Showing papers in "Siam Review in 1995"

Using linear algebra for intelligent information retrieval

Abstract: Currently, most approaches to retrieving textual materials from scientific databases depend on a lexical match between words in users’ requests and those in or assigned to documents in a database. ...

A survey of the maximum principles for optimal control problems with state constraints

Abstract: This paper gives a survey of the various forms of Pontryagin’s maximum principle for optimal control problems with state variable inequality constraints. The relations between the different sets of optimality conditions arising in these forms are shown. Furthermore, the application of these maximum principle conditions is demonstrated by solving some illustrative examples.

Historical development of the Newton-Raphson method

Abstract: This expository paper traces the development of the Newton-Raphson method for solving nonlinear algebraic equations through the extant notes, letters, and publications of Isaac Newton, Joseph Raphson, and Thomas Simpson. It is shown how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to give a general formulation, in terms of fluxional calculus, applicable to nonpolynomial equations. Simpson's extension of the method to systems of equations is exhibited.

Displacement structure: theory and applications

Abstract: In this survey paper, we describe how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what we call displacement structure. In particular, a fast triangularization procedure can be developed for such matrices, generalizing in a striking way an algorithm presented by Schur (1917) [J. Reine Angew. Math., 147 (1917), pp. 205–232] in a paper on checking when a power series is bounded in the unit disc. This factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra. We mention, among others, inverse scattering, analytic and unconstrained rational interpolation theory, digital filter design, adaptive filtering, and state-space least-squares estimation.

Korn's inequalities and their applications in continuum mechanics

Abstract: Korn’s inequalities have played a central role in the development of linear elasticity, not only in connection with the basic theoretical issues such as existence and uniqueness, but also in a variety of applications. The Korn inequalities, and other related inequalities for integrals of quadratic functionals, also arise in the analysis of viscous incompressible fluid flow. The dimensionless optimal constants appearing in these inequalities, the Korn constants, depend only on the shape of the domains of concern. Information on the geometric dependence of these constants is essential in applications. In this review article, we summarize the major results on Korn’s inequalities for bounded domains in two and three dimensions, with emphasis on results concerning the Korn constants. Some applications in continuum mechanics are also described.

Convergence rates for Markov chains

Abstract: This is an expository paper that presents various ideas related to nonasymptotic rates of convergence for Markov chains. Such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science. They also have applications to analysis of card shuffling and other areas.In this paper, we attempt to describe various mathematical techniques that have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible.

Anti-plane shear deformations in linear and nonlinear solid mechanics

Abstract: The intent of this expository paper is to draw the attention of the applied mathematics community to an interesting two-dimensional mathematical model arising in solid mechanics involving a single second-order linear or quasi-linear partial differential equation. This model has the virtue of relative mathematical simplicity without loss of essential physical relevance. Anti-plane shear deformations are one of the simplest classes of deformations that solids can undergo. In anti-plane shear (or longitudinal shear, generalized shear) of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate. Thus anti-plane shear, with just a single scalar axial displacement field, may be viewed as complementary to the more complicated (yet perhaps more familiar) plane strain deformation, with its two in-plane displacements. In recent years, considerable attention has been paid to the analysis of anti-plane shear deformations within the context of variou...

Software libraries for linear algebra computations on high performance computers

Abstract: This paper discusses the design of linear algebra libraries for high performance computers. Particular emphasis is placed on the development of scalable algorithms for multiple instruction multiple data (MIMD) distributed memory concurrent computers. A brief description of the EISPACK, LINPACK, and LAPACK libraries is given, followed by an outline of ScaLAPACK, which is a distributed memory version of LAPACK currently under development. The importance of block-partitioned algorithms in reducing the frequency of data movement between different levels of hierarchical memory is stressed. The use of such algorithms helps reduce the message startup costs on distributed memory concurrent computers. Other key ideas in our approach are the use of distributed versions of the Level 2 and Level 3 basic linear algebra subprograms (BLAS) as computational building blocks, and the use of basic linear algebra communication subprograms (BLACS) as communication building blocks. Together the distributed BLAS and the BLACS c...

Mathematical morphology: a modern approach in image processing based on algebra and geometry

Abstract: Mathematical morphology is a theory of image transformations and image functionals which is based on set-theoretical, geometrical, and topological concepts. The methodology is particularly useful for the analysis of the geometrical structure in an image. The main goal of this paper is to give an impression of the underlying philosophy and the mathematical theories which are relevant to this field. The following topics are discussed: introduction to mathematical morphology; generalization to complete lattices; morphological filters and their construction by iteration; geometrical aspects of morphology (e.g., convexity, distance, geodesic operators, granulometries, metric dilations, distance transform, cost functions); and extension of binary operators to grey-scale images.

