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
Previous Issue Next Issue

1999

Volume 41, Issue 2, pp. 197-373


Survey and Review

Nick Trefethen, Section Editor

SIAM Rev. 41, pp. 197-197 (1 page)

Online Publication Date: August 04, 2006

Full Text: | Download PDF

multimedia

Show Abstract
The two Survey and Review articles in this issue illustrate that applied mathematics comes in extraordinarily varied flavors.
James Sethian's article, "Fast marching methods," has the style of algorithmic mathematics. We are given a class of geometrical problems and asked, how can they be solved quickly and accurately? The article presents a collection of related algorithms, including a new method for achieving second-order accuracy even near a line of discontinuous derivatives. Readers might want to preview the article by looking at the remarkable collection of applications in section 6, including shape-offsetting, photolithographic development, seismic migration, geodesic paths on surfaces, motion planning, and scene visibility calculations. See section 4.3, especially Figure 14, for an explanation of how fast marching methods, the subject of this article, differ from level set methods, which Sethian has developed extensively and helped to make famous in his book Level Set Methods. These two methods form the core of an expanded new book from Cambridge University Press entitled Level Set Methods and Fast Marching Methods.
The article by Michael Berry and Jonathan Keating, "The Riemann zeros and eigenvalue asymptotics," has the flavor of physics. Some mathematicians may feel uneasy with this kind of presentation that avoids theorems, focusing on asymptotic formulas whose convergence properties are not always known. Others may feel that after an article like this, any other style seems thin. We urge the reader to remember that discovery often precedes rigor and to enjoy the ride! Berry and Keating believe that there are deep connections between the zeros of the Riemann zeta function, orbits of Hamiltonian systems, and eigenvalues of random matrices. They make a compelling case that this is not just an analogy at a philosophical level but a close agreement of numerical quantities. Look, for example, at Figures 2, 3, or 6. Here are partial sums of divergent series that capture the detailed behavior of the Riemann zeros better than you know your own driveway. This article is dense, filled with fruits of an extraordinary decade of discoveries by Berry and Keating and their colleagues.
The four Survey and Review articles published so far in the "new" SIAM Review have come from the areas of probability, AIDS modeling, computational algorithms, and the physics of number theory. This diversity of topics and styles will continue in the issues ahead, and we encourage readers to contact the Section Editor at any time with suggestions of topics or authors to be sought out for future volumes, or with proposals to contribute manuscripts of their own.

Fast Marching Methods

J. A. Sethian

SIAM Rev. 41, pp. 199-235 (37 pages) | Cited 142 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations. Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas.

The Riemann Zeros and Eigenvalue Asymptotics

M. V. Berry and J. P. Keating

SIAM Rev. 41, pp. 236-266 (31 pages) | Cited 21 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 41, pp. 267-268 (2 pages)

Online Publication Date: August 04, 2006

Full Text: | Download PDF

multimedia

Show Abstract
Our second installment of Problems and Techniques features articles on the controllability of the Navier--Stokes equations and the parallel partitioning of irregular graphs.
Fernández-Cara surveys theoretical results on controlling the incompressible Navier--Stokes equations. Needless to say, flow control has important applications in several areas, including reducing turbulence and enhancing the performance of flight vehicles or turbo-machinery, or increasing the efficiency of flow-dominated manufacturing processes.
Motivated by a conjecture of J. L. Lions, Fernández-Cara examines approximate and null controllability. The former refers tofinding a control function such that the fluid's velocity at a given time is (arbitrarily) close to a prescribed velocity. The latter implies the existence of a control such that the velocity vanishes at a given time. Lions conjectured the approximate controllability of the Navier--Stokes equations, but this and null controllability remain unanswered in general situations. Fernández-Cara reviews known results and also examines the controllability of finite-dimensional approximations of the Navier-Stokes equations through Galerkin's method. Nice analyses of some very difficult problems are presented.
In our second article, Karypis and Kumar present a multilevel strategy for partitioning irregular graphs. Thus, they seek to divide a graph into a number of subgraphs having the same number of vertices such that the number of edges connecting vertices in the different subgraphs is minimal. This problem has numerous applications in parallel scientific computation including the solution of sparse linear algebraic systems and the load balancing of unstructured-grid finite element computations. With this latter application, which is near and dear to my heart, elements may comprise graph vertices while connections of elements through common faces correspond to edges.
The graph partitioning problem is NP-complete, so various approximation techniques have been developed, including graph-theoretic strategies and partitionings based on spectral and geometric decompositions. From a software perspective, Hendrickson and Leland set the stage with their CHACO system. Many approaches involve a multilevel strategy that coarsens the original graph until it can easily be partitioned and then projects these partitionings onto finer graphs. This is akin to the popular multigrid strategies for iterative solutions of linear algebraic systems. Multilevel strategies are central to the popular and very successful METIS system that was developed by our authors.
So what's the problem? Most partitioning algorithms execute in serial mode. This is often acceptable; however, serial partitioning may not be possible for very large problems and it certainly would not be possible for algorithms such as adaptive finite element computation, which typically require a dynamic re-partitioning with each mesh or method alteration. Hence there is a need for parallel partitioning procedures, which are the subject of the article. Coarsening, partitioning, and prolongation procedures are described and applied to a suite of graphs obtained from finite element computation. These procedures have good speed-ups and efficiencies close to the best serial procedures. The Karypis and Kumar article should be of interest to both discrete mathematicians and computational scientists.
Please join me in thanking our authors, Enrique Fernández-Cara, George Karypis, and Vipin Kumar, for providing two excellent articles on subjects of widespread interest.
We're always eager to receive exciting articles on recent mathematical developments, so please send them our way. We would also be happy to have input on subject matter that you would like to see in the section.

