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Showing papers in "Siam Review in 1997"


Journal Articleā€¢DOIā€¢
TL;DR: The goal of this documentation is to summarize the essential applications of the nonlinear complementarity problem known to date, to provide a basis for the continued research on the non linear complementarityproblem, and to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.
Abstract: This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

1,016Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: The Gibbs phenomenon is reviewed from a different perspective and it is shown that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case.
Abstract: The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting. The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation.

747Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: Ten examples of computed pseudospectra of thirteen highly nonnormal matrices arising in various applications are presented, each chosen to illustrate one or more mathematical or physical principles.
Abstract: If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ||An|| or ||exp(tA)||. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ||(zI - A)-1||. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

507Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: The area of combinatorial Gray codes is surveyed, recent results, variations, and trends are described, and some open problems are highlighted.
Abstract: The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

488Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: The history of the algorithmic approach to this problem is recalled, some successful solution algorithms are reviewed, and some algorithms of 1995 are outlined that solve this problem at a surprisingly low computational cost.
Abstract: The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.

361Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: The various mathematical problems arising, some deficiencies of the current models and algorithms, and possible (past and future) attacks to arrive at solutions to the protein-folding problem are discussed.
Abstract: This paper discusses the mathematical formulation of and solution attempts for the so-called protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein. From a mathematical point of view, there are several main sides to the static problem: -- the selection of an appropriate potential energy function; -- the parameter identification by fitting to experimental data; and -- the global optimization of the potential. The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation) or (in case of constrained molecular dynamics) of differential-algebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs. The present paper gives a self-contained introduction to the necessary background from physics and chemistry and surveys some of the literature. It also discusses the various mathematical problems arising, some deficiencies of the current models and algorithms, and possible (past and future) attacks to arrive at solutions to the protein-folding problem.

198Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: It is concluded that the behavior of the residuals in inverse iteration is governed by the departure of the matrix from normality rather than by the conditioning of a Jordan basis or the defectiveness of eigenvalues.
Abstract: The purpose of this paper is two-fold: to analyze the behavior of inverse iteration for computing a single eigenvector of a complex square matrix and to review Jim Wilkinson's contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. In the case of normal matrices we show that residual norms decrease strictly monotonically. For eighty percent of the starting vectors a single iteration is enough. In the case of non-normal matrices, we show that the iterates converge asymptotically to an invariant subspace. However, the residual norms may not converge. The growth in residual norms from one iteration to the next can exceed the departure of the matrix from normality. We present an example where the residual growth is exponential in the departure of the matrix from normality. We also explain the often significant regress of the residuals after the first iteration: it occurs when the non-normal part of the matrix is large compared to the eigenvalues of smallest magnitude. In this case computing an eigenvector with inverse iteration is exponentially ill conditioned (in exact arithmetic). We conclude that the behavior of the residuals in inverse iteration is governed by the departure of the matrix from normality rather than by the conditioning of a Jordan basis or the defectiveness of eigenvalues.

154Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: This work investigates the global convergence of Euler's and Halley's methods by using their geometric interpretation and a generalization of these methods is discussed.
Abstract: We investigate the global convergence of Euler's and Halley's methods by using their geometric interpretation. A generalization of these methods is also briefly discussed.

86Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: An algorithm due to Descloux and Tolley is described that blends singular finite elements with domain decomposition and it is shown that, with a modification that doubles its accuracy, this algorithm can be used to compute efficiently the eigenvalues for polygonal regions.
Abstract: Recently it was proved that there exist nonisometric planar regions that have identical Laplace spectra. That is, one cannot "hear the shape of a drum." The simplest isospectral regions known are bounded by polygons with reentrant corners. While the isospectrality can be proven mathematically, analytical techniques are unable to produce the eigenvalues themselves. Furthermore, standard numerical methods for computing the eigenvalues, such as adaptive finite elements, are highly inefficient. Physical experiments have been performed to measure the spectra, but the accuracy and flexibility of this method are limited. We describe an algorithm due to Descloux and Tolley [Comput. Methods Appl. Mech. Engrg., 39 (1983), pp. 37--53] that blends singular finite elements with domain decomposition and show that, with a modification that doubles its accuracy, this algorithm can be used to compute efficiently the eigenvalues for polygonal regions. We present results accurate to 12 digits for the most famous pair of isospectral drums, as well as results for another pair.

78Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: To illustrate various mathematical methods which may be used to solve problems of engineering, mathematical physics, and mathematical biology, Volterra's model for population growth of a species in a closed system is solved using several methods familiar to junior- or senior-level students in applied mathematics.
Abstract: To illustrate various mathematical methods which may be used to solve problems of engineering, mathematical physics, and mathematical biology, Volterra's model for population growth of a species in a closed system is solved using several methods familiar to junior- or senior-level students in applied mathematics. Volterra's model is a first-order integro-ordinary differential equation where the integral term represents the effect of toxin accumulation on the species. The solution methods used are (i) numerical methods for solving a first-order initial value problem supplemented with numerical integration, (ii) numerical methods for solving a coupled system of two first-order initial value problems, and (iii) phase-plane analysis. A singular perturbation solution previously presented is also outlined. While conclusions drawn using the four methods are correlated, the student may analyze and solve the problem using any of the methods independently of the others.

73Ā citations


Journal Articleā€¢DOIā€¢
TL;DR: A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented, finding that one based on approximate wavelet decompositions and the other on a different direct multileVEL splitting of the finite element discretization space lead to optimal relative condition numbers of the multileel preconditionsers.
Abstract: A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the two-level methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomially-based inner iteration method. The latter gives rise to hybrid-type multilevel cycles. This is the so-called (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the so-called wavelet multilevel decomposition based on $L^2$-projections, which in practice must be approximated. Both approaches---the AMLI one and the one based on approximate wavelet decompositions---lead to optimal relative condition numbers of the multilevel preconditioners.

Journal Articleā€¢DOIā€¢
TL;DR: The commercial software package MAJIQTRIM solves this combined skiving and cutting stock problem using both heuristic and linear programming (LP) methods.
Abstract: Skiving is the process of joining smaller rolls to form larger rolls. When this process is combined with traditional roll-cutting technology, profitable solutions to once "hopeless" one-dimensional cutting stock problems can be found. A generalization of the classic cutting stock problem models the combined skiving and cutting stock problem. The commercial software package MAJIQTRIM solves this combined problem using both heuristic and linear programming (LP) methods.

Journal Articleā€¢DOIā€¢
TL;DR: Using an elementary analysis of Lienard's equation and Lyapunov's direct method, sufficient conditions are derived for the global asymptotic stability of the disease-free and endemic equilibria in this model.
Abstract: An S ---> I ---> R ---> I epidemic model with vital dynamics, temporary immunity, and nonlinear incidence rate is analyzed. This model consists of a system of nonlinear ordinary differential equations and can be seen as an extension of the model proposed by Tudor [SIAM Rev., 32 (1990), pp. 136--139] for herpes viral infections in which R is a latent class. Using an elementary analysis of Lienard's equation and Lyapunov's direct method, sufficient conditions are derived for the global asymptotic stability of the disease-free and endemic equilibria in this model.

Journal Articleā€¢DOIā€¢
TL;DR: This expository paper develops the principal known results on the stable matching of marriage games in the language of directed graphs and yields a new algorithm and a new proof for the existence of stable matchings and new proofs for many known facts.
Abstract: This expository paper develops the principal known results (and some new ones) on the stable matchings of marriage games in the language of directed graphs. This both unifies and simplifies the presentation and renders it more symmetric. In addition, it yields a new algorithm and a new proof for the existence of stable matchings, new proofs for many known facts, and some new results (notably concerning players' strategies and the properties of the stable matching polytope).

Journal Articleā€¢DOIā€¢
TL;DR: This note proposes a new proof of Farkas' lemma which is based on elementary arguments, and is not a trivial conclusion, and its proof contains certain difficulties.
Abstract: Farkas' lemma is one of the key results in optimization. Yet, it is not a trivial conclusion, and its proof contains certain difficulties. In this note we propose a new proof which is based on elementary arguments.

