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2002

Volume 44, Issue 4, pp. 523-747


Survey and Review

Nick Trefethen, Section Editor

SIAM Rev. 44, pp. 523-523 (1 page)

Online Publication Date: August 04, 2006

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What do these algorithms have in common?
The answer is that they are all infinite algorithms for finite problems. We could solve (1) by Gaussian elimination, (2) by adding up the pairwise forces, (3) by nested dissection, or (4) by the simplex method, and in each case we would get the exact answer, apart from rounding errors, in a finite number of steps. But it may be quicker to use methods that take an infinite number of steps! These algorithms are approximate and iterative, but they converge so fast that they are often the best way to solve large-scale problems if you want "only," say, 15 digits of accuracy. (1)--(4) and their relatives have changed scientific computing profoundly since the 1970s and 1980s.
The following article by Forsgren, Gill, and Wright has its roots in (4). But once you depart from finite algorithms and embrace iteration, the finiteness of the underlying problem loses its importance: you gain robustness as well as speed. Suddenly linear programming is not so different from nonlinear, and the subject of this article is the huge subject of interior methods for nonlinear optimization that has blossomed since Karmarkar proposed his famous algorithm in 1984.
The move to fast continuous algorithms has other aspects. Computer scientists have found that although a discrete combinatorial problem may be NP-hard if you want an exact solution, near-optimal solutions may be computable via continuous algorithms such as those reviewed here. The problem of finding large cuts in graphs, for example, can be solved in polynomial time if you'll settle for a cut that might be 13% smaller than the largest possible. Thus interior methods are playing a role in drawing together discrete and continuous applied mathematicians.
SIAM Review draws mathematicians together too, and it has been a privilege to serve as Section Editor for four years. In January, I will be succeeded in this post by Randy LeVeque of the University of Washington.

Interior Methods for Nonlinear Optimization

Anders Forsgren, Philip E. Gill, and Margaret H. Wright

SIAM Rev. 44, pp. 525-597 (73 pages) | Cited 86 times

Online Publication Date: August 04, 2006

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Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization.
This picture changed dramatically with Karmarkar's widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 44, pp. 599-599 (1 page)

Online Publication Date: August 04, 2006

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Most places have legalized lotteries and numbers games. Our first article deals with managing risk for the game operators of four-digit number games. I know that New York has a daily "Win 4" number game. However, the games studied by Teo and Leong, the authors of our first article, are more complex and are representative of games played in the Far East in places such as Singapore and Malaysia. Teo and Leong develop a control strategy to restrict play on certain "hot numbers" to avoid having operators sustain losses while optimizing the number of bets. I certainly found this article interesting; however, those of you who know me know that I prefer other games of chance.
Statistical analyses frequently require fitting planes to multivariate data. This includes least squares (L2) and least absolute (L1) deviations as well as principal components fitting, which involves finding the plane spanned by the eigenvectors of the covariance or correlation matrix corresponding to the largest eigenvalue. In our second article, Steven Ellis analyzes these problems in the simplest case of fitting a line to three or four points in a plane. His article illustrates and explains the complex behavior that may result when fitting "unstable" data sets where small perturbations in the data can produce large changes in the resulting line fit. The author characterizes instability in relation to the "singularities" of the methods. The singularities of the least squares method involve collinear data sets. Quite naturally, instabilities occur in the vicinity of the singularities.

Managing Risk in a Four-Digit Number Game

Chung-Piaw Teo and Siew Meng Leong

SIAM Rev. 44, pp. 601-615 (15 pages) | Cited 1 time

Online Publication Date: August 04, 2006

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The four-digit number game is a popular game of chance played in Southeast Asia. The players in this game choose a four-digit number and place their bets on it. In this paper, we study the design of a control mechanism for managing bets in this game. Our objective is to design a control mechanism to decide whether bets should be accepted or rejected. We propose a nonlinear optimization model for this problem and provide the mathematical justification for the control mechanism used by several operators in this region. We also suggest a simple improved control mechanism. Using data provided by a company in the region, we show that our control mechanism can accept more money per draw, while the risk exposure of the proposed mechanism can be considerably smaller than the current system.

