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2007

Volume 49, Issue 2, pp. 177-367

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 49, pp. 177-177 (1 page)

Online Publication Date: April 30, 2007

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Cancer is scary business. Just think of the U.S. statistics—around 1.5 million new cases per year, and over half a million deaths. It is also expensive business. The NIH estimated the cost of cancer to the U.S. economy to be over $200 billion, of which $78 billion is direct medical costs and the rest is loss of productivity. It is so much in our consciousness that it is reflected in popular TV shows. In the first two seasons of the hit hospital drama “House,” almost twenty episodes dealt with patients diagnosed with or suspected to have a tumor or cancer.
There is good news. The five‐year relative survival rate has gone up from 51% for the years 1975–1977 to 66% for 1996–2002. Through research, scientists have developed better diagnostic methods and treatments for cancer. Mathematical modeling is central in the search for the cure for cancer. It can guide laboratory investigations and provide valuable insights into the development of effective treatments.
This issue’s Survey and Review article, “Mathematical Models of Avascular Tumor Growth” by Roose, Chapman, and Maini, provides a valuable introduction to the subject of tumor growth modeling. The reader is given a state‐of‐the‐art review of the two main approaches to modeling—continuous and discrete. The continuous model involves a system of partial differential equations, while the discrete model is a cellular automaton. We hope that this well‐written paper will inspire more mathematicians to join the effort to find a cure for cancer.

Mathematical Models of Avascular Tumor Growth

Tiina Roose, S. Jonathan Chapman, and Philip K. Maini

SIAM Rev. 49, pp. 179-208 (30 pages) | Cited 54 times

Online Publication Date: May 01, 2007

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This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.
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Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 49, pp. 209-209 (1 page)

Online Publication Date: April 30, 2007

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The Super Bowl is the annual championship game of professional American football, the National Football League (NFL). A coin toss determines which team kicks off and which one receives. This coin toss is big business in Las Vegas, spawning bets worth millions of dollars. In the past 12 years, 9 tosses came up tails and only 3 came up heads. Why so many tails and so few heads? Persi Diaconis, Susan Holmes, and Richard Montgomery may have found an explanation in their paper Dynamical Bias in the Coin Toss, based on a fascinating mixture of mechanics, probability distributions, and high speed cameras. If you want to be a millionaire, SIAM Review is here to help!
In the second paper, A Hybrid Approach for Efficient Robust Design of Dynamic Systems, Martin Mönnigmann and his five coauthors are concerned with optimizing models for chemical processes such as fermentation (think of yeast producing alcohol in the brewing of beer). If such a process operates on a continuous basis, one needs to find optimal equilibrium solutions for the underlying model. Say, for example, that one would like to determine the yeast concentration that maximizes the amount of alcohol in the beer. If bacteria can affect the yeast concentration, one also needs to be assured that the fermentation process remains stable, even when yeast concentrations fluctuate slightly. With the help of bifurcation theory, interval arithmetic, and nonlinear solvers, the authors derive a strategy for computing such solutions. Prost! (That’s German for “cheers.”)
A Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization by Jean‐Baptiste Hiriart‐Urruty is one of the more unusual papers in the Problems and Techniques section. The fourteen very different problems include a question posed by Newton in 1686 (what is the shape of a body that offers minimal resistance in a fluid?); issues in matrix theory (which real symmetric matrices can be simultaneously diagonalized with a single congruence transformation?); and solutions to eikonal equations (which smooth functions $f$ over which domains satisfy $\|\nabla f(x)\| = 1$?). Each problem comes with a precise description, a short history, and a digestible set of references.

Dynamical Bias in the Coin Toss

Persi Diaconis, Susan Holmes, and Richard Montgomery

SIAM Rev. 49, pp. 211-235 (25 pages) | Cited 10 times

Online Publication Date: May 01, 2007

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We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high‐speed photography are reported. For natural flips, the chance of coming up as started is about .51.

