Showing papers in "Siam Review in 1994"
TL;DR: The definition of a multiresolution analysis is given and how wavelets fit into it is shown and the essentials of two major applications are outlined: data compression and compression of linear operators.
Abstract: In this paper an overview of wavelet based multiresolution analyses is presented. First, the continuous wavelet transform in its simplest form is discussed. Then, the definition of a multiresolution analysis is given and how wavelets fit into it is shown. The authors take a closer look at orthogonal, biorthogonal and semiorthogonal wavelets. The fast wavelet transform, wavelets on an interval, multidimensional wavelets and wavelet packets are discussed. Several examples of wavelet families are introduced and compared. Finally, the essentials of two major applications are outlined: data compression and compression of linear operators.
535 citations
TL;DR: In the classical form of the finite element method called the hversion, piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy as discussed by the authors.
Abstract: In the classical form of the finite element method called the hversion, piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy. In this paper, we discuss the fundamental theoretical ideas behind the relatively recent p version and h-p version. In the p version, a fixed mesh is used and p is allowed to increase. The h-p version combines both approaches. The authors describe and explain the basic properties and characteristics of these newer versions, especially in areas where their behavior is significantly different from that of the h version. Simplified proofs of key concepts are included and computational illustrations of several results are provided. A benchmark comparison between the various versions in included.
418 citations
TL;DR: The author presents results concerning scaling exponents in turbulence and estimates the average dissipation rate, the average dimension of level sets, and a class of two-dimensional equations that are useful models of incompressible dynamics.
Abstract: The author presents results regarding certain average properties of incompressible fluids derived from the equations of motion. The author estimates the average dissipation rate, the average dimension of level sets. The role played by the field of direction of vorticity in the three-dimensional Euler and Navier-Stokes equations is discussed and a class of two-dimensional equations that are useful models of incompressible dynamics is described. The author presents results concerning scaling exponents in turbulence.
316 citations
TL;DR: Average case and probabilistic settings are considered and it turns out that in these two settings, worst case unsolvable or intractable problems become solvable or tractable.
Abstract: analyzed. Roughly speaking, when the residual criterion is used, the error of an approximate solution is measured by how much it violates the equation rather than by its distance from the true solution. The worst case setting is the most reassuring setting since the cost and errors of algorithms are defined by their worst case performance. However, for some problems it is too pessimistic and it causes the respective worst case complexity to be huge. This is especially the case for multivariate problems. For ill-posed problems, the situation is even worse since the worst case complexity is infinite. To cope with such intractability (or unsolvability) analyzing the problems in other settings is in order. Therefore, in Chapter 7, average case and probabilistic settings are considered. In these settings, the space of functions is equipped with a probability measure (Gaussian measures are assumed through out the book). Then the error and cost of algorithms are measured by their expectations (average case setting) or probability of being too large (probabilistic setting) instead of by their maximal values. It turns out that in these two settings, worst case unsolvable or intractable problems become solvable or tractable. Another two settings, asymptotic and randomized, are considered in Chapter 8. Both settings are closely related to the worst case setting; however, in the asymptotic setting sequences of algorithms with increasing number of information pieces are studied. A winner is the one with the fastest rate of convergence. In the randomized setting, nondeterministic (random) information and algorithms (e.g., Monte Carlo methods) are allowed. am not quoting concrete results on complexities for various problems and/or settings; in particular, how much complexity reduction is gained by switching from the worst case setting to other settings. Doing so would be like telling the conclusion of a suspense novel to the reader who is about to start reading it. Hence, instead, let me strongly recommend this valuable book.
220 citations
TL;DR: A simple proof of the probability integral transform theorem in probability and statistics is given that depends only on probabilistic concepts and elementary properties of continuous functions and yields the theorem in its fullest generality.
Abstract: A simple proof of the probability integral transform theorem in probability and statistics is given that depends only on probabilistic concepts and elementary properties of continuous functions. This proof yields the theorem in its fullest generality. A similar theorem that forms the basis for the inverse method of random number generation is also discussed and contrasted to the probability integral transform theorem. Typical applications are discussed. Despite their generality and far reaching consequences, these theorems are remarkable in their simplicity and ease of proof.