On the Approximate and Null Controllability of the Navier--Stokes Equations

Enrique Fernández-Cara

SIAM Rev. 41, pp. 269-277 (9 pages) | Cited 5 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
This paper presents some known results on the approximate and null controllability of the Navier--Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.

Parallel Multilevel series k-Way Partitioning Scheme for Irregular Graphs

George Karypis and Vipin Kumar

SIAM Rev. 41, pp. 278-300 (23 pages) | Cited 25 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
In this paper we present a parallel formulation of a multilevel k-way graph partitioning algorithm. A key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k-way partitioning algorithm. In particular, the time taken by our parallel graph partitioning algorithm is only slightly longer than the time taken for re-arrangement of the graph among processors according to the new partition. Experiments with a variety of finite element graphs show that our parallel formulation produces high-quality partitionings in a short amount of time. For example, a 128-way partitioning of graphs with one million vertices can be computed in a little over two seconds on a 128-processor Cray T3D. Furthermore, the quality of the partitions produced is comparable (edge-cuts within 5%) to those produced by the serial multilevel k-way algorithm. Thus our parallel algorithm makes it feasible to perform frequent repartitioning of graphs in dynamic computations without compromising the partitioning quality.

SIGEST

SIAM Rev. 41, pp. 301-301 (1 page)

Online Publication Date: August 04, 2006

Full Text: | Download PDF

multimedia

Show Abstract
The SIGEST paper in this issue, ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,' by Peter Shor, first appeared in late 1997 in SIAM Journal on Computing, volume 26. This paper, a landmark in the field of quantum computation, describes the first significant algorithms for a hypothetical quantum computer. Since his presentation of these algorithms in 1994 at the Symposium on Foundations of Computer Science (FOCS), there has been an explosion of interest in quantum computing, with Shor the leading figure. His contributions to this area have been widely honored; in the summer of 1998, he received the prestigious Nevanlinna Award for outstanding work in computer science by a researcher under the age of 40. The paper is a masterpiece of clear exposition, creative mathematics, and fascinating insights. For the paper's original appearance, Shor devoted particular attention to making the introduction understandable by a broad scientific audience. He has extensivelyupdated and expanded the paper for the SIGEST version to describe the content and implications of more recent work; the SIAM Review editors are grateful to him for this extra effort. We hope that our readers will savor the experience of reading the seminal paper in quantum computing.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

Peter W. Shor

SIAM Rev. 41, pp. 303-332 (30 pages) | Cited 43 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

Education

Bobby Schnabel, Section Editor

SIAM Rev. 41, pp. 333-333 (1 page)

