Showing papers in "Siam Review in 1998"
TL;DR: It is shown that the basic regularization procedures for finding meaningful approximate solutions of ill-conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course.
Abstract: It is shown that the basic regularization procedures for finding meaningful approximate solutions of ill-conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course. Apart from rewriting many known results in a more elementary form, we also derive a new two-parameter family of merit functions for the determination of the regularization parameter. The traditional merit functions from generalized cross validation (GCV) and generalized maximum likelihood (GML) are recovered as special cases.
655 citations
TL;DR: A method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described.
Abstract: A method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described. Numerical examples illustrating the power of the approximation are presented.
521 citations
TL;DR: This paper is a review of work on thin fluid films where surface tension is a driving mechanism and discusses asymptotic results, travelling waves, stability, and similarity solutions, as well as analytical work on the resultant equations.
Abstract: This paper is a review of work on thin fluid films where surface tension is a driving mechanism. Its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations.
In general the introduction of surface tension into standard lubrication theory leads to a fourth-order nonlinear parabolic equation
$$
\pad{h}{t}+\pad{}{x}\left(C \frac{h^3}{3}\frac{\partial^3 h}{\partial x^3} +f(h,h_x,h_{xx})\right) = 0 ,\label{abeq1}
$$
where $h=h(x,t)$ is the fluid film height. For steady situations this equation may be integrated once and a third-order ordinary differential equation is obtained. Appropriate forms of this equation have been used to model fluid flows in physical situations such as coating, draining of foams, and the movement of contact lenses.
In the introduction a form of the above equation is derived for flow driven by surface tension, surface tension gradients, gravity, and long range molecular forces. Modifications to the equation due to slip, the effect of two free surfaces, two phase fluids, and higher dimensional forms are also discussed. The second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given. The third section reviews analytical work on the model equations as well as the "generalized lubrication equation"
$$
\pad{h}{t}+\pad{}{x}\left(h^n h_{xxx}\right) = 0.
$$
In particular the discussion focusses on asymptotic results, travelling waves, stability, and similarity solutions. Numerical work is also discussed, while for analytical results the reader is directed to existing literature.
433 citations
TL;DR: A collection of inverse eigenvalue problems are identified and classified according to their characteristics, and current developments in both the theoretic and the algorithmic aspects are summarized and reviewed.
Abstract: A collection of inverse eigenvalue problems are identified and classified according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserve further study. An extensive bibliography of pertinent literature is attached.
355 citations
TL;DR: The emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty.
Abstract: This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
340 citations
TL;DR: The use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations.
Abstract: We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.
327 citations
TL;DR: For arbitrarily spaced grids, this work describes a computationally very inexpensive numerical algorithm that can be found very conveniently with a two-line algorithm when using a symbolic language such as Mathematica.
Abstract: The classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope (e.g., specialized recursions for centered and staggered approximations, for Adams--Bashforth-, Adams--Moulton-, and BDF-formulas for ODEs, etc.). Two recent algorithms overcome these problems. For equispaced grids, such weights can be found very conveniently with a two-line algorithm when using a symbolic language such as Mathematica (reducing to one line in the case of explicit approximations). For arbitrarily spaced grids, we describe a computationally very inexpensive numerical algorithm.
321 citations
TL;DR: A unified investigation of a class of nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions and several new variants of the algorithms are derived.
Abstract: The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuring--type methods and the Neumann--Neumann-type methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include local-global and global-local techniques. The analyses for both two- and three-dimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived.
211 citations
TL;DR: This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler, and Shmoys and focuses on special cases of the traveling salesman problem that have been obtained during the decade 1985--1995.
Abstract: The traveling salesman problem (TSP) belongs to the most basic, most important, and most investigated problems in combinatorial optimization. Although it is an ${\cal NP}$-hard problem, many of its special cases can be solved efficiently in polynomial time. We survey these special cases with emphasis on the results that have been obtained during the decade 1985--1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler, and Shmoys [The Traveling Salesman Problem---A Guided Tour of Combinatorial Optimization, Wiley, Chichester, pp. 87--143].
161 citations
TL;DR: Several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed and they are compared with regard to their accuracy and their computational efficiency.
Abstract: Several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed. The algorithms can be viewed as approximate generalizations of the fast Fourier transform (FFT), and they are compared with regard to their accuracy and their computational efficiency.
124 citations
TL;DR: The strong stability of general fully discrete RK methods governed by coercive approximations is proved, which is able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.
Abstract: The integration of semidiscrete approximations for time-dependent problems is encountered in a variety of applications. The Runge--Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines.
These methods of lines are governed by possibly ill-conditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations.