193 citations
169 citations
TL;DR: Some new techniques, based on error-reducing transformations of the integrand, are described that have been shown to be useful both in estimating high-dimensional integrals and in solving integral equations.
Abstract: Much of the recent work dealing with quasi-random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite-dimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasi-random methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finite-dimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on error-reducing transformations of the integrand, are described that have been shown to be useful both in estimating high-dimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasi-Monte Carlo method certain devices that have proven to be very valuable in statistical (pseudoran...
132 citations
TL;DR: Numerical methods for stochastic control problems in continuous time to help people facing with some harmful virus inside their desktop computer to read a good book with a cup of coffee in the afternoon.
Abstract: Thank you very much for downloading numerical methods for stochastic control problems in continuous time. As you may know, people have look numerous times for their chosen readings like this numerical methods for stochastic control problems in continuous time, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some harmful virus inside their desktop computer.
117 citations
TL;DR: The elementary algorithm presented here fits readily in a first course in numerical linear algebra and is suitable for use in research on total least squares regression.
Abstract: Total least squares regression (TLS) fits a line to data where errors may occur in both the dependent and independent variables. In higher dimensions, TLS fits a hyperplane to such data. The elementary algorithm presented here fits readily in a first course in numerical linear algebra.
112 citations
TL;DR: This paper contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of perturbed planar analytic systems and the general theory is illustrated by a number of examples from the literature.
Abstract: This paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems. The basic theme throughout the paper is that the number, positions, and multiplicities of the limit cycles that bifurcate under perturbations are related to the number, positions, and multiplicities of the zeros of the Melnikov function for the system. The general theory is illustrated by a number of examples from the literature, some of which are extended to include new results.
107 citations
TL;DR: The paper reviews the current state of the mathematical theory of the equations resuiting from differential models and discusses a recently developed technique of integral estimates called Saint Venant’s type, which will be of interest to researchers in partial differential equations who have an interest in fluid mechanics.
Abstract: (4) Mathematical problems arising in differential modelsfor viscoelastic fluids, by C. Guillop6 and J.-C. Saut (29 pages). The characteristic property of viscoelastic fluids is that the stress depends not only on the current velocity field, but on the history of the motion. One way to model such a history dependence is by a system of ordinary differential equations along particle trajectories which relate stress and velocity gradient. Such models are referred to as \"differential\" as opposed to \"integral\" models that express the stress in terms of one or several integrals along particle trajectories. The paper reviews the current state of the mathematical theory of the equations resuiting from differential models. The topics discussed include loss of evolution and change of type, existence of steady flows, inflow boundary conditions, existence for initial value problems, and flow stability. (5) Weak solutionsfor thermoconvectiveflows ofBoussinesq-Stefan type, by J.-E Rodrigues (24 pages). This paper considers a solid-liquid interface problem with a phase transition occurring at the interface. The equations are the heat equation in each phase and the Navier-Stokes equation in the fluid phase. A generalized Newtonian constitutive law with temperature dependent viscosity is allowed for the fluid. The existence of steadystate solutions is established under appropriate assumptions. (6) Boundaryand initial-boundary valueproblemsfor the Navier-Stokes equations in domains with noncompact boundaries, by V. A. Solonnikov (46 pages). In comparison to flows in interior or exterior domains, there are two new issues when the boundary extends to infinity. First, in addition to the usual initial and boundary conditions there needs to be some prescription of fluxes or pressure drops when the flow domain has several \"exits to infinity.