SIAM Digital Library
 
 
 

SIAM Review

BROWSE VOLUMES

Year Range: 

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS

2012

Volume 54, Issue 1, pp. 1-208

FREE

Survey and Review

Desmond J. Higham, Section Editor

SIAM Rev. 54, pp. 1-1 (1 page)

Full Text: | Download PDF

Show Abstract
The first article of this section, “Thin-Layer Solutions of the Helmholtz and Related Equations,” by Ockendon and Tew, is firmly rooted in the customs of applied mathematical analysis. The survey considers four intimately related elliptic PDEs: Helmholtz, modified Helmholtz, convection-diffusion, and complex Helmholtz. The focus is on singularly perturbed solutions in two dimensions that arise in the asymptotic limit where the Laplacian term vanishes. This work was “compiled over the past six years” but reflects many more years of experience gained by the authors in the arts of matched asymptotic expansions, the Wentzel–Kramers–Brillouin (WKB) method, and multiple scales. Here you will find a catalogue of interrelated asymptotic thin-layer solutions that apply when there are rapid transverse variations in one dependent variable. The authors comment authoritatively on the relative merits of the asymptotic tools of the trade and show how phenomena associated with one PDE can be translated across to the others, creating a valuable resource in the best traditions of applied mathematics.
The second article, “Modeling Growth in Biological Materials,” by Jones and Chapman, echoes the first in (a) having an author from the Oxford Mathematical Institute and (b) exploiting the presence of small parameters wherever possible. But it has a different, application-oriented flavor, focusing on mathematical models of tissue growth. This is a field where mathematical advances can considerably improve both biological knowledge and therapeutics. The survey covers a great deal of ground, quoting experimental results on tennis players' bones and rabbits' lungs, touching on some classic work of D'Arcy Thompson and John Conway, and drawing together a range of complementary modeling approaches. Using surprisingly few equations, it gives equal weight to the two broad themes of macro and micro.
The continuum, macroscale models reviewed by Jones and Chapman use the familiar applied mathematical tools of kinematics, mechanics, and constitutive laws to produce PDEs for bulk behavior. At the other end of the scale, they look at the idea of starting with individual cells and introducing rules that govern their interactions. Such cell-based models can be more flexible and easier to calibrate and evaluate, but are less amenable to analysis. Their use is also limited by the availability of computing power. It is natural to ask whether these two distinct approaches could be combined into a single unified theory. Is there a modeling journey that begins with precise micro-level interactions, moves through one or more multiscale layers, and emerges with a macroscale description? This philosophy of putting together rather than taking apart, which perhaps comes more naturally to applied mathematicians than experimental life scientists, has been a driving force behind a range of high-profile activities in the area of systems biology. However, the authors explain here that even in the well-defined field of tissue growth, rigorous application of the underlying mathematical principles of homogenization and localization are far from trivial; many inconsistencies need to be ironed out and fundamental challenges remain.

Thin-Layer Solutions of the Helmholtz and Related Equations

J. R. Ockendon and R. H. Tew

SIAM Rev. 54, pp. 3-51 (49 pages)

Online Publication Date: February 08, 2012

Full Text: | Download PDF

Show Abstract
This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection–diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.

Modeling Growth in Biological Materials

Gareth Wyn Jones and S. Jonathan Chapman

SIAM Rev. 54, pp. 52-118 (67 pages)

Online Publication Date: February 08, 2012

Full Text: | Download PDF

Show Abstract
The biomechanical modeling of growing tissues has recently become an area of intense interest. In particular, the interplay between growth patterns and mechanical stress is of great importance, with possible applications to arterial mechanics, embryo morphogenesis, tumor development, and bone remodeling. This review aims to give an overview of the theories that have been used to model these phenomena, categorized according to whether the tissue is considered as a continuum object or a collection of cells. Among the continuum models discussed is the deformation gradient decomposition method, which allows a residual stress field to develop from an incompatible growth field. The cell-based models are further subdivided into cellular automata, center-dynamics, and vertex-dynamics models. Of these the second two are considered in more detail, especially with regard to their treatment of cell–cell interactions and cell division. The review concludes by assessing the prospects for reconciliation between these two fundamentally different approaches to tissue growth, and by identifying possible avenues for further research.
FREE

