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2002

Volume 44, Issue 1, pp. 1-165


Survey and Review

Nick Trefethen, Section Editor

SIAM Rev. 44, pp. 1-1 (1 page)

Online Publication Date: August 04, 2006

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Richard Dawkins writes of "Climbing Mount Improbable." Chemists speak of minimization on potential energy surfaces. Computer scientists talk of traveling salesman tours. These are examples of problems with "fitness landscapes" in which the goal is to find a maximum or minimum among a set of states that in most applications will be almost unimaginably vast. Further examples come from Ising models in physics, or board positions in chess.
In the following article, Christian Reidys and Peter Stadler propose tools for mapping the mountain. At first one might ask, What can be new in this subject, since it is neither more nor less than the usual problem of optimization? The answer is that in fact it is more than the usual, for the focus here is not just on the domain \textit{X} and objective function \textit{f} : \textit{X} $\rightarrow \mathbb{R}$ that define the optimization problem, but equally on the geometry of the "moves" that are made from point to point in \textit{X}. In biology, mutations are not random but highly constrained by the details of DNA or RNA. In chemistry, one structure may be converted to another by a simple bond rotation, whereas two other equally similar structures have no ready pathway between them. In the traveling salesman problem, one candidate tour may be improved to another by a local path transposition.
Reidys and Stadler put forward a general mathematical procedure for describing some of the topographies that may arise in large-scale combinatorial problems like these. They argue that this analysis may shed light on the behavior of nonstandard human methods for optimization, such as genetic algorithms or simulated annealing, as well as on the algorithms used by complex systems in nature. Despite this article's 253 cited references, ideas are moving fast at this triple point of science and computing and nonlinear mathematics; these are early days yet.

Combinatorial Landscapes

Christian M. Reidys and Peter F. Stadler

SIAM Rev. 44, pp. 3-54 (52 pages) | Cited 28 times

Online Publication Date: August 04, 2006

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Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance, or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the ``static' properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 44, pp. 55-55 (1 page)

Online Publication Date: August 04, 2006

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The Rayleigh quotient iteration (RQI) that we know and love is used to calculate an eigenvalue-eigenvector pair of a symmetric matrix. There are numerous situations when multiple eigenvalues and the subspace spanned by the corresponding eigenvectors are required. Our first article, by Absil, Mahony, Sepulchre, and Van Dooren, describes a generalization of RQI to address this problem. While other algorithms for this problem exist, they either converge at a slow linear \textit{O}(\textit{n}) rate in the dimension \textit{n} of the matrix or have a high \textit{O}(\textit{n}$^{\mbox{\font size{6}{6}\selectfont 3}}$) cost per iteration. The present generalized RQI procedure has cubic convergence and an \textit{O}(\textit{np}$^{\mbox{\font size{6}{6}\selectfont 2}}$) unit cost for invariant subspaces of dimension \textit{p}. The key involves the solution of a Sylvester system working with a condensed form of the original matrix and the theory relies onthe structure of Grassmann manifolds.
The current situation in Afghanistan would seem to make the article by Hassan Sedaghat on the dynamics of a discrete-time model of combat particularly relevant. Sedaghat examines the behavior of a "ground version" of a combat model proposed by Epstein by neglecting air support. He emphasizes that realistic combat models used for "war game" simulations would include this as well as deterministic and stochastic effects due to, e.g., weather, terrain, psychology, and logistics. Nevertheless, the simplified deterministic ground model, consisting of three nonlinear difference equations, has a rich structure that may form the core of a more complex model. The key ingredient of the ground model is a withdrawal strategy that introduces a jump discontinuity into the model. Sedaghat's asymptotic and transient analyses of the ground model reveal that steady, oscillatory, and chaotic behaviors are all possible. The latter effect sounds realistic to me!
Please enjoy these very interesting articles.

A Grassmann--Rayleigh Quotient Iteration for Computing Invariant Subspaces

P. A. Absil, R. Mahony, R. Sepulchre, and P. Van Dooren

SIAM Rev. 44, pp. 57-73 (17 pages) | Cited 9 times

Online Publication Date: August 04, 2006

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The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.

Convergence, Oscillations, and Chaos in a Discrete Model of Combat

Hassan Sedaghat

SIAM Rev. 44, pp. 74-92 (19 pages) | Cited 1 time

Online Publication Date: August 04, 2006

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A piecewise smooth mapping of the three-dimensional Euclidean space is derived from a discrete-time model of combat. The mathematical analysis of this mapping focuses on the effects of discontinuity caused by the defender's withdrawal strategy---a prime component of the original model. Both the asymptotics and the transient behavior are discussed, and all the behavior types noted in the title are established as possible outcomes.

