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

2007

Volume 49, Issue 4, pp. 543-732

FREE

Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 49, pp. 543-543 (1 page)

Full Text: | Download PDF

Show Abstract
Imagine you had two lives—Dr. Jekyll and Mr. Hyde—and you can switch between these two personas. The question is, “Will your life ever be out of control no matter how you switch back and forth?” The answer, in the case of the character from the well-known story by Robert Louis Stevenson, is “no.” This issue's Survey and Review article is about such a dynamical system.
In the article “Stability Criteria for Switched and Hybrid Systems,” the authors consider a system which can switch between several dynamical states. In a control application, one could imagine driving such a system to a desired state by transitions which follow a control law. The main question investigated in the paper is stability. That is, independent of how you switch from mode to mode, is the trajectory of the system stable?
The authors consider the case where each of the modes is described by a linear autonomous system. Even if the individual modes are exponentially stable, the stability criterion for the switched system is surprisingly subtle and difficult. The analysis relies on the existence of a certain Lyapunov function which can be verified algebraically in some special cases.
Another surprising fact about hybrid systems, discussed in the present review, is that it is possible to devise a switching rule that stabilizes a hybrid system where the individual dynamical mode is unstable. So, perhaps there is hope for Dr. Jekyll and Mr. Hyde after all.

Stability Criteria for Switched and Hybrid Systems

Robert Shorten, Fabian Wirth, Oliver Mason, Kai Wulff, and Christopher King

SIAM Rev. 49, pp. 545-592 (48 pages) | Cited 9 times

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
FREE

Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 49, pp. 593-593 (1 page)

Full Text: | Download PDF

Show Abstract
In this issue, the Problems and Techniques section features papers concerned with parallel matrix vector multiplication, polynomial interpolation, and propagation of electromagnetic waves.
The first paper, “Revisiting Hypergraph Models for Sparse Matrix Partitioning” by Bora Uçar and Cevdet Aykanat, is concerned with parallelizing matrix vector multiplication $y=Ax$, where the matrix $A$ is sparse and rectangular. The authors propose a hypergraph model to describe dependences among the data and model communication among processors.
In the hypergraph model, the elements of $x$ and $y$ and the rows of $A$ are represented by vertices of a graph. If element $a_{i,j}$ of $A$ is nonzero, then the multiplication $a_{ij}x_j$ contributes a nonzero summand and requires $x_j$ to be present in the processor holding row $i$. This dependence is accounted for with an undirected edge connecting $x_j$ with row $i$ of $A$. A so-called net collects all dependences for a vertex: For instance, the net $n_x(j)$ contains vertex $x_j$ as well as all rows of $A$ that have an edge with $x_j$. The problem of minimizing communication among processors can now be formulated as allocating the vertices to processors, so that each net has its vertices spread over as few processors as possible. The authors illustrate the effectiveness of hypergraph models when constraints are put on the allocation of the data to processors.
When the German mathematician, physicist, and spectroscopist Carl David Tolmé Runge (1856–1927) interpolated the function $f(x)=1/(1+25x^2)$ at $n+1$ equally spaced points in the interval $[-1,1]$ by a polynomial of degree $n$, he made a startling observation: With increasing degree $n$, the polynomials approximate $f$ less accurately—instead of, as one would have hoped, more accurately. Although the polynomials agree with $f$ at the interpolation points, they oscillate between the interpolation points, and the oscillations worsen as the polynomial degree grows. Thence polynomial interpolation at equally spaced points fell into disrepute.
It turns out, however, that interpolation at equally spaced points cannot be avoided altogether. It is necessary, for instance, in computational fluid dynamics, when one has to solve hyperbolic partial differential equations. Moreover, the interpolant may also have to retain the positivity, monotonicity, and boundedness of the underlying function. How to meet these demands in the face of Runge's phenomenon? This is the subject of Martin Berzins's paper “Adaptive Polynomial Interpolation on Evenly Spaced Meshes.”
The third paper, “Uniform Asymptotics Applied to Ultrawideband Pulse Propagation” by Natalie Cartwright and Kurt Oughstun, deals with a problem that was first studied by the German theoretical physicist Arnold Sommerfeld (1868–1951) and his student, the French physicist Léon Brillouin (1889–1969): the propagation of electromagnetic waves in dispersive materials. The problem at hand is complicated by the fact that the pulse is an ultrawide band consisting of a wide range of frequencies. The propagation of the pulse is represented as a complex integral, and the authors derive a continuous asymptotic expansion of this integral by applying a combination of saddle point methods.
FREE