\" Secondly, the solutions of interest often have infinite energy integrals. The paper discusses a recently developed technique of integral estimates to deal with this problem. These estimates are called \"Saint Venant’s type\" because the method was first used in the study of Saint Venant’s principle in elasticity. (7) Stability problems in electrohydrodynamics, ferrohydrodynamics and thermoelectric magnetohydrohynamics, by B. Straughan (30 pages). This paper studies the stability of a number of problems that involve the coupling of fluid mechanics with electrodynamic and thermal effects. Both linear and nonlinear stability are considered. The main technique for the nonlinear case is the energy method. (8) Mathematical results for compressible flows, by A. Valli (37 pages). Results on the existence of smooth solutions for compressible fluid mechanics are reviewed. Both inviscid and Newtonian fluids are considered. The paper discusses results on initial value problems as well as steady and time periodic flows. The book contains the following contributed papers: O. Ban, Une solution numrique des quations de Navier-Stokes stationnaires; L. Consiglieri, Stationary solutions for a Binghamflow with nonlocalfriction; S. Fabre, E’tude de la stabilit du couplage des quations d’Euler et Maxwell ?t tree dimension d’espace; E James, Diphasic equilibrium and chemical engineering; M. R. Levitin, Vibrations ofa viscous compressible fluid in bounded and unbounded domains; Yu.I. Skrynnikov, Shock wave in resonant despersion media; A. Skvortsov, Solitary vorticesna new exact solution ofhydrodynamic equations. The book will be of interest to researchers in partial differential equations who have an interest in fluid mechanics. It differs in a pleasant way from the flood of proceedings volumes that look like they are written with a xerox machine and a stapler. The reader will find wellwritten expository articles on various areas of current research. They will be of value to anyone who would like to work in one of these areas or who would simply like to find out about the current state of the art.
TL;DR: Newton’s Method is based on a linear approximation of the function whose roots are to be determined taken at the current point, and the resulting algorithm is known to converge quadratically.
Abstract: Newton’s Method is based on a linear approximation of the function whose roots are to be determined taken at the current point, and the resulting algorithm is known to converge quadratically. In a procedure to increase the rate of convergence the author modifies the target function in such a way that Newton’s Method applied to the modified function will yield a faster rate of convergence.
TL;DR: This work first gives a formal presentation of the notion of indicators for identifying zero variables, and then studies various indicators proposed in the literature for use with interior-point methods for linear programming.
Abstract: This study is concerned with constrained optimization problems where the only inequality constraints are nonnegativity constraints on the variables. In these problems, the ability to identify zero variables (binding constraints) early on in an iterative method is of considerable value and can be used to computational advantage. This work first gives a formal presentation of the notion of indicators for identifying zero variables, and then studies various indicators proposed in the literature for use with interior-point methods for linear programming. Both theory and experimentation are presented that speak strongly against the use of the variables as indicators; perhaps the most frequently used indicator in the literature. This study implies that an indicator proposed by Tapia is particularly effective in the context of primal-dual interior-point methods. The local rate of convergence for several indicators is also studied.
TL;DR: The decomposition method is applied to the solution of the Lotka–Volterra equations modelling the dynamic behaviour of an arbitrary number of species, where the n term is given by a recurrence relation.
Abstract: The decomposition method is applied to the solution of the Lotka–Volterra equations modelling the dynamic behaviour of an arbitrary number of species. The analytical solution derived is an infinite power series for each species, where the n term is given by a recurrence relation. As particular examples; the cases of one, two, and three species are considered. For these cases, comparisons between the present semi-analytical solution and a fully numerical solution (or an exact one for one species) show exc ellent agreement.
TL;DR: This on-going work of numerical stability for wider classes of problems admitting more complicated dynamics is unified and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theories for the older problems are given.
Abstract: In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics. Whilst this has resulted in a coherent and self-contained body of knowledge, it has not thoroughly addressed the problems of real interest in applications. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and possible directions for future work are outlined. In particular, striking similarities between this new developing stability theory and the classical non-linear stability theory are emphasised. The classical theories of $A$, $B$, and algebraic stability for Runge-Kutta methods are briefly reviewed, and it is emphasised that the classes of equations to which these theories apply - linear decay and contractive problems - only admit trivial dynamics. Four other categories of equations - gradient, dissipative, conservative and Hamiltonian systems - are considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to fully chaotic solutions, are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications. Runge-Kutta schemes which preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role. The effects of error control are considered, and multi-step methods are discussed briefly. Finally, various open problems are described.