SIGEST

SIAM Rev. 44, pp. 93-93 (1 page)

Online Publication Date: August 04, 2006

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This issue's SIGEST paper, "Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing," is taken from SIAM Journal on Discrete Mathematics (SIDMA). Bin packing is a classical NP-hard problem with many practical applications. Given a list of items of different sizes and a set of fixed-capacity bins, how can the items be packed into the bins with a minimal amount of waste? If a CD contains at most an hour of music, how many CDs will it take to record a large collection of musical numbers of different duration? If expensive wood is supplied in fixed-length boards, how many boards are needed to produce a specified array of different-sized shelves? How should data files of varying dimension be stored into fixed-size blocks on disk?
The original version of this paper first appeared in volume 13 of SIDMA in 2000, with the title "Bin Packing with Discrete Item Sizes, Part I: Perfect Packing Theorems and the Average Case Behavior of Optimal Packings" and seven distinguished coauthors---E. G. Coffman, Jr., C. Courcoubetis, M. R. Garey, D. S. Johnson, P. W. Shor, R. R. Weber, and M. Yannakakis. The SIGEST paper is substantially different from its first incarnation, in large part because certain proofs in the original paper have been superseded by the authors' subsequent work on a new "sum of squares algorithm," cited in the present paper; the authors have accordingly omitted the relevant sections from the original paper. In addition, the authors have expanded and updated their summary of the latest results for discrete distributions.
The paper gives a readable and short introduction not just to the classical bin packing problem, but also to the mathematical differences that arise when the item sizes must be drawn from a finite set---the typical case in real-world applications---compared to the often-made assumption in theoretical contexts that the item sizes are chosen based on a continuous probability distribution. Combinatorial questions disappear in the limit of the continuous case---but is something lost in moving from discrete to continuous models? The answer is "yes," for reasons explained and illustrated through analyzing "online" algorithms, in which items are assigned to bins in the order of their appearance in the item list.
The concluding section of the paper, written mainly for the SIGEST version, presents a clear and complete survey of knowledge about average-case behavior of bin packing algorithms under discrete distributions, along with a characterization of the differences between results for continuous and discrete distributions. It also describes an array of challenging open problems.
We are grateful to the authors for giving us an impressive and engrossing SIGEST paper.

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

E. G. Coffman, Jr., C. Courcoubetis, M. R. Garey, D. S. Johnson, P. W. Shor, R. R. Weber, and M. Yannakakis

SIAM Rev. 44, pp. 95-108 (14 pages) | Cited 5 times

Online Publication Date: August 04, 2006

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We consider the one-dimensional bin packing problem under the discrete uniform distributions $U\{j,k\}$, $1 \leq j \leq k-1$, in which the bin capacity is $k$ and item sizes are chosen uniformly from the set $\{1,2,\ldots,j\}$. Note that for $0 < u = j/k \leq 1$ this is a discrete version of the previously studied continuous uniform distribution $U(0,u]$, where the bin capacity is 1 and item sizes are chosen uniformly from the interval $(0,u]$. We show that the average-case performance of heuristics can differ substantially between the two types of distributions. In particular, there is an online algorithm that has constant expected wasted space under $U\{j,k\}$ for any $j,k$ with $1 \leq j < k-1$, whereas no online algorithm can have $o(n^{1/2})$ expected waste under $U(0,u]$ for any $0 < u \leq 1$. Our $U\{j,k\}$ result is an application of a general theorem of Courcoubetis and Weber that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of $n$ items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, or $O(1)$, depending on whether certain ``perfect' packings exist. The perfect packing theorem needed for the $U\{j,k\}$ distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper. We also survey other recent results comparing the behavior of heuristics under discrete and continuous uniform distributions.

Education

Bobby Schnabel, Section Editor

SIAM Rev. 44, pp. 109-109 (1 page)

Online Publication Date: August 04, 2006

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This issue's Education section contains a wonderfully rich example of mathematical modeling. Newbury and Spiteri, in the paper "Inverting Gravitational Lenses," provide a module that combines a fascinating application from astrophysics, gravitational lenses, with two types of mathematical models, forward and inverse, and numerical techniques for solving these models, including MATLAB routines available over the Internet. As the authors state, one of the beauties of this topic is that it is an exciting, recent development about which some nontrivial issues can be explored using fairly basic mathematical modeling techniques. This paper gives students the opportunity to learn a little about cosmology, to derive and utilize several mathematical models, and to understand via this example the issues associated with inverse modeling of ill-posed problems.
The paper is written in a style that is easy and enjoyable to follow, with numerous figures that help illustrate the concepts. It is particularly suitable for courses in mathematical modeling or inverse problems but, as the authors state, could also be used as a fascinating module in a more general course in scientific computing. The paper also serves as an excellent example to prospective authors who are considering writing modules for the Education section of SIAM Review.
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Inverting Gravitational Lenses

Peter R. Newbury and Raymond J. Spiteri

SIAM Rev. 44, pp. 111-130 (20 pages)

Online Publication Date: August 04, 2006

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Gravitational lensing provides a powerful tool to study a number of fundamental questions in astrophysics. Fortuitously, one can begin to explore some nontrivial issues associated with this phenomenon without a lot of very sophisticated mathematics, making an elementary treatment of this topic tractable even to senior undergraduates. In this paper, we give a relatively self-contained outline of the basic concepts and mathematics behind gravitational lensing as a recent and exciting topic for courses in mathematical modeling or scientific computing. To this end, we have designed and made available some interactive software to aid in the simulation and inversion of gravitational lenses in a classroom setting.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 44, pp. 131-165 (35 pages)

Online Publication Date: August 04, 2006

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Andrew Stuart provides us with a timely featured review of the annual Acta Numerica, now publishing its tenth issue, the brainchild of its highly original editor Prof. Arieh Iserles of Cambridge University. Like the Annual Reviews in many fields of application, Acta Numerica has become the first place to look when one needs to get up to date on a particular branch of numerical analysis.
This issue of SIAM Review also contains reviews of over twenty other new books, spanning probability and turbulence to history and computational fluid dynamics. Their authors are experts, who generally give reasons to praise these publications. After submitting a reserved review published below, however, one reviewer drolly quotes Dorothy Parker, writing "This is not a book to be tossed aside lightly. It should be thrown with full force."
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