Revisiting Hypergraph Models for Sparse Matrix Partitioning

Bora Uçar and Cevdet Aykanat

SIAM Rev. 49, pp. 595-603 (9 pages) | Cited 3 times

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
We provide an exposition of hypergraph models for parallelizing sparse matrix-vector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for the parallel matrix-vector multiply based on one-dimensional (1D) matrix partitioning. In the elementary model, the vertices represent the data of a matrix-vector multiply, and the nets encode dependencies among the data. We then apply a recently proposed hypergraph transformation operation to devise models for 1D sparse matrix partitioning. The resulting 1D partitioning models are equivalent to the previously proposed computational hypergraph models and are not meant to be replacements for them. Nevertheless, the new models give us insights into the previous ones and help us explain a subtle requirement, known as the consistency condition, of hypergraph partitioning models. Later, we demonstrate the flexibility of the elementary model on a few 1D partitioning problems that are hard to solve using the previously proposed models. We also discuss extensions of the proposed elementary model to two-dimensional matrix partitioning.
FREE

Adaptive Polynomial Interpolation on Evenly Spaced Meshes

M. Berzins

SIAM Rev. 49, pp. 604-627 (24 pages) | Cited 1 time

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches the oscillations may be contained and the resulting polynomials are data-bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each subinterval. Computational results for a number of challenging functions including a number of problems similar to Runge's function with as many as 511 points per interval are shown.

Uniform Asymptotics Applied to Ultrawideband Pulse Propagation

Natalie A. Cartwright and Kurt E. Oughstun

SIAM Rev. 49, pp. 628-648 (21 pages) | Cited 3 times

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
A canonical problem of central importance in the theory of ultrawideband pulse propagation through temporally dispersive, absorptive materials is the propagation of a Heaviside step-function signal through a medium that exhibits anomalous dispersion. This problem is rich in the use of asymptotic theory. Sommerfeld and Brillouin provided the first (qualitatively accurate but quantitatively inaccurate) closed-form approximations of the dynamic evolution of this waveform through a single-resonance Lorentz model dielectric based upon Debye's method of steepest descent. An improved approximation has since been provided by Oughstun and Sherman using modern, uniform asymptotic methods that rely upon the saddle-point method. An accurate, uniform asymptotic approximation describing the dynamical evolution of the unit step-function modulated sine wave signal through a single-resonance Lorentz model dielectric is presented here based upon their work. This refined asymptotic description results in a continuous evolution of the propagated field for all space-time points.
FREE

SIGEST

The Editors

SIAM Rev. 49, pp. 649-649 (1 page)

Full Text: | Download PDF

Show Abstract
This issue's SIGEST paper, from the SIAM Journal on Optimization (SIOPT), is an excellent example of deep mathematical analysis, involving a range of mathematical techniques, which is having significant practical algorithmic and computational consequences.
The title of the paper, “A Sum of Squares Approximation of Nonnegative Polynomials,” by Jean B. Lasserre, combines two concepts that are known to all SIAM readers: sums of squares, the types of functions that arise in many data fitting and other applications; and nonnegative polynomials, polynomials whose values are greater than or equal to zero for all inputs. It is obvious that any polynomial that can be written as a sum of squares is nonnegative. It is far from obvious whether any nonnegative polynomial has a representation as, or arbitrarily close to, a sum of squares of polynomials. In this SIGEST paper, Lasserre shows that this is the case. The nomination from the editors of SIOPT captures the immediate and more general contributions of this paper eloquently, in the context of a series of three papers of which this is the second:
“Jean Lasserre, in a series of papers (starting from the first paper in SIOPT Vol. 11, pp. 796–817, and concluding with the third paper in SIOPT Vol. 15, pp. 383–393) by exploiting deep results from real algebraic geometry and theory of moments for multivariate polynomials, showed that the optimal value of a nonconvex polynomial optimization problem (i.e., an optimization problem whose objective function and constraints are specified by multivariate polynomials) can asymptotically be approximated by an increasing sequence of optimal values from increasing tighter semidefinite programming relaxation problems. Even though the result is asymptotic in the most general case, fortuitously, practical experience has revealed that very often, the first few levels of semidefinite programming relaxations typically can achieve the global optimum value.
The theoretical connection that Jean Lasserre (and Pablo Parrilo at about the same time from the viewpoint of sum of squares representations of multivariate polynomials) had established between polynomial optimizations with real algebraic geometry and theory of moments has led to exciting developments in combinatorial optimization. In particular, many hard combinatorial problems such as the maximum clique problem of a graph and the quadratic assignment problem are now actively revisited by researchers under the framework of polynomial optimization to derive increasingly tighter bounds.”
That is, a consequence of this and related work is the ability to use a sequence of semidefinite programming problems to (nearly) solve much more difficult optimization problems. Semidefinite programming problems are problems with a linear objective function and linear constraints in a matrix variable $X$, plus the additional constraint that $X$ be positive semidefinite. Since a rich array of efficient algorithms for solving semidefinite programming problems has been developed in recent years, reducing more general problems to a sequence of semidefinite programming problems is an important advance.
We hope that readers of SIAM Review will enjoy this glimpse into the sophisticated mathematics behind an important algorithmic innovation in optimization.