A rank-one reduction formula and its applications to matrix factorizations

Abstract: Let $A \in R^{m \times n} $ denote an arbitrary matrix. If $x \in R^n $ and $y \in R^m $ are vectors such that $\omega = y^T Ax e 0$, then the matrix $B: = A - \omega ^{ - 1} Axy^T A$ A has rank exactly one less than the rank of A. This Wedderburn rank-one reduction formula is easy to prove, yet the idea is so powerful that perhaps all matrix factorizations can be derived from it. The formula also appears in places such as the positive definite secant updates BFGS and DFP as well as the ABS methods. By repeatedly applying the formula to reduce ranks, a biconjugation process analogous to the Gram–Schmidt process with oblique projections can be developed. This process provides a mechanism for constructing factorizations such as ${\text{LDM}}^T $, QR, and SVD under a common framework of a general biconjugate decomposition $V^T AU = \Omega $ that is diagonal and nonsingular. Two characterizations of biconjugation provide new insight into the Lanczos method and its breakdown. One characterization shows that ...

A chaotic exploration of aggregation paradoxes

Abstract: Paradoxes from statistics and decision sciences form amusing, yet intriguing mathematical puzzles. On deeper examination, they constitute serious problems that could cause us, unintentionally, to adopt inferior alternatives. It is indicated here how ideas from “dynamical chaos” and orbits of symmetry groups can be modified and combined to create a mathematical theory to understand, classify, and find new properties of these puzzling phenomena.

A unified proof for the convergence of Jacobi and Gauss-Seidel methods

Abstract: We present a new unified proof for the convergence of both the Jacobi and the Gauss–Seidel methods for solving systems of linear equations under the criterion of either (a) strict diagonal dominance of the matrix, or (b) diagonal dominance and irreducibility of the matrix. These results are well known. The proof for criterion (a) makes use of Gersgorin’s theorem, while the proof for criterion (b) uses Taussky’s theorem that extends Gersgorin’s work. Hence the topic is interesting for teaching purposes.

Stabilization of the inverted linearized pendulum by high frequency vibrations

Abstract: In this note we give a simple geometrical picture that explains why an inverted pendulum is stabilized by high frequency vibrations.

Classroom notes

Abstract: An algorithm is developed that will select a hospital's charge structure to maximize reimbursements. Although the algorithm relies on the Kuhn–Tucker theory of nonlinear programming, it may be implemented using elementary sorting techniques.

Case study from industry

Abstract: The shoulder of a packaging machine is a developable surface that guides the packing material without stretching or tearing. The shoulder is traditionally manufactured by bending a flexible plate along a given bending curve, also without stretching or tearing. In this paper, the shoulder geometry is described mathematically by methods from classical differential geometry. For a given bending curve the generating lines of the (developable) shoulder surface are completely specified. It is shown that the bending curve can be chosen such that the resulting shoulder contains a planar triangle. The special case of a conical shoulder is also discussed, and the underlying bending curve is determined explicitly.

On floating-point summation

Abstract: In this paper we focus on some general error analysis results in floating-point summation. We emphasize analysis that is useful from both a scientific and a teaching point of view.

Representation and Control of Infinite Dimensional Systems, Vols. 1 and 2 (A. Bensonssan, G. Da Prato, M. Delfour, and S. Mitter)

Abstract: One ofthe most important developments in statistics over the last 20 years has been the counting process approach to event history analysis. The impetus for event history analysis originally came from medical statistics, where there is a need to analyze censored survival data, often with explanatory variables. It was realized in the early 1970’s that a natural way of specifying models for such data was in terms of hazard functions (intensities or transition rates). In 1975, O. O. Aalen showed that the basic nonparametric problems for censored data could be studied in terms of a counting process that records events as time proceeds. Many models of statistical interest could then be formulated in terms of the intensity of the underlying counting process. Furthermore, powerful methods from continuous-time martingale theory were then available for the development of the statistical theory. The statistical theory of event history analysis is now well developed and the monograph under review represents a timely and comprehensive account of it. The book is certain to become a standard reference in the field for many years to come. Appealing primarily to mathematically inclined statisticians and biostatisticians, and not intended to be a textbook, it would nevertheless be suitable as the basis of a second-year graduate course in survival analysis. The book would also be helpful for statisticians who are mainly interested in applications, given a reading of the (15page) survey of the basic properties of counting processes and their associated martingales at the start of Chapter 2. The style is relaxed and there are numerous practical examples and plots, yet there is careful attention to mathematical rigor. There are no apparent inconsistencies in style or notation, which is unusual for a book with four authors. The book is organized as follows. Chapter gives an overview of the development of the subject, and a number of interesting specific exampies, mainly from medicine and biology, that are used as motivation for the subsequent theoretical development, Chapter 2 provides a helpful survey of necessary mathematical background material, in which the authors describe the basic tools of stochastic calculus, collect likelihood and partial likelihood formulae for counting processes, and summarize key material on product-integrals and the functional delta-method. In Chapter 3 it is shown how right censoring and left truncation can be handled by a multiplicative intensity model. Chapter 4, dealing with nonparametric estimation, is the core of the book. It examines in considerable detail the large sample properties of the Nelson-Aalen estimator for the cumulative hazard function, and also the product-limit estimator for the transition matix of a nonhomogeneous Markov process (including the famous Kaplan-Meier estimator), along with related estimators. Product integration and martingale central limit theory play a crucial role. The NelsonAalen estimator is interpreted as a nonparametric maximum likelihood estimator (NPMLE). Continuing in the context of the multiplicative intensity model, Chapter 5 considers hypothesis testing and Chapter 6 studies the fitting and checking of parametric models of this type. Chapter 7 is concerned with regression models, in which the intensity of the counting process is allowed to depend on (possibly timedependent) covariates, as well as unknown parameters or functions. It gives a thorough treatment of the immensely popular Cox proportional hazards model, and of Aalen’s additive risk model, which has not been covered in any previous monograph. In Chapter 8 it is shown that many of the estimators studied in the earlier chapters are asymptotically efficient. The authors note that the easier it is to compute an NPMLE, the more likely it is to be asymptotically efficient. They caution, however, that there are lots of ways the NPMLE can fail to work, even in nice models--one interesting example (but not in the book) is the standard fight censorship model with failure indicators missing at random. Chapters 9 and 10 contain surveys of frailty models and multivariate time scales, topics that have been the subject ofmuch recent research and are still a long way from being settled. This book is essential reading for anyone who is interested in the state of the art of event history analysis.