Journal Articleā€¢DOIā€¢
TL;DR: A new proof of the Coe--Tang result that the at-risk divisors have six consecutive ones in positions 5 through 10 is supplied and it is proved that the worst-case absolute error for arguments in [1,2] is on the order of 1e--5.
Abstract: Despite all of the publicity surrounding the Pentium bug of 1994, the mathematical details of the bug are poorly understood. We discuss these details and supply a new proof of the Coe--Tang result that the at-risk divisors have six consecutive ones in positions 5 through 10. Also, we prove that the worst-case absolute error for arguments in [1,2) is on the order of 1e--5.

Journal Articleā€¢DOIā€¢
TL;DR: This review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface, and discusses the relevance of traditional ideas from the theories of moving boundary problems, singular integral equations, variational inequalities, and stability.
Abstract: This review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface. Such situations can often be approximated by mixed boundary value problems in which the boundary data switches across a ``codimension-two'' free boundary, namely, the edge of the region obtained by projecting the free boundary normally onto the prescribed surface. As in the parent problem, the codimension-two free boundary needs to be determined as well as the solution of the relevant field equations, but no systematic methodology has yet been proposed for nonlinear problems of this type. After presenting some examples to illustrate the surprising behavior that can sometimes occur, we discuss the relevance of traditional ideas from the theories of moving boundary problems, singular integral equations, variational inequalities, and stability. Finally, we point out the ways in which further refinement of these techniques is needed if a coherent theory is to emerge.

Journal Articleā€¢DOIā€¢
TL;DR: A rigorous description of the Lagrange--Charpit method used to find a complete integral of a nonlinear p.d. e.s. adapted for a university course in differential equations.
Abstract: We give a rigorous description of the Lagrange--Charpit method used to find a complete integral of a nonlinear p.d. e. adapted for a university course in differential equations.

Journal Articleā€¢DOIā€¢
TL;DR: This note shows some useful eigenvalue and eigenvector properties of matrices with two symmetries, such as matrices which are symmetric and persymmetric.
Abstract: A matrix is called persymmetric (Golub and Van Loan, Matrix Computations, The Johns Hopkins University Press, 1989) if it is symmetric across its lower-left to upper-right diagonal, with similar definitions for per-antisymmetric and per-Hermitian matrices. This note shows some useful eigenvalue and eigenvector properties of matrices with two symmetries, such as matrices which are symmetric and persymmetric.

Journal Articleā€¢DOIā€¢
TL;DR: A novel two-phase flow model is proposed to study the formation and migration of the macrovoids, a major manufacturing problem, and it is demonstrated that the model is in good agreement with experimental results.
Abstract: Resin transfer molding (RTM) has drawn interest in recent years as an attractive technique for the manufacture of advanced fiber-reinforced composite materials. A major issue in this new manufacturing process is the reduction of voids during the resin fill process so that products with high quality are manufactured. Process modeling is particularly useful in understanding, designing, and optimizing the process conditions. The purpose of this paper is to illustrate the importance of mathematical and numerical modeling to this industrial problem. First, an overview of the RTM process, its manufacturing problems, and related background issues is given. A survey of various RTM models developed in recent years by researchers in this field are then presented. Finally, as an application, a novel two-phase flow model, developed recently by the authors, is proposed to study the formation and migration of the macrovoids, a major manufacturing problem. The unique feature of this model is the identification of local pressure as a major mobilization factor of these macrovoids. It is demonstrated that the model is in good agreement with experimental results.

Journal Articleā€¢DOIā€¢
TL;DR: A new general purpose solution algorithm which obviates the use of artificial variables is presented which searches for a feasible segment of a boundary hyperplane by using rules similar to the ordinary simplex.
Abstract: The simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point. We present a new general purpose solution algorithm which obviates the use of artificial variables. The algorithm searches for a feasible segment of a boundary hyperplane (a face of feasible region or an intersection of several faces) by using rules similar to the ordinary simplex.