Research Spotlights

Ilse Ipsen, Section Editor

SIAM Rev. 54, pp. 119-119 (1 page)

Full Text: | Download PDF

Show Abstract
A new section with a new direction. Research Spotlights replaced Expository Research Papers on January 31, 2012.
Papers in Research Spotlights should cover topics in applied and computational mathematics of particularly wide interest and importance. Contributions must also be accessible to the broad and diverse SIAM Review readership. The principal theme of Research Spotlights is author flexibility with the intent that an expanded format promotes creativity and spurs innovative articles. Authors now have latitude in terms of considering a standard research paper or else a nontraditional article such as a mini-survey, a timely communication, a software description, or a new mathematical perspective within an application area.
Prospective authors are encouraged to first consult with the section editor Ray Tuminaro (rstumin@sandia.gov) about potential contributions—especially those that lie outside the scope of traditional SIAM articles. While they can have a more relaxed format, ultimately articles must be of broad interest, must be accessible to the community, and must meet the technical review standards of SIAM journals.
Focus groups provide feedback on potential new products: Which juice is more appealing to you, the slimy green or the yucky yellow one? Systems of sensors process signals to decide: Is there an intruder or not? These are examples of “group decision making,” a process where individuals work together to make a collective decision. The problem of figuring out how the collective arrives at a decision occurs in areas as varied as cognitive psychology, economics, political science, and signal processing.
Margot Kimura and Jeff Moehlis in their paper “Group Decision-Making Models for Sequential Tasks” consider the “two-alternative forced-choice test,” where one must choose between two hypotheses: slimy green or yucky yellow; intruder or no intruder. Decisions must be made quickly and can only tolerate certain error rates. This means that there are limits on how often the sensors are allowed to miss an intruder, or signal a false alarm.
The model in this paper assumes $N$ independent decision makers, each of whom receives observations, sequentially, one at a time. Each decision maker continues to process the observations until s/he is able to make a decision. The incoming observations are represented by independent random variables, with known prior probabilities for each decision. The processing consists of applying the “sequential probability ratio test” to each new observation. Based on the prespecified error rates for the number of misses and false alarms, this test either reports a decision or continues to process the next observation. The authors also consider a continuous version of this test, which becomes a drift-diffusion model as the time between observations goes to zero.
Once a decision maker has come up with a decision, s/he reports it to the “fusion center,” which is responsible for arriving at a collective decision. The fusion center can operate in one of three modes: race (report only the very first decision that arrives), majority-total (wait until all $N$ decisions have arrived and then report the majority), and majority-first (wait until $N/2$ identical decisions have arrived, and then report this smallest possible majority).
For each such mode, the authors derive probability distribution functions for the collective error rates and decision times, from the error rates and decision times of the individual decision makers. Simulations are presented to compare the different modes. Which mode turns out to be the most efficient is not at all obvious and depends on the scenario at hand, whether decision makers can have different error rates or can malfunction. Finally, the authors extend their analysis to more general modes, where the fusion center makes a collective decision based on the first $\eta$ decisions that arrive.
The approach presented here has many advantages. It is general and applies to many situations that require collective decision making based on sequential observations, including even “cybernetic groups” with human observers and nonhuman detectors. Furthermore it is systematic and elegant, because it provides a clear path for deriving the efficiency of collective decisions from those of individual decision makers. Especially appreciated is a list of acronyms thoughtfully included by the authors at the beginning of the paper, which makes it easy to decipher the many acronyms in this area.