A Hybrid Approach for Efficient Robust Design of Dynamic Systems

Martin Mönnigmann, Wolfgang Marquardt, Christian H. Bischof, Thomas Beelitz, Bruno Lang, and Paul Willems

SIAM Rev. 49, pp. 236-254 (19 pages) | Cited 8 times

Online Publication Date: May 01, 2007

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We propose a novel approach for the parametrically robust design of dynamic systems. The approach can be applied to system models with parameters that are uncertain in the sense that values for these parameters are not known precisely, but only within certain bounds. The novel approach is guaranteed to find an optimal steady state that is stable for each parameter combination within these bounds. Our approach combines the use of a standard solver for constrained optimization problems with the rigorous solution of nonlinear systems. The constraints for the optimization problems are based on the concept of parameter space normal vectors that measure the distance of a tentative optimum to the nearest known critical point, i.e., a point where stability may be lost. Such normal vectors are derived using methods from nonlinear dynamics. After the optimization, the rigorous solver is used to provide a guarantee that no critical points exist in the vicinity of the optimum, or to detect such points. In the latter case, the optimization is resumed, taking the newly found critical points into account. This optimize‐and‐verify procedure is repeated until the rigorous nonlinear solver can guarantee that the vicinity of the optimum is free from critical points and therefore the optimum is parametrically robust. In contrast to existing design methodologies, our approach can be automated and does not rely on the experience of the designing engineer. A simple model of a fermenter is used to illustrate the concepts and the order of activities arising in a typical design process.

Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization

Jean‐Baptiste Hiriart‐Urruty

SIAM Rev. 49, pp. 255-273 (19 pages) | Cited 4 times

Online Publication Date: May 01, 2007

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We present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization. These problems can be classified into three groups: problems of pure mathematical interest, problems motivated by scientific computing and applications, and problems whose solutions are known but for which we would like to know better proofs. For each problem we provide a succinct presentation, a list of appropriate references, and a view of the state of the art of the subject.
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Sigest

SIAM Rev. 49, pp. 275-275 (1 page)

Online Publication Date: April 30, 2007

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In applied mathematics and scientific computation, we commonly generate mathematical models of great size and complexity. These can consist of hundreds of thousands or millions of parameters and algebraic or differential equations. Our field consistently makes great advances in the ability to solve larger and larger models both through improved algorithms and effective utilization of ever faster computers.
Still, in many situations, it is very important to be able to produce simplified mathematical models that represent the problem well. The reasons for using simplified models range from the efficiency of solving them to their ability to focus upon the primary characteristics of the problem. An overall approach of producing such simplified models is called reduced‐order models, and one of the key techniques for generating reduced‐order models is proper orthogonal decomposition (POD). POD is applied in a variety of fields and applications under a variety of names. For example, in statistics, principal component analysis techniques are used to transform a large set of possibly correlated variables into a smaller set of uncorrelated variables. The method of empirical orthogonal functions performs a related transformation in the area of models of geophysical fluid dynamics.
Naturally, a key question is how well reduced‐order models reflect the original mathematical model. A variety of existing research has addressed this question. In this issue’s SIGEST paper, “Error Estimation for Reduced‐Order Models of Dynamical Systems,” by C. Homescu, L. Petzold, and R. Serban, which originally appeared in the SIAM Journal on Numerical Analysis in 2005, the authors take this question one step further. They pose, and address, the question of how well the results of the reduced‐order model generated by POD reflect the solution to perturbations of the original model. These perturbations could come from changes to the initial conditions, or to the model parameters.
Based on a combination of the small‐sample statistical condition estimation method and error estimation using the adjoint method, the authors are able to establish ranges of perturbations to the original model for which the estimates produced by the reduced‐order model retain a reasonable degree of validity. Their approach produces a type of condition number that can be calculated a priori and that indicates the sensitivity of the results of the reduced‐order model to changes in the parameters of the original model. Their mathematical analysis is followed by several numerical examples that validate their mathematical estimates.
We hope that this paper will help introduce SIAM readers to the important area of reduced‐order models, as well as to important new research about the applicability of these models.

