around the ring. The object is to minimize the maximum load on the cycle, where the load of an edge is the sum of the demands routed through that edge.
We provide a fast, simple algorithm which achieves a load that is guaranteed to exceed the optimum by at most 3/2 times the maximum demand, and that performs even better in practice. En route we prove the following curious lemma: for any $x_1, \dots, x_n \in [0,1]$ there exist $y_1, \dots, y_n$ such that for each $k$, $|y_k|=x_k$ and $$ \left| \sum_{i=1}^k y_i - \sum_{i=k+1}^n y_i \right| \le 2. $$
[This article is reprinted here (with updates) from SIAM J. Discrete Math., 11 (1998), pp. 1--14. New developments include a $1+\varepsilon$ approximation algorithm and a variation of ring loading in the setting of wavelength division multiplexing; remarks added for this printing, about these and other issues, are enclosed in brackets.]

Education

Bobby Schnabel, Section Editor

SIAM Rev. 41, pp. 793-793 (1 page)

Online Publication Date: August 02, 2006

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This is the fourth issue of the new Education section in SIAM Review. It is probably worth restating the purpose of this new section once more (there will be a test at the end of the semester). The education section provides modules that students and teachers of applied mathematics and scientific computation can use in courses that study these fields and beyond. The main types of modules are applications, special topics, descriptions of software packages, and historical modules. Generally, these modules are written to students.
The first year of this section has demonstrated the diversity of modules in the education section, including descriptions of web-based software packages, new interpretations of topics in applied mathematics, explanations of elementary and advanced applications of mathematics and computation, and appealing examples. We hope all have shared a fresh, accessible writing style that will make them a pleasure for students.
This issue's article by Kruk and Wolkowicz will stand as an excellent example of the style that the Education section aspires to. It combines several attributes: presentation of a topic that readily can be included as supplementary material in linear or nonlinear programming courses; incorporation of an interesting real-world example that will enrich such courses; provision of online code supporting the example; and a refreshing writing style. The article shows, rigorously but simply, how and why a nonlinear programming problem is transformed to a fractional programming formulation and in turn can then be solved using Simplex method techniques.
It may be useful to mention once more to prospective authors that unsolicited submissions form the primary source of papers for the Education section. Guidelines for the section are available at http://www.siam.org/journals/sirev/Revguide.htm. As the style and goals of this section are rather different from traditional journal articles, the editorial board and section editor are more than happy to discuss ideas or preview manuscripts before they are formally submitted.
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Pseudolinear Programming

Serge Kruk and Henry Wolkowicz

SIAM Rev. 41, pp. 795-805 (11 pages) | Cited 3 times

Online Publication Date: August 02, 2006

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This short note revisits an algorithm previously sketched by Mathis and Mathis [SIAM Rev., 37 (1995), pp. 230--234] and used to solve a nonlinear hospital fee optimization problem. An analysis of the problem structure reveals how the Simplex algorithm, viewed under the correct light, can be the driving force behind a successful algorithm for an almost linear problem. This presentation is intended for students who have been exposed to the Simplex method for linear programming and are progressing, via the Karush--Kuhn--Tucker conditions, toward nonlinear optimization.

Book Reviews Introduction

R. Bruce Kellogg, Section Editor

SIAM Rev. 41, pp. 807-807 (1 page)

Online Publication Date: August 02, 2006

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As book review editor, I have become very aware of the many books in control theory that are appearing. The subject uses sophisticated mathematical tools in functional analysis and at the same time has great practical importance. Our thanks go to Joe Ball for giving a valuable survey of some recent offerings in control theory.

Book Reviews

SIAM Rev. 41, pp. 809-851 (43 pages)

Online Publication Date: August 02, 2006

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