We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coefficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations.
Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.
TL;DR: The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugates, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed and six approximations are discussed in a systematic way.
Abstract: The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor $\rho \le 1$ can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.
TL;DR: The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high-dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi -random sequences.
Abstract: The need to simulate stochastic processes numerically arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper addresses the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, the valuation of a 30-year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high-dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi-random sequences.
TL;DR: A new scaling law for wall-bounded turbulence is proposed, and a reinterpretation of the Kolmogorov--Obukhov scaling for the local structure of turbulence is reinterpreted, which has been of paramount interest in both theory and applications.
Abstract: Intermittency, a basic property of fully developed turbulent flow, decreases with growing viscosity; therefore classical relationships obtained in the limit of vanishing viscosity must be corrected when the Reynolds number is finite but large. These corrections are the main subject of the present paper. They lead to a new scaling law for wall-bounded turbulence, which is of key importance in engineering, and to a reinterpretation of the Kolmogorov--Obukhov scaling for the local structure of turbulence, which has been of paramount interest in both theory and applications. The background of these results is reviewed, in similarity methods, in the statistical theory of vortex motion, and in intermediate asymptotics, and relevant experimental data are summarized.
TL;DR: Scaling, translation, and the spiral group of transformations are applied to well-known problems in mathematical physics, such as the boundary layer equations, the wave equation, andThe heat conduction equation.
Abstract: The importance of similarity transformations and their applications to partial differential equations is discussed. The theory has been presented in a simple manner so that it would be beneficial at the undergraduate level. Special group transformations useful for producing similarity solutions are investigated. Scaling, translation, and the spiral group of transformations are applied to well-known problems in mathematical physics, such as the boundary layer equations, the wave equation, and the heat conduction equation. Finally, a new transformation including the mentioned transformations as its special cases is also proposed.
TL;DR: Methods, which are based on the Cayley--Hamilton theorem, for the computation of An for nonsingular A are presented and it is shown that the value of A is determined by the inequality of the following type:
Abstract: Methods, which are based on the Cayley--Hamilton theorem, for the computation of An for nonsingular A are presented.
TL;DR: This review uses the one-dimensional sine-Gordon (sG) equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems and focuses on the collective coordinate approach as one of the most useful tools.
Abstract: Perturbed, one-dimensional, integrable (i.e., soliton-bearing) equations arise in many applied contexts when trying to improve the (usually highly idealized) description of problems of interest in terms of the purely integrable equations. In particular, when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects, perturbations depending on the spatial coordinate must be added to the original equation. In this review, we use the one-dimensional sine-Gordon (sG) equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems. To place the work in context, we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools. We introduce several versions of this perturbative method and relate them to more involved procedures. We analyze in detail the application to the sG equation, but the procedure is very general. To illustrate this other examples of the application of collective coordinates are briefly revisited. In our case study, this approach helps us identify perturbative and nonperturbative regimes, yielding a very simple picture of the former. Beyond perturbative calculations, the same example of the inhomogeneous sG equation allows us to introduce a phenomenon, termed length scale competition, which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems, as other instances discussed in the review show. Such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem; numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition. To complement our description of the techniques employed in these studies, computational details of our numerical simulations are also included. Finally, the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder.
TL;DR: Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.
Abstract: Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.
TL;DR: A simple formula is established that relates the trace of the resolvent with the characteristic polynomial of a matrix, which will lead to a novel and elegant proof of the recursive relations that compute the coefficients of the characteristicPolynomial used in the Leverrier--Faddeev method.
Abstract: A simple formula is established that relates the trace of the resolvent with the characteristic polynomial of a matrix. This will lead to a novel and elegant proof of the recursive relations that compute the coefficients of the characteristic polynomial used in the Leverrier--Faddeev method.
TL;DR: In this paper, the authors investigate the effect of damping on the fractal structure of the basin boundaries of a simple polynomial z 3 - 1 = 0 in the complex plane.
Abstract: An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z 3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newton's method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.
TL;DR: It is pointed out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems.
Abstract: The aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. Furthermore we show that free boundary problems arise also via a similarity analysis of moving boundary hyperbolic problems and they can be obtained as approximations of boundary value problems defined on infinite intervals. Most of the theoretical content of this survey is original: it generalizes and unifies results already available in literature. As far as applications of the proposed approach are concerned, three problems of interest are considered and numerical results for each of them are reported.
TL;DR: The purpose of this paper is to rectify this shortcoming in the state of the art here and to discuss, by way of a review, issues that have not been previously articulated.