A motivational example for the numerical solution of two-point boundary-value problems

Abstract: This paper presents an example of a two-point boundary-value problem which can be used to motivate the study of numerical techniques for solving such problems by undergraduates. The problem is referred to as the putting problem and illustrates the shooting method for two-point boundary-value problems. The differential equations are developed and numerical results are given.

The approximation problem for drift-diffusion systems

Abstract: This review surveys a significant set of recent ideas developed in the study of nonlinear Galerkin approximation. A significant role is played by the Krasnosel’skii calculus, which represents a generalization of the classical inf-sup linear saddlepoint theory. A description of a proper extension of this calculus and the relation to the ii-sup theory are part of this review. The general study is motivated by steady-state, self-consistent, drift-diffusion systems. The mixed boundary value problem for nonlinear elliptic systems is studied with respect to defining a sequence of convergent approximations, satisfying requirements of: (i) optimal convergence rate; (ii) computability; and, (iii) stability. It is shown how the fixed point and numerical fixed point maps of the system, in conjunction with the Newton–Kantorovich method applied to the numerical fixed point map, permit a solution of this approximation problem. A critical aspect of the study is the identification of the breakdown of the Newton–Kantorovi...

The Fuzzy Systems Handbook: A Practitioner’s Guide to Building, Using, and Maintaining Fuzzy Systems (Earl Cox)

Abstract: The Fuzzy Systems Handbook is a practitioner’s guide as the title suggests. The first 270 pages contain a general introduction to fuzzy logic: alpha-level sets, fuzzy quantifiers and hedges, set intersection and union, compensatory set operators, and the compositional rules of inference. These operations are a prelude to fuzzy function approximation. This book leads the reader gently up to what is now known as a fuzzy controller or fuzzy rule base. It is not a math text. Its target audience is the business programmer solving problems in product pricing and risk assessment. Even so, the book does lightly cover examples of engineering control albeit with a nonmath slant. The discussion is friendly and the reader soon discovers that the book is a manual for the software supplied with the book on an IBM compatible diskette. There are no demos in the usual sense of the term. There are examples. For the reader, it’s a programmer’s world. A fuzzy system is a set of if-then rules that maps inputs to outputs. The rule base may contain a number of these rules and may include unconditional rules. Conditional rules fit the pattern ifX is Aj then Y is Bj. For example, if temperature is near zero then 02 production is very high describes the optimal temperature for a catalase reaction. The rules can be descriptive as in the previous case or prescriptive as in if temperature is very high then make cooling very high. Each fuzzy rule defines a fuzzy Cartesian patch Aj x Bj C X x Y. If-part set Aj C R\" has the (2) ya(x)b(y)dy

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (V. L. Kocic and G. Ladas)

Abstract: Difference equations, which used to be considered only as discretizations of differential equations, are attracting considerable attention, mainly because the adaptation from the continuous case to the discrete case is not direct, but requires special devices. Further, discrete problems find applications in almost every branch of science, for example, probability theory, stochastic time series, economics, psychology, electrical networks, geometry, etc. In fact, in the last few years, several books 1]-[10] and research papers (see ], which covers over 400 articles) have been written. Mainly these books deal with the classical theory, local as well as global behavior, and computational aspects of the solutions of difference equations. The volume under review is thus in the present trend and will be useful to the scientific community. It deals with the global behavior of solutions of nonlinear scalar ordinary difference equations of order greater than 1. In particular, it offers a detailed study of concepts such as global asymptotic stability, periodicity, permanence and persistence, and semicycles of the solutions. The subject matter has been arranged as follows: Chapter begins with several simple difference equations which occur in diverse fields. These examples provide the motivation of this monograph and are nicely illustrated. Here some definitions and basic results are also collected, which are used throughout the monograph. Chapter 2 presents general results on the global asymptotic stability and global attractivity of the solutions ofsome nonlinear difference equations.