Showing papers in "Siam Review in 2007"

Stability Criteria for Switched and Hybrid Systems

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.

A Direct Formulation for Sparse PCA Using Semidefinite Programming

Abstract: Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming-based relaxation for our problem We also discuss Nesterov's smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem The method has complexity $O(n^4 \sqrt{\log(n)}/\epsilon)$, where $n$ is the size of the underlying covariance matrix and $\epsilon$ is the desired absolute accuracy on the optimal value of the problem

A Survey of the S-Lemma

Abstract: In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry.

Mathematical Models of Avascular Tumor Growth

Abstract: 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.

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Abstract: Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex Alternatively, for large systems (eg, a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem Partition function and concentration information can then be used to calculate equilibrium base-pairing observables The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality

Generalized Chebyshev Bounds via Semidefinite Programming

Abstract: A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev’s inequality for scalar random variables. Two semidefinite programming formulations are presented, with a constructive proof based on convex optimization duality and elementary linear algebra.

A Sum of Squares Approximation of Nonnegative Polynomials

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 $\epsilon$.

Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization

Abstract: 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.

Error Estimation for Reduced-Order Models of Dynamical Systems

Abstract: 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.

Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme

Abstract: Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form. The closure is obtained on demand through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation. Such decisions include (a) the highest spatial derivative active in the coarse equation, (b) whether the equation satisfies certain conservation laws, or (c) whether the coarse dynamics is Hamiltonian or dissipative. These decisions affect the number and type of boundary conditions as well as the algorithms employed. In the absence of an explicit formula for the temporal derivative, we propose, implement, and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We believe that such schemes constitute an important part of the equation-free approach to multiscale computation.

The Mathematics of Phylogenomics

Abstract: The grand challenges in biology today are being shaped by powerful high-throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes, and detailed information about variation within populations. We are therefore able to ask, for the first time, fundamental questions about the evolution of genomes, the structure of genes and their regulation, and the connections between genotypes and phenotypes of individuals. The answers to these questions are all predicated on progress in a variety of computational, statistical, and mathematical fields. The rapid growth in the characterization of genomes has led to the advancement of a new discipline called phylogenomics. This discipline results from the combination of two major fields in the life sciences: genomics, i.e., the study of the function and structure of genes and genomes; and molecular phylogenetics, i.e., the study of the hierarchical evolutionary relationships among organisms and their genomes. The objective of this article is to offer mathematicians a first introduction to this emerging field, and to discuss specific mathematical problems and developments arising from phylogenomics.

A Reduced-Order Kalman Filter for Data Assimilation in Physical Oceanography

Abstract: A central task of physical oceanography is the prediction of ocean circulation at various time scales. Mathematical techniques are used in this domain not only for the modeling of ocean circulation but also for the enhancement of simulation through data assimilation. The ocean circulation model of concern here, namely, HYCOM, is briefly presented through its variables, equations, and specific vertical coordinate system. The main part of this paper focuses on the Kalman filter as a data assimilation method, and especially on how this mathematical technique, usually associated with a prohibitively high computing cost for operational sciences, is simplified in order to make it applicable to the simulation of realistic ocean circulation models. Some practical issues are presented, such as a brief explanation about ocean observation systems, together with examples of data assimilation results.

Revisiting Hypergraph Models for Sparse Matrix Partitioning

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.

Uniform Asymptotics Applied to Ultrawideband Pulse Propagation

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.

A Hybrid Approach for Efficient Robust Design of Dynamic Systems

Abstract: 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.

Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods

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.

Adaptive Polynomial Interpolation on Evenly Spaced Meshes

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.

A Classification of Spikes and Plateaus

Abstract: In this paper we classify local maxima into spikes and plateaus. We give analytic definitions for spikes and plateaus in terms of a nonlocal gradient and a fourth order derivative. In higher dimensions the Hesse matrix of $\Delta f(x)$ is of relevance. This classification is applied to pattern formation models in mathematical physics and mathematical biology, including Cahn-Hilliard equations, chemotaxis equations, reaction-diffusion equations, Gierer-Meinhardt models, and Gray-Scott models. We show for some of these examples that the stability of spatial patterns depends on the spike versus plateau type of the solution. We prove, for example, that scalar reaction-diffusion equations in any spatial dimension cannot have stable spike steady states.

A Matrix Perturbation View of the Small World Phenomenon

Abstract: We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204].

Fundamental Solutions for Some Partial Differential Operators from Fluid Dynamics and Statistical Physics

Abstract: 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.

As Flat As Possible

Abstract: How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students build geometric and analytic understanding of solutions of certain elliptic and parabolic partial differential equations.

Determining Sets for the Discrete Laplacian

Abstract: We define a notion of determining sets for the discrete Laplacian in a domain O. 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 O. It is shown that there exist determining sets that are fairly evenly distributed in O. A number of basic properties of determining sets are derived.

Buchberger's Algorithm and the Two-Locus, Two-Allele Model

Abstract: The present paper uses results from algebraic geometry to study the classical model describing the equilibrium frequencies of the four gametic types in the two-locus, two-allele genetic model under the assumption that the fitnesses are constant and that the cis- and trans- fitnesses are equal. It shows that there is a finite process for determining an upper bound on the maximum number of equilibrium frequencies for almost all choices of recombination and fitness parameters, and that this number is finite. It also shows that the solutions are locally continuous for almost all choices of parameters. The paper studies the equilibrium frequencies as functions of the recombination parameter, with attention to the cases when the recombination becomes infinite and when one of the equilibrium frequencies becomes infinite.