Fitting a Line to Three or Four Points on a Plane

Steven P. Ellis

SIAM Rev. 44, pp. 616-628 (13 pages)

Online Publication Date: August 04, 2006

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Linear regression and principal components analysis are examples of plane fitting methods. Plane fitting is a very important activity in multivariate statistical analysis. The geometry of plane fitting is surprisingly complex, but general insights into it can be gained by considering the problem of fitting a line ("one-dimensional plane") to only three or four bivariate, quantitative data points. Graphical analysis reveals that any line fitting method must have "singularities," i.e., data sets near which the line fitting method is unstable. (For example, collinear data sets are the singularities of least squares linear regression.) Singularities can be classified according to the effects they have on the behavior of the line fitting method and those effects can be quantified as well. The dimension of ("degrees of freedom" in) the set of all singularities of a line fitting method is related to the probability of getting a data set near a singularity. These ideas are illustrated in principal component analysis and least squares and least absolute deviation linear regression.

SIGEST

SIAM Rev. 44, pp. 629-629 (1 page)

Online Publication Date: August 04, 2006

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This issue's SIGEST paper brings together two prominent themes in numerical analysis---finite element methods and adaptivity.
Some date the ideas of the finite element method to the Hurwitz--Courant book of 1922; the first mathematical application of the finite element method is often attributed to Richard Courant's 1943 work on solving a torsion problem using piecewise polynomials. Engineers independently began using finite element techniques during the 1950s, primarily for structural analysis, and the name "finite element method" was coined in 1960 by Ray Clough, a professor of structural engineering at the University of California, Berkeley. Since then, finite element methods and their variations have been widely studied and generalized, becoming in the process a solution technique that is almost ubiquitous in scienceand engineering.
The concept of an adaptive numerical method---meaning, broadly speaking, one that uses information acquired while solving a problem to shape the solution procedure---is similarly woven into today's scientific computing. Adaptive methods are especially popular for problems containing singularities, sharp boundaries, or disparate scales. The buzzword "multiscale" frequently appears in recent characterizations of the most challenging problems, and is naturally linked with adaptivity.
In 1978, the combination of these ideas---adaptivity and the finite element method---began with the seminal paper "Error Estimates for Adaptive Finite Element Computations," by Ivo Babuska and Werner Rheinboldt [SIAM Journal on Numerical Analysis, 15 (1978), pp.~736--754]. Thus it is particularly appropriate that this issue's SIGEST paper from that same journal, "Convergence of adaptive finite element methods," by Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, also involves adaptive finite element methods.
As the authors observe, adaptive methods can succeed only with good a~posteriori error estimates to help guide changes in the mesh. This paper tackles the mostly open problem of analyzing convergence in more than one dimension for the adaptive method paradigm of repeated solution, error estimation, and refinement. Based on a measure called data oscillation, which captures information that can be missed in averaging, the authors present an algorithm and prove its fundamental error reduction property. The nonexpert reader will find a clear introduction to the a posteriori error estimates on which adaptive finite elements are founded, followed by presentation of a new algorithm and a proof of its error reduction property. Many illuminating numerical examples are presented throughout, and two of the many colorful figures appear on this issue's cover.
We thank the authors for their contribution to SIAM Review.

Convergence of Adaptive Finite Element Methods

Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert

SIAM Rev. 44, pp. 631-658 (28 pages) | Cited 58 times

Online Publication Date: August 04, 2006

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Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Education

Bobby Schnabel, Section Editor

SIAM Rev. 44, pp. 659-659 (1 page)