Online Publication Date: August 04, 2006

Full Text: | Download PDF

multimedia

Show Abstract
The Education section is intended to provide modules that teachers and students of applied mathematics and scientific computation can use directly in studying these fields, in courses and beyond. These modules are likely to fall into one or more of four broad categories: applications, special topics, software, and historical modules. The first issue of this section contained one paper describing a Web-based software package and a second discussing a modern special topic in applied mathematics. In this, the second issue, there are two different types of papers.
Michael Berry (no relation to the Sir Michael Berry of the article with Keating earlier in this issue), Zlatko Drmac, and Liz Jessup describe the application of linear algebra to problems in information retrieval. In particular, they highlight an important recent technique in information retrieval called latent semantic indexing that itself draws heavily upon linear algebra techniques including the singular value decomposition. The early parts of the paper can serve as a very nice motivating example for students and teachers in both linear algebra and numerical computation courses, while the more detailed and advanced parts of the paper can constitute a unit in a graduate or advanced undergraduate course in numerical linear algebra that combines real applications with serious numerical mathematics.
P. Broadbridge, G.R. Fulford, N.D. Fowkes, D.T.C. Chan, and C. Lassig provide a simple yet compelling use of mathematical modeling to understand a very practical problem, the appearance of bubbles in gummed wine labels. Using techniques that are understandable by undergraduates and a delightful writing style, they demonstrate the insight provided by mathematical modeling on a problem that is simple to describe yet perplexing to explain. The paper will make an excellent module in differential equation and mathematical modeling courses and can even be used as a motivating example in earlier courses. Both articles are written in the style that is meant to be the standard for this section: informal, easy to understand, and written to students.
Unsolicited submissions form the primary source of potential papers for the Education section, although the editorial board also will recruit submissions when it is aware of good prospects. The goals and style of this section are somewhat different than is standard for academic journals and even for educationally-oriented papers in the previous version of SIAM Review. For these reasons, prospective authors are strongly encouraged to consult the section guidelines, which were published in this preface in the previous issue of the journal and are available on the web at http://www.siam.org/journals/sirev/revguide.htm. Prospective authors are also welcome to contact the Section Editor or Associate Editors with ideas and questions about planned submissions before making a formal submission.
FREE

Matrices, Vector Spaces, and Information Retrieval

Michael W. Berry, Zlatko Drmac, and Elizabeth R. Jessup

SIAM Rev. 41, pp. 335-362 (28 pages) | Cited 58 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
The evolution of digital libraries and the Internet has dramatically transformed the processing, storage, and retrieval of information. Efforts to digitize text, images, video, and audio now consume a substantial portion of both academic and industrial activity. Even when there is no shortage of textual materials on a particular topic, procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking. Recently developed information retrieval technologies are based on the concept of a vector space. Data are modeled as a matrix, and a user's query of the database is represented as a vector. Relevant documents in the database are then identified via simple vector operations. Orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself. The purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections.
FREE

Bubbles in Wet, Gummed Wine Labels

P. Broadbridge, G. R. Fulford, N. D. Fowkes, D. Y. C. Chan, and C. Lassig

SIAM Rev. 41, pp. 363-372 (10 pages) | Cited 3 times

Online Publication Date: August 04, 2006

Full Text: | Download PDF

Show Abstract
It is shown that bubbling on wine bottle labels is due to absorption of water from the glue, with subsequent hygroscopic expansion. Contrary to popular belief, most of the glue's water must be lost to the atmosphere rather than to the paper. A simple lubrication model is developed for spreading glue piles in the pressure chamber of the labelling machine. This model predicts a maximum rate for application of labels. Buckling theory shows that the current arrangement of periodic glue strips can indeed accommodate paper expansion. This project provides interesting applications of various areas of undergraduate mathematics, such as trigonometry, Maclaurin series, dimensional analysis, and fluid mechanics. It illustrates that simple mathematical modelling may provide insight into a complicated real-world problem.

Book Reviews

R. Bruce Kellogg, Section Editor

SIAM Rev. 41, pp. 373-373 (1 page)

Online Publication Date: August 04, 2006

Full Text: | Download PDF

multimedia

Show Abstract
The related subjects of dynamical systems and chaos have received great attention in recent years. Not only have there been frequent (occasionally sensational) articles in the popular press, but there have also been major advances in mathematical theory and numerical algorithms, along with important applications in science, engineering, medicine, and communications. The combination of wide interest and lively research has produced a number of books on dynamical systems and chaos, some of which are suitable as textbooks. In the book review essay for this issue, Professor Hogan discusses books that can be used to put together a course on these subjects that appeals to students both within and outside mathematics departments.
Close

Your Recent History Recent History Help

Recently Viewed

  1. The FFT as a Multigrid Algorithm
  2. Random Number Generators on Vector Supercomputers and Other Advanced Architectures
  3. Ill-Conditioning and Computational Error in Interior Methods for Nonlinear Programming
  4. Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions
  5. Maximum Block Improvement and Polynomial Optimization
© 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