Showing papers on "Asymptotic analysis published in 1992"

Exact Mean Integrated Squared Error

Abstract: An exact and easily computable expression for the mean integrated squared error (MISE) for the kernel estimator of a general normal mixture density, is given for Gaussian kernels of arbitrary order. This provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis. The family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes. A number of applications of this method giving important new insights into kernel density estimation are presented. Among these is the discovery that the usual asymptotic approximations to the MISE can be quite inaccurate, especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes. 1. Introduction. Substantial research has been devoted to kernel density estimation. This is because it provides a simple, yet appealing, context in which to study problems and issues that arise in all types of nonparametric curve estimation. This includes regression, spectral density and hazard estimation, and also a variety of other estimators, including histograms, splines and orthogonal series. Three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis, simulation and numerical calculation of error criteria. Each of these methods provides many useful insights into the complicated structure present in the study of curve estimation. However, each has its limitations as well. rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples, through general results applying to entire classes of settings. The weakness of asymptotics is that they only describe behavior in the limit. This is still very useful in many situations because the asymptotics describe the actual situation quite well. However, it is less useful when the asymptotics have not yet kicked in