Online Publication Date: August 04, 2006

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The papers in the Education section of this issue of SIAM Review cover two different topics: construction of efficient numerical algorithms, and mathematical modeling.
The title of Desmond Higham's paper, "Nine Ways to Implement the Binomial Method for Option Valuation in MATLAB," may be a play on the SIAM Review golden oldie by Cleve Moler and Charlie Van Loan, "Nineteen Dubious Methods for Computing the Matrix Exponential." The content is related to an invention of Moler's---MATLAB---but more generally to fundamental principles in the construction of efficient numerical algorithms. Through an interesting financial example, Higham illustrates several basic principles that have very general applicability, beyond any particular problem or programming language:
Any course in scientific or numerical computation, or algorithm development, that wishes to teach these principles will find Higham's paper an excellent vehicle for doing so.
"An Elementary Model for Control of a Semiconductor Etching Process," by Madhukar, Parent, Rosen, and Wang, is an excellent example of mathematical modeling. The project from which it is derived illustrates applied mathematics in action: a collaboration by a team of materials scientists and applied mathematicians to design, test, and implement optimal control and estimation schemes for the manufacture of advanced semiconductor devices. The authors show that by using no more than sophomore-level chemistry, physics, and differential equations, they can derive a three-equation dynamic model of the etching process of gallium arsenide. The three equations model the chemical reaction of the etching process, the dynamics of the pressure in the etching chamber, and the dynamics of a throttle valve that plays a key role in the chamber. The paper describes the model, the estimation of the model parameters from experimental data, validation of the model, and application of the model to process control. It will make an excellent case study in mathematical modeling courses, as well as being a contribution to the study of semiconductor etching.
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Nine Ways to Implement the Binomial Method for Option Valuation in MATLAB

Desmond J. Higham

SIAM Rev. 44, pp. 661-677 (17 pages) | Cited 3 times

Online Publication Date: August 04, 2006

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In the context of a real-life application that is of interest to many students, we illustrate how the choices made in translating an algorithm into a high-level computer code can affect the execution time. More precisely, we give nine MATLAB programs that implement the binomial method for valuing a European put option. The first program is a straightforward translation of the pseudocode in Figure 10.4 of The Mathematics of Financial Derivatives, by P. Wilmott, S. Howison, and J. Dewynne, Cambridge University Press, 1995. Four variants of this program are then presented that improve the efficiency by avoiding redundant computation, vectorizing, and accessing subarrays via MATLAB's colon notation. We then consider reformulating the problem via a binomial coefficient expansion. Here, a straightforward implementation is seen to be improved by vectorizing, avoiding overflow and underflow, and exploiting sparsity. Overall, the fastest of the binomial method programs has an execution time that is within a factor 2 of direct evaluation of the Black--Scholes formula. One of the vectorized versions is then used as the basis for a program that values an American put option. The programs show how execution times in MATLAB can be dramatically reduced by using high-level operations on arrays rather than computing with individual components, a principle that applies in many scientific computing environments. The relevant files are downloadable from the World Wide Web.
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An Elementary Model for Control of a Semiconductor Etching Process

A. Madhukar, T. Parent, I. G. Rosen, and C. Wang

SIAM Rev. 44, pp. 678-695 (18 pages)

Online Publication Date: August 04, 2006

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A dynamic model for the thermal chlorine etching of gallium arsenide is formulated and validated. The model consists of three ordinary differential equations. One models the chemical reaction between the chlorine gas and the gallium arsenide substrate being etched. The second equation, which is based on an inflow/outflow paradigm, models the dynamics of the pressure in the etching chamber. The third equation models the dynamics of a throttle valve which controls the chamber pressure. The entire model is based upon a combination of empirical and first principle physics-based reasoning, and is formulated using sophomore-level elementary chemistry, physics, and differential equations. Spectroscopic ellipsometry, a nondestructive optically based technique for real-time in-situ characterization of materials, is described. Offline and real-time ellipsometry measurements of sample thickness are used to identify or estimate otherwise unmeasurable parameters which appear in the model and to verify or validate our model via comparison with simulation results based upon the model. The use of the model in the design, implementation, and testing of a feedback control system for the etching process is discussed.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 44, pp. 697-747 (51 pages)

Online Publication Date: August 04, 2006

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This issue begins with John Stillwell's charming review of Yandell's book on Hilbert's problems and their solvers. If you're like me, it'll convince you to read not only Yandell but also other works by Stillwell.
The other reviews are diverse and full of expert opinions. The topics range from deep mathematics to a variety of applications, sometimes even combinations of theory, practice, and computation. We are especially grateful for a few longer-than-usual reviews that present their conclusions as somewhat discursive essays. We also note the recurrent interest in history and good exposition, as well as criticisms of incompleteness and inadequate editing.
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