TL;DR: If the function $\mu$ is time-limited (compactly supported) and nonsingular, it is proven that this deconvolution problem is ill-posed, and the problem of solving for f is well-posed if the set of convolvers satisfies the condition of being what is called strongly coprime.
Abstract: Linear translation invariant systems (e.g., sensors, linear filters) are modeled by the convolution equation $s = f *\mu $, where f is the input signal, $\mu$ is the system impulse response function (or, more generally, impulse response distribution), and s is the output signal. In many applications, the output s is an inadequate approximation of f, which motivates solving the convolution equation for f, i.e., deconvolving f from $\mu$. If the function $\mu$ is time-limited (compactly supported) and nonsingular, it is proven that this deconvolution problem is ill-posed.A theory of solving such equations has been developed by Berenstein et al. It circumvents ill-posedness by using a multichannel system. If the signal f is overdetermined by using a system of convolution equations, $s_i = f * \mu _i ,\,i = 1, \ldots ,n$, the problem of solving for f is well-posed if the set of convolvers $\{ \mu _i\}$ satisfies the condition of being what is called strongly coprime. In this case, there exist compactly suppor...
TL;DR: P Phenomenological interpretations of the bifurcation behavior of the model equations are offered, based on the pattern formational aspects of convective instabilities in hydrodynamics, morphological instability in solidification, and diffusive instabilitiesIn ecology.
Abstract: Comparisons are made between the results obtained by the application of various methods of weakly nonlinear stability analysis to several prototype reaction-diffusion equations which among them reproduce most of the salient features representative of such investigations of model systems for dissipative phenomena. Emphasis is placed upon the subtleties involved in scaling these equations and deducing asymptotically valid solutions. Phenomenological interpretations of the bifurcation behavior of the model equations are offered, based on the pattern formational aspects of convective instabilities in hydrodynamics, morphological instabilities in solidification, and diffusive instabilities in ecology.
TL;DR: The present note gives an elementary proof of Garabedian's proof that the Green function for the biharmonic boundary value problem with vanishing Dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse.
Abstract: P R. Garabedian showed in 1951 that the Green function for the biharmonic boundary value problem with vanishing Dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse. This refuted a conjecture made by J. Hadamard in 1908. The proof of Garabedian was based on kernel functions; the present note gives an elementary proof.
TL;DR: Under mild assumptions, the classical Karush–Kuhn–Tucker approach to Lagrange multiplier theory is extended to an infinite programming formulation and the result is used to obtain necessity conditions for a well-known problem from the statistical literature on probability density estimation.
Abstract: Under mild assumptions, the classical Karush–Kuhn–Tucker approach to Lagrange multiplier theory is extended to an infinite programming formulation. The main result generalizes the usual first-order necessity conditions to address problems in which the domain of the objective function is Hilbert space and the number of constraints is arbitrary. The result is used to obtain necessity conditions for a well-known problem from the statistical literature on probability density estimation.
TL;DR: The author summarizes the applications of matching a global and a local approximation to specific problems in the nineteenth century by a number of well-known natural philosophers, starting with Laplace in 1805.
Abstract: Ludwig Prandtl is properly credited with the development of the boundary-layer idea in viscous flow, which was generalized to the method of matched asymptotic expansions. However, the idea of matching a global and a local approximation was previously applied to specific problems in the nineteenth century by a number of well-known natural philosophers, starting with Laplace in 1805. The author summarizes their applications in hydrostatics, hydrodynamics, elasticity, electrostatics, and acoustics, with particular attention to the process of matching.
TL;DR: A numerical package of the Mathieu and modified Mathieu functions is developed to actually construct profiles of a sequence of eigenfunctions in order to visualize special eigenmodes and others, and a new type of transition state named "focusing modes" is observed herein.