Abstract: The recent technological advances in communications, manufacturing, and transportation systems have made networks the mainstay of modern life. Consequently, the reliability of networks has become an important issue and much progress has been made in its assessment. However, the state of the art here suffers from a serious limitation. It assumes that the underlying node and arc probabilities are known with certainty. The purpose of this paper is to rectify this shortcoming and to discuss, by way of a review, issues that have not been previously articulated.
To accomplish our goal, we undertake three tasks. The first is to develop a joint prior distribution for the reliabilities of the individual components. This distribution is to be defined on the unit hypercube and should make provision for the incorporation of dependencies. Several strategies for specifying such a prior distribution are proposed. The second task is to generate samples from either the prior distribution or the resulting posterior distribution that is obtained when data from tests on the components is available. Because of the high dimensionality of the prior, or the posterior, this task is best accomplished via simulation techniques such as Gibbs sampling. The third task pertains to simulating the reliability of the network by using the samples obtained from the second task as inputs to an algorithm for network reliability calculations. The entire exercise of assessing network reliability is therefore computer intensive. The same is also true of fault tree analysis.
TL;DR: Properties of solutions Z to the Wiener--Hopf integral equation, involving random walks, supermartingales, coupling, the Hewitt--Savage 0--1 law, ladder heights, and exponential change of measure are discussed by purely probabilistic methods.
Abstract: Existence, uniqueness, and asymptotic properties of solutions Z to the Wiener--Hopf integral equation $Z(x)$ $ =$ $z(x)+\int_{-\infty}^xZ(x-y)F(dy)$, $x\ge 0$, are discussed by purely probabilistic methods, involving random walks, supermartingales, coupling, the Hewitt--Savage 0--1 law, ladder heights, and exponential change of measure.
TL;DR: This note is devoted to simplifying one method to calculate the exponential of a matrix presented by I. E. Leonard in [SIAM Rev., 38 (1996), pp. 507--512].
Abstract: This note is devoted to simplifying one method to calculate the exponential of a matrix presented by I. E. Leonard in [SIAM Rev., 38 (1996), pp. 507--512].
TL;DR: The classical transmission line equations for the distributed parameters model can be obtained from standard two-port network matrices avoiding the explicit use of partial differential equations through a lemma derived directly from the Cayley--Hamilton theorem.
Abstract: As shown in this paper, the classical transmission line equations for the distributed parameters model can be obtained from standard two-port network matrices avoiding the explicit use of partial differential equations. This is performed through a lemma derived directly from the Cayley--Hamilton theorem. The main advantage of this approach is that the modeling arises naturally in the frequency domain, allowing the consideration of frequency-dependent parameters (as, for instance, the resistance and inductance variations caused by the skin effect), normally not taken into account in time domain models.
TL;DR: It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions, and boundary-value methods are developed to overcome these instabilities.
Abstract: An investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank. We introduce a classification of the solutions into two types, explicit and implicit. For the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached. No such sector exists for implicit solutions. In consequence, the two types of solution have different uniqueness properties. Another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions.
We also investigate the computation of the solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions. To overcome these instabilities we develop boundary-value methods, complete with error analysis.
Numerical examples illustrate the computation of both explicit and implicit solutions, and also the associated Stokes multipliers.
TL;DR: It is shown that the equation of motion of a simple pendulum cannot be derived from Newton's laws of particle mechanics without hidden assumptions.
Abstract: It is shown that the equation of motion of a simple pendulum cannot be derived from Newton's laws of particle mechanics without hidden assumptions. The exposure of these assumptions leads to an interesting inverse problem.
TL;DR: It is shown how to solve inhomogeneous linear differential algebraic systems with constant coefficients with coefficients of constant coefficients.
Abstract: We show how to solve inhomogeneous linear differential algebraic systems with constant coefficients.
TL;DR: All second-order linear ordinary differential equations are shown in an elementary way to possess the symmetry algebra ${s}\ell(2,R)$.
Abstract: All second-order linear ordinary differential equations are shown in an elementary way to possess the symmetry algebra ${s}\ell(2,R)$.
TL;DR: This paper discusses mathematical models for the response of a small air bubble in water to an ultrasound pulse, a context that arises in the modern treatment for kidney stones, and applies asymptotics to describe large-amplitude solutions of Rayleigh's equations.
Abstract: This paper discusses mathematical models for the response of a small air bubble in water to an ultrasound pulse, a context that arises in the modern treatment for kidney stones. The paper reviews Rayleigh's 1917 theory for bubble response, applies asymptotics to describe large-amplitude solutions of Rayleigh's equations, and briefly discusses effects neglected in the simple model. The style is expository, intended both to introduce this application to mathematicians and to illustrate the use of asymptotic methods to nonmathematicians.