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2009

Volume 51, Issue 4, pp. 659-807

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 51, pp. 659-659 (1 page)

Online Publication Date: November 04, 2009

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Data are often modeled, or explained, by standard distributions. The most common one, the normal distribution, has the most widespread use. It is a simple model that captures an average behavior and a measure of deviations from this average. The title of a controversial book published in the 1990s, The Bell Curve, comes from the observation that the IQ scores have a normal distribution. Thus IQ data suggest that there is an average intelligence and a standard deviation from this average. Not all data are so easily summarized. There are data sets for which you cannot fit a normal distribution to the associated histogram.
The subject of this issue's Survey and Review article is power-law distributions. Data obeying a power-law distribution are very interesting and rich. They exhibit self-similarity—the data viewed at one scale has a similar structure when viewed at another scale. For example, Internet packet data viewed in a 1-hour window have similar features when viewed in a 1-day window. These data also have “fat tails,” meaning that large deviation events occur more often than is expected. Benoît Mandelbrot made the observation that financial data are better explained with fat-tailed distributions. It has been suggested that modeling financial data with normal distribution can underestimate the risk associated with an asset. The recent economic crisis has taught us the dangers of underestimating risk.
The review paper, by Clauset, Shalizi, and Newman, is a very accessible mathematical introduction to power-law distribution. It provides a “recipe” on how to assess whether a data set is of the power-law type and, if so, what its salient properties are. The authors apply their analytical approach to a set of real-world data from different fields. This thoughtful and charming exercise demonstrates the prevalence of power-law data and the effectiveness of the authors' approach to characterizing them.

Power-Law Distributions in Empirical Data

Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman

SIAM Rev. 51, pp. 661-703 (43 pages) | Cited 8 times

Online Publication Date: November 06, 2009

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Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.
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Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 51, pp. 705-705 (1 page)

Online Publication Date: November 04, 2009

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This issue features two papers involving optimization problems, one targeted at minimizing temperature differences across a plate, and the other at multidimensional integration.
1. Want to turn the graphics card in your laptop or desktop into a personal supercomputer? Eddie Wadbro and Martin Berggren show you how, in their paper “Megapixel Topology Optimization on a Graphics Processing Unit.” The advantage of graphics cards is that they are cheap, relatively speaking, and that they allocate more resources to data processing than a CPU would. Graphics cards are also becoming easier to program (especially for those of us who grew up with Connection Machines and FPS-164s). Graphics processing units are natural hardware platforms for problems that are highly data parallel. This includes the topology optimization problem considered here, where a limited amount of high-conductivity material is to be distributed across a heated plate so that its temperature field is as even as possible. The authors express this as an area-to-point flow optimization problem. A subsequent finite element discretization gives a symmetric positive-definite linear system that is solved by a diagonally preconditioned conjugate gradient method. As it turns out, the optimal distribution of the high-conductivity material emanates like a root from the heat sink, with increasing girth and finer branches as the discretization is refined.
2. In their paper “Approximate Volume and Integration for Basic Semialgebraic Sets,” Didier Henrion, Jean Bernard Lasserre, and Carlo Savorgnan are concerned with deterministic techniques for difficult multidimensional integration, of the type where only brute force Monte Carlo methods have a chance at producing acceptable approximations. The bodies can be disconnected or nonconvex, and are described by sets of polynomial inequalities (i.e., semialgebraic sets).
The foundation for this work was laid more than a hundred years ago, when Chebyshev, Markov, and Stieltjes showed how to approximate one-dimensional integrals by sequences of moments. In this paper, the authors formulate the multidimensional integration as an infinite-dimensional linear programming problem, and approximate the required moments by a hierarchy of semidefinite programming problems. Numerical examples illustrate that the approach produces accurate approximations in two and three dimensions.

Megapixel Topology Optimization on a Graphics Processing Unit

Eddie Wadbro and Martin Berggren

SIAM Rev. 51, pp. 707-721 (15 pages)

Online Publication Date: November 06, 2009

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We show how the computational power and programmability of modern graphics processing units (GPUs) can be used to efficiently solve large-scale pixel-based material distribution problems using a gradient-based optimality criterion method. To illustrate the principle, a so-called topology optimization problem that results in a constrained nonlinear programming problem with over 4 million decision variables is solved on a commodity GPU.

Approximate Volume and Integration for Basic Semialgebraic Sets

D. Henrion, J. B. Lasserre, and C. Savorgnan

SIAM Rev. 51, pp. 722-743 (22 pages)

Online Publication Date: November 06, 2009

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Given a basic compact semialgebraic set $\mathbf{K}\subset\mathbb{R}^n$, we introduce a methodology that generates a sequence converging to the volume of $\mathbf{K}$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on $\mathbf{K}$ can be approximated as closely as desired, which permits the approximation of the integral on $\mathbf{K}$ of any given polynomial; the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.
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SIGEST

The Editors

SIAM Rev. 51, pp. 745-745 (1 page)