Error Estimation for Reduced‐Order Models of Dynamical Systems

Chris Homescu, Linda R. Petzold, and Radu Serban

SIAM Rev. 49, pp. 277-299 (23 pages) | Cited 6 times

Online Publication Date: May 01, 2007

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The use of reduced‐order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.
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Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 49, pp. 301-301 (1 page)

Online Publication Date: April 30, 2007

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Our knowledge in mathematics often expands by taking familiar problems and revisiting them with a new twist. In this issue we have two papers that examine problems that are usually first encountered in an introductory PDE course, though shown from a new perspective.
Our first contribution, “Fundamental Solutions for Some Partial Differential Operators from Fluid Dynamics and Statistical Physics,” by Jorge Aarão, offers an answer to a wearisome question that often arises in introductory PDE classes, namely, can one find only the fundamental solution for a few special problems such as the heat equation and Laplace’s equation? The author describes a method for computing the fundamental solution for a set of PDEs that arise, among other places, in fluid mechanics, where they describe diffusion and advection by a linear flow. The methodology described here should be accessible to students in an introductory course on PDEs, and the examples could form the foundation for a supplemental lecture or an independent research project.
The second paper, “Determining Sets for the Discrete Laplacian,” by Aviad Rubinstein, Jacob Rubinstein, and Gershon Wolansky, adds a nice twist to the idea of solving a discrete approximation for Laplace’s equation which yields a discrete analogue of a harmonic function. A continuous harmonic function $u (x, y)$ is determined by the mean‐value property; that is, its value at a point is equal to the average of the values of the function over the set of points a fixed distance away. A discrete harmonic function defined on a lattice, $u (i, j)$, has the same property, except the average is now taken over the nearest neighbors. It is well known in both cases that specifying the function on the boundary of, say, a square domain uniquely specifies the function. However, the discrete problem allows a fascinating generalization: can we specify the function at some other set of points which still determines the function uniquely? The answer is yes, and this idea has a natural application in a phase detection problem that comes out of optics. This paper nicely ties to problems that arise in PDEs, numerical methods, and linear algebra with a soupçon of physical applications, yet is written at a level accessible to any student with a solid understanding of linear algebra and has enough open questions to be an ideal springboard for undergraduate research.
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Fundamental Solutions for Some Partial Differential Operators from Fluid Dynamics and Statistical Physics

Jorge Aarão

SIAM Rev. 49, pp. 303-314 (12 pages)

Online Publication Date: May 01, 2007

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We present a method to find fundamental solutions for a class of partial differential equations that often arise in fluid dynamics and in transport problems. The method is elementary in the sense that it uses only linear algebra and ODEs.
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Determining Sets for the Discrete Laplacian

Aviad Rubinstein, Jacob Rubinstein, and Gershon Wolansky

SIAM Rev. 49, pp. 315-324 (10 pages)

Online Publication Date: May 01, 2007

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We define a notion of determining sets for the discrete Laplacian in a domain Ω. A set $D$ is called determining if harmonic functions are uniquely determined by providing their values on $D$, and if $D$ has the same size as the boundary of Ω. It is shown that there exist determining sets that are fairly evenly distributed in Ω. A number of basic properties of determining sets are derived.

Book Reviews

Bob O’Malley, Section Editor

SIAM Rev. 49, pp. 327-367 (41 pages)

Online Publication Date: April 30, 2007

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I’m very grateful to board member Arieh Iserles for writing a featured review of Donald Knuth’s latest fascicle of Volume 4 of The Art of Computer Programming: Generating All Trees. Arieh wrote this on short notice, since the designated reviewer of another important feature selection failed to deliver. Though book reviewers, no doubt for a variety of good reasons, sometimes don’t do their job in a timely manner, this is the first time in memory that a featured review didn’t materialize, even through extended deadlines. The situation makes us realize how fortunate we are to have so many loyal and conscientious reviewers. Thanks very much to all of you.
In addition to combinatorics, this issue features books in mathematical biology, software, fluid and solid mechanics, probability and statistics, differential equations, pure and applied geometry, analysis, and even the history of mathematics.
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