Journal Articleā€¢DOIā€¢
TL;DR: Two methods for finding the center and radius of a circular starting line of a racetrack in an ancient Greek stadium are presented and compared and it is shown that the first method yields a circle whose radius is somewhat longer than the radius determined by the least-squares method.
Abstract: Two methods for finding the center and radius of a circular starting line of a racetrack in an ancient Greek stadium are presented and compared The first is a method employed by the archaeologists who surveyed the starting line and the second is a least-squares method leading to a maximum-likelihood circle We show that the first method yields a circle whose radius is somewhat longer than the radius determined by the least-squares method and propose reasons for this difference A knowledge of the center and radius of the starting line is useful for determining units of length and angle used by the ancient Greeks, in addition to providing information on how ancient racetracks were laid out

Journal Articleā€¢DOIā€¢
TL;DR: The matrix Riccati equation may be transformed into the system of linear second-order ordinary differential equations (ODEs), known in differential geometry as the Jacobi equation, generalizing a familiar result in the scalar case.
Abstract: The matrix Riccati equation may be transformed into the system of linear second-order ordinary differential equations (ODEs), known in differential geometry as the Jacobi equation, generalizing a familiar result in the scalar case.

Journal Articleā€¢DOIā€¢
TL;DR: This note presents a rigorous proof of the fact that in a renewal process, the interarrival interval containing a specific time point is stochastically larger than a "typical" interar Arrival interval in the process.
Abstract: This note presents a rigorous proof of the fact that in a renewal process, the interarrival interval containing a specific time point is stochastically larger than a "typical" interarrival interval in the process. The key of the proof is a very useful and easily proved covariance inequality.

Journal Articleā€¢
TL;DR: In this paper, the first and second derivatives associated with such problems and the relation of these derivatives to implicit differentiation and equality constrained optimization are discussed. But they are not discussed in this paper.
Abstract: This paper discusses the calculation of sensitivities, or derivatives, for optimization problems involving systems governed by differential equations and other state relations. The subject is examined from the point of view of nonlinear programming, beginning with the analytical structure of the first and second derivatives associated with such problems and the relation of these derivatives to implicit differentiation and equality constrained optimization. We also outline an error analysis of the analytical formulae and compare the results with similar results for finite-difference estimates of derivatives. We then attend to an investigation of the nature of the adjoint method and the adjoint equations and their relation to directions of steepest descent. We illustrate the points discussed with an optimization problem in which the variables are the coefficients in a differential operator.

Journal Articleā€¢DOIā€¢
TL;DR: It turns out that these conservation laws are intimately related to a geometric theory---the geometry of webs---that originated more than 60 years earlier, which helped discover how many conservation laws a nonlinear wave system can have.
Abstract: Recently, unexpected conservation laws have been discovered for various nonlinear wave systems. Among these systems is the system of Rossby waves, which describes the global dynamics of the atmosphere and the ocean. It turns out that these conservation laws are intimately related to a geometric theory---the geometry of webs---that originated more than 60 years earlier. This relation helped discover how many conservation laws a nonlinear wave system can have.

Journal Articleā€¢DOIā€¢
Jon Lee1ā€¢
TL;DR: A geometry is described for understanding Hoffman's example that demonstrated that Dantzig's simplex method for linear programming can cycle (unless special precautions are taken).
Abstract: In 1951, Alan J. Hoffman provided the first example that demonstrated that Dantzig's simplex method for linear programming can cycle (unless special precautions are taken). This paper describes a geometry for understanding Hoffman's example---something that has been lacking for over two-fifths of a century.


Journal Articleā€¢DOIā€¢
TL;DR: A method of presenting optimization to first-semester calculus students is demonstrated, based on the Lagrange multiplier approach, and the geometry of a given problem becomes the central focus of the effort to find a solution.
Abstract: A method of presenting optimization to first-semester calculus students is demonstrated. The method is based on the Lagrange multiplier approach, and the geometry of a given problem becomes the central focus of the effort to find a solution. The method proposed here deviates from the traditional approach in that the student is asked to examine a constrained optimization problem and the geometry of the problem remains the central theme.

Journal Articleā€¢DOIā€¢
TL;DR: A simple epidemiological model whose dynamics are described by a pair of nonlinearly coupled Lotka--Volterra oscillators is shown to have a two-dimensional center manifold that turns out to be identical to the center eigenspace and is thus analytically determinable.
Abstract: A simple epidemiological model whose dynamics are described by a pair of nonlinearly coupled Lotka--Volterra oscillators is shown to have a two-dimensional center manifold. This center manifold turns out to be identical to the center eigenspace and is thus analytically determinable. On the center manifold, the system is reduced to a single Lotka--Volterra oscillator.