Abstract: Analysis of eigenfunctions and eigenvalues of the Laplacian is important in the understanding of distributed parameter vibration systems and quantum mechanics. The elliptical domain is advantageous for such work because it allows the explicit representation of eigenfunctions as products of the Mathieu functions. In (Ann. Phys., 9 (1960), pp. 24-751, Keller and Rubinow used wave propagation, geometrical optics, and WKB methods to show that two special types of eigenmodes, whispering gallery and bouncing ball modes, exist on a general convex domain, and they illustrated the case of an elliptical domain. In this paper, we develop a numerical package of the Mathieu and modified Mathieu functions to actually construct profiles of a sequence of eigenfunctions in order to visualize such special eigenmodes and others. In the process, we have also been able to observe a new type of transition state named "focusing modes" herein, which seem to have a complementary behavior to the bouncing ball modes. Numerical eigenvalues are tabulated for comparison and some discussions on the "focusing modes" are presented.
TL;DR: This short survey does not pretend to be absolutely complete and, as with all surveys, is to a certain degree subjective because most attention is paid to scientific work closest to the interests of the author.
Abstract: This short survey does not pretend to be absolutely complete and, as with all surveys, is to a certain degree subjective because most attention is paid to scientific work closest to the interests of the author Nevertheless we make every effort to give an adequate picture of the development of this scientific field The systematic study of singularly perturbed systems began in the Soviet Union forty years ago The basic articles are by A N Tikhonov [8]-[10] Before then, some sporadic papers of foreign authors, for instance [1], were known In his first paper [8], Tikhonov considered the initial value problem
TL;DR: This part of the book gets somewhat technical, but when the unbounded component G of the spectrum of L is simply connected, everything reduces to a study of a Riemann conformal map, and more familiar results show up.
Abstract: the spectrum of L to be able to construct admissible Green’s functions (which somehow reveals a certain similarity to pseudospectra). In this part of the book, the argumentation gets somewhat technical, but when the unbounded component G of the spectrum of L is simply connected, everything reduces to a study of a Riemann conformal map, and more familiar results show up. For instance, asymptotically optimal polynomials can then be obtained by means of Faber polynomials. Recall that Faber polynomials can be defined in terms of their generating function. In fact, this is one of Nevanlinna’s favorite concepts throughout the entire book, and is stimulated by the resolvent, which is the generating function for the successive approximation polynomials p. (L) L:
TL;DR: Expect Kedem's monograph to be influential: it is accessible, innovative, and interesting; however HOC methodology is still but a small section of the time-series territory.
Abstract: intuitive explanations. The writing style is articulate and reflects Dr. Kedem’s exuberant personality and his enthusiasm for his subject is contagious. His references are comprehensive, but not exhaustive. (He has been generous in citing contributions of this reviewer, but the omitted work [2] deserves priority for Theorem 7.1 and the conjectures about theCM approach having O (N -3/2) precision.) The author does have the tendency to draw the reader through many steps of the evolution of his thinking on the HOC problem; your reviewer would have preferred a more direct path to the end product. This enterprising and unconventional monograph will be a useful resource for specialists in frequency estimation and as a source of fundamental and interesting problem areas for mathematically inclined researchers. The theory for discrete-time zero crossings is nothing like as rich and detailed as its continuous-time counterpart (e.g., [3]), yet in view of the needs of digital information processing, this is the more important case. Kedem has documented that there are serious difficulties with known computational methods achieving maximum-likelihood (ML) rates of convergence. As the author explains, there is theoretical indication that a version of the HOC approach also achieves ML precision for some cases. The book provides a springboard for investigators to explore this approach more carefully, and to polish up results and algorithms in Chapter 7. At the very least, unless the noise power is negligible, the CM method is more effective than some of the O(N)-FLOP algorithms distributed in computer packages. In summary, expect Kedem’s monograph to be influential: it is accessible, innovative, and interesting; however HOC methodology is still but a small section of the time-series territory.
TL;DR: Lone wolf: magic of magnamund amazon s3 mathematicians: the new artists?
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TL;DR: The author reviews the foundations of the two mainstreams in matching: the intermediate matching and the asymptotic matching principles and the interrelations between the two procedures are discussed.
Abstract: The author reviews the foundations of the two mainstreams in matching: the intermediate matching and the asymptotic matching principles and the interrelations between the two procedures are discussed.