A Sum of Squares Approximation of Nonnegative Polynomials

Jean B. Lasserre

SIAM Rev. 49, pp. 651-669 (19 pages) | Cited 1 time

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
We show that every real nonnegative polynomial $f$ can be approximated as closely as desired (in the $l_1$-norm of its coefficient vector) by a sequence of polynomials $\{f_\epsilon\}$ that are sums of squares. The novelty is that each $f_\epsilon$ has a simple and explicit form in terms of $f$ and $\epsilon$.
FREE

Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 49, pp. 671-671 (1 page)

Full Text: | Download PDF

Show Abstract
For most of us, using MATLAB to solve the linear system $Ax = b$, where $A$ is an $N \times N$ square matrix, is as simple as typing x = A, which invokes a standard algorithm (row-pivoted Gaussian elimination). Yet, amazingly, it is possible to construct examples with $N$ as small as 60 where this algorithm fails miserably. While this seems like a sad state of affairs, in fact the algorithm is sound because the examples where it fails are extremely rare. As the authors of this issue's contribution to the Education section make amply clear, your chances of winning the lottery multiple times in a single day are much greater than randomly finding such a snark-like matrix.
Our article this issue, “Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods," by Tobin A. Driscoll and Kara L. Maki, describes how to estimate the frequency of these extremely rare examples. Clearly, with events this unusual, standard random sampling will never adequately sample our state space. The authors outline a clever strategy for “Climbing Mount Improbable" (to borrow a phrase from noted evolutionary biologist Richard Dawkins). By biasing their random walk in state space toward larger growth factors (a measure of how much rounding error will be amplified by the linear solver) and tracking the effect of the bias, they are able to determine the distribution of these exceedingly rare beasts.
While this paper is pitched at a fairly high level, it would make an ideal starting point for a project in advanced probability, statistical mechanics, or numerical linear algebra. The paper also is an ideal case study on the use of Markov chain Monte Carlo methods, an area of very active research.
So, if you are in the mood for a snark hunt, read on....
FREE

Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods

Tobin A. Driscoll and Kara L. Maki

SIAM Rev. 49, pp. 673-692 (20 pages)

Online Publication Date: November 01, 2007

Full Text: | Download PDF

Show Abstract
The growth factor of a matrix quantifies the amount of potential error growth possible when a linear system is solved using Gaussian elimination with row pivoting. While it is an easy matter [N. J. Higham and D. J. Higham, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155–164] to construct examples of $n\times n$ matrices having any growth factor up to the maximum of $2^{n-1}$, the weight of experience and analysis [N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996], [L. N. Trefethen and R. S. Schreiber, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335–360], [L. N. Trefethen and I. D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997] suggest that matrices with exponentially large growth factors are exceedingly rare. Here we show how to conduct numerical experiments on random matrices using a multicanonical Monte Carlo method to explore the tails of growth factor probability distributions. Our results suggest, for example, that the occurrence of an $8\times 8$ matrix with a growth factor of 40 is on the order of a once-in-the-age-of-the-universe event.

Book Reviews

Edited by Robert E. O'Malley,, Jr.

SIAM Rev. 49, pp. 695-732 (38 pages)

Full Text: | Download PDF

Show Abstract
I'm unusually pleased with the variety of reviews in this issue, and I hope you will agree with me.
I wrote the featured review, which relates three very different books on asymptotics. One might hope that applied mathematicians could converge on a list of core topics to teach students about this subject, but we'll have to instead continue to content ourselves with a variety of well-written and useful texts that cover alternative perspectives.
The remaining reviews by a distinguished list of experts range from coverage of monographs on deblurring and multiple scattering to textbooks on real analysis and airplane flight dynamics. We thank them all for their valuable recommendations.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Two-Term Asymptotic Approximation of a Cardiac Restitution Curve
  2. Spatiospectral Concentration on a Sphere
  3. The Structure and Function of Complex Networks
  4. Interior Methods for Nonlinear Optimization
  5. Convergence, Oscillations, and Chaos in a Discrete Model of Combat
© 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