Group Decision-Making Models for Sequential Tasks

Margot Kimura and Jeff Moehlis

SIAM Rev. 54, pp. 121-138 (18 pages)

Online Publication Date: February 08, 2012

Full Text: | Download PDF

Show Abstract
The sequential probability ratio test (SPRT) and related drift-diffusion model (DDM) are optimal for choosing between two hypotheses using the minimal (average) number of samples and relevant for modeling the decision-making process in human observers. This work extends these models to group decision making. Previous works have focused almost exclusively on group accuracy; here, we explicitly address group decision time. First, we derive explicit solutions for the error rate and probability distribution function of decision times for a group of independent, (possibly) nonidentical decision makers using one of three simple rules: Race, Majority Total, and Majority First. We illustrate our solutions with a group of $N$ i.i.d. decision makers who each make an individual decision using the SPRT-based DDM, then compare the performance of each group rule under different constraints. We then generalize these group rules to the $\eta$-Total and $\eta$-First schemes, to demonstrate the flexibility and power of our approach in characterizing the performance of a group, given the performance of its individual members.
FREE

SIGEST

The Editors

SIAM Rev. 54, pp. 139-139 (1 page)

Full Text: | Download PDF

Show Abstract
Alcuin of York was a famous scholar, monk, and teacher in the late eighth century. He was the head of the emperor Charlemagne's Palace School and developed a precursor of our modern alphabet. Alcuin died in 804 A.D. while serving as the Abbot of St. Martin of Tours.
This issue's SIGEST paper “The Alcuin Number of a Graph and Its Connections to the Vertex Cover Number” by Péter Csorba, Cor A. J. Hurkens, and Gerhard J. Woeginger, from the SIAM Journal on Discrete Mathematics, takes its title from a problem from one of his books:
A man had to transport to the far side of a river a wolf, a goat, and a bundle of cabbages. The only boat he could find was one which would carry only himself and one of them. For that reason he sought a plan which would enable them all to get to the far side unhurt. Let he who is able say how it could be possible to transport them safely.
The authors have done some research on the history of this problem, and report that similar problems are in folklore worldwide.
Most readers of SIAM Review should be able to figure this one out pretty readily. However, as the authors of the paper happily point out, this problem is an early example of a reachability problem, a kind of combinatorial optimization problem. In the 1200 years since, many people have studied reachability problems. This paper resolves several open questions and poses some new ones.
Reachability problems are scheduling problems. The paper lists examples from computer science, operations research, and, of course, puzzles. In the terms of Alcuin's problem, one has a set of objects, some of which cannot be transported together, and a boat of finite capacity. The Alcuin number $A$ is the smallest possible capacity for which there is a feasible schedule. In Alcuin's problem, the number is 1 (so the problem, as originally posed, has a solution).
The mathematical formulation expresses the problem in terms of a graph, so the objective is to understand the computational complexity of computing $A(G)$, where $G$ is the graph of the problem. Among the new results are a new proof that computing the Alcuin number is NP-hard and a new bound on the length of a feasible schedule. Even a novice can appreciate the very well written introduction to the paper, and may be motivated to attack some of Alcuin's other problems, such as the problem of the three jealous husbands (Google it!).

The Alcuin Number of a Graph and Its Connections to the Vertex Cover Number

Péter Csorba, Cor A. J. Hurkens, and Gerhard J. Woeginger

SIAM Rev. 54, pp. 141-154 (14 pages)

Online Publication Date: February 08, 2012

Full Text: | Download PDF

Show Abstract
We consider a planning problem that generalizes Alcuin's river crossing problem to scenarios with arbitrary conflict graphs. This generalization leads to the so-called Alcuin number of the underlying conflict graph. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around the Alcuin number. Our technical main result is an NP-certificate for the Alcuin number. It turns out that the Alcuin number of a graph is closely related to the size of a minimum vertex cover in the graph, and we unravel several surprising connections between these two graph parameters. We provide hardness results and a fixed parameter tractability result for computing the Alcuin number. Furthermore we demonstrate that the Alcuin number of chordal graphs, bipartite graphs, and planar graphs is substantially easier to analyze than the Alcuin number of general graphs.
FREE