Online Publication Date: November 04, 2009

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The calculation of the matrix exponential e$^{\mbox{\scriptsize{\textit{A}}}}$ may be one of the best known matrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” published in this journal in 1978 (and revisited in this journal in 2003). The matrix exponential is utilized in a wide variety of numerical methods for solving differential equations and many other areas.
It is somewhat amazing given the long history and extensive study of the matrix exponential problem that one can improve upon the best existing methods in terms of both accuracy and efficiency, but that is what the SIGEST selection in this issue does. “The Scaling and Squaring Method for the Matrix Exponential Revisited” by N. Higham, originally published in the SIAM Journal on Matrix Analysis and Applications in 2005, applies a new backward error analysis to the commonly used scaling and squaring method, as well as a new rounding error analysis of the Padé approximant of the scaled matrix. The analysis shows, and the accompanying experimental results verify, that a Padé approximant of a higher order than currently used actually results in a more accurate and efficient algorithm, due to the need to perform fewer matrix multiplications and fewer squarings.
SIGEST papers are expected to combine important research, broad applicability, and excellent exposition. In addition to the first two qualities that are evident from the comments above, it is no surprise that this paper exhibits the latter, since the author literally wrote the book—that is, the popular SIAM book, Handbook of Writing for the Mathematical Sciences. The paper is a true pleasure to read, and the author has added an extensive preamble for the SIGEST version that makes the topic even more accessible to new audiences as well as commenting upon subsequent work.
This paper should become an excellent choice in teaching numerical analysis as well as essential reading for researchers in numerical linear algebra. Its combination of practical algorithmic issues such as scaling, numerical analysis techniques such as backward error analysis, and a carefully conducted and presented computational study make it a true exemplar for our field. One question remains, however, after all these years: should we still assert that all the methods for calculating the matrix exponential are dubious?

The Scaling and Squaring Method for the Matrix Exponential Revisited

Nicholas J. Higham

SIAM Rev. 51, pp. 747-764 (18 pages) | Cited 3 times

Online Publication Date: November 06, 2009

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The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in the MATLAB function expm. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give a new rounding error analysis that shows the computed Padé approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Padé approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than the expm function in MATLAB 7.0 when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Padé approximation to the function $x \coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis.
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Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 51, pp. 765-765 (1 page)

Online Publication Date: November 04, 2009

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In the parlance of mathematical finance, an option is a contract that gives the holder the right (but not the obligation) to buy an asset at a particular price during a specified future time range. In this form of legalized gambling, if the asset's value falls below the agreed upon price, the option may be worthless, but if the price rises, the option has locked in your price of choice and becomes worth the difference between the actual and agreed upon prices. The question of how to value various types of options is part of the foundation of mathematical finance, notably driving the development of the Black–Scholes model (which was recognized by a Nobel Prize in 1997); however, understanding this breakthrough requires a knowledge of the calculus describing stochastic processes, putting it beyond the reach of many mathematics students.
This issue's Education section offering, “Pricing American Perpetual Warrants by Linear Programming” by Robert J. Vanderbei and Mustafa Ç. Pınar, replaces the continuous random walk of a stochastic process with a discrete one and turns the pricing problem into an exercise in linear programming. The result is a natural entry point into the world of mathematical finance from a discrete linear programming perspective. This paper is easily accessible as a case study for a first course in linear programming.
Although the turmoil in the financial markets has momentarily tarnished the allure of a career in mathematical finance, the truth of the matter is that it is still a growth industry and a steady consumer of recent graduates in the mathematical sciences. This paper provides an easy pathway to seeing how linear programming ideas are naturally applicable in finance. While an understanding of this problem may not help your flagging retirement portfolio, it may very well let students know that one option they have is to pursue a career in financial mathematics.
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Pricing American Perpetual Warrants by Linear Programming

Robert J. Vanderbei and Mustafa Ç. Pınar

SIAM Rev. 51, pp. 767-782 (16 pages)

Online Publication Date: November 06, 2009

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A warrant is an option that entitles the holder to purchase shares of a common stock at some prespecified price during a specified interval. The problem of pricing a perpetual warrant (with no specified interval) of the American type (that can be exercised any time) is one of the earliest contingent claim pricing problems in mathematical economics. The problem was first solved by Samuelson and McKean in 1965 under the assumption of a geometric Brownian motion of the stock price process. It is a well-documented exercise in stochastic processes and continuous-time finance curricula. The present paper offers a solution to this time-honored problem from an optimization point of view using linear programming duality under a simple random walk assumption for the stock price process, thus enabling a classroom exposition of the problem in graduate courses on linear programming without assuming a background in stochastic processes.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 51, pp. 785-807 (23 pages)

Online Publication Date: November 04, 2009

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Jon Borwein, as expected, has written an enthusiastic featured review of the popular and impressive Princeton Companion to Mathematics by Timothy Gowers and others. However, Bob Beezer also wrote a review of the open-source software Sage, which our readers should also be pleased to learn about, so we've decided to have two featured reviews in this issue! Other reviews are about stochastic differential equations, formal languages, the traveling salesman, ODEs, MIT mathematics, stochastic processes, and Markov processes. Some interesting opinions are contained.
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