Education

Louis F. Rossi, Section Editor

SIAM Rev. 54, pp. 155-155 (1 page)

Full Text: | Download PDF

Show Abstract
Back in the late 80s when I learned to fly airplanes, my instructor required all the student pilots to fly without the benefit of noise attenuating headsets. In the air, one of the first lessons I learned was that the sound or spectrum of the engine tells a story. If one listens carefully, it can tell us if the mixture needs to be adjusted or if a ring seal is worn loose. Indeed, you can hear many features of an engine, a fact that is crudely implemented with automatic sensors in modern automobile engines. Of course, this type of deduction is not unique to engines, and deducing the features of systems from their spectrum gives rise to a wealth of wonderful mathematical questions known as inverse problems. In the current Education installment, authors Steven Cox, Mark Embree, and Jeffrey Hokanson explore a fascinating inverse problem arising from a vibrating string with beads attached to it, and it is their laboratory apparatus, a monochord, which is featured on the cover of this issue.
In the broad field of inverse problems, there are a host of “Can you...?” questions. Perhaps the most familiar of these is “Can you hear the shape of a drum?” The answer was proven to be “no” via counterexample, and the quest to resolve the problem and many of its offshoots led to a large body of excellent mathematics. In this issue's feature, the authors tipped their hand by choosing the title “One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem.” Specifically, the authors chose an intriguing inverse problem consisting of beads threaded through a massless string of known length under high tension. If we pluck this system, can we use its spectrum to determine masses and positions of the beads?
This issue's module provides the instructor or the avid student with all four pillars of applied mathematics: modeling, analysis, computation, and experimentation, though the main thrust is on the latter three. The authors discuss the model and then carefully describe their monochord apparatus for exploring this system experimentally. After collecting displacement data from a single point on the monochord, they use a short MATLAB script to verify that indeed the model captures the essential behavior of the system. The hard part is going the other way: Can we map the eigenvalues of the time series back to bead positions and masses? The authors have already told us that we can, and I will not spoil your fun by revealing too many details, except to say that the authors exploit connections between orthogonal polynomials and the characteristic polynomials of symmetric tridiagonal matrices. The resulting algorithm is robust and, as the results will attest, effective. The article is perfect for advanced undergraduates or graduate students, and it might even inspire some to build a monochord ... or at least turn down the car stereo and listen to the engine from time to time.
FREE

One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem

Steven J. Cox, Mark Embree, and Jeffrey M. Hokanson

SIAM Rev. 54, pp. 157-178 (22 pages)

Online Publication Date: February 08, 2012

Full Text: | Download PDF

Show Abstract
To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical finite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem for which a variety of elegant algorithms exist. After presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates, we confirm its efficacy through physical experiment. We encourage readers to conduct their own explorations using the numerous data sets we provide.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 54, pp. 181-208 (28 pages)

Full Text: | Download PDF

Show Abstract
Even long-time readers of SIAM Review may wonder what a featured review is. I, too, sometimes wonder. Occasionally, two or three related books are reviewed simultaneously. More often, the objective is to highlight an important new book or topic. It takes planning to arrange such a review for each issue. Sometimes other deserving books appear in the meantime or a lovely review or two will be otherwise submitted.
For this issue, I asked Jim Simmonds to write a featured review of Audoly and Pomeau's Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells. As reported, it features very novel applications and a remarkable combination of physical insight and analytical adeptness. Roy's brilliant new monograph on Sources in the Development of Mathematics appears to be so central to our understanding of calculus that I couldn't pass up the opportunity to highlight it as well.
The issue also includes reviews by long-time favorite authors on such diverse subjects as boundary elements, combinatorics, classical analysis, discrete integrable systems, dynamical systems, estimation, fluid mechanics, quantum mechanics, and probability and stochastics. As always, our expert reviewers deserve our thanks and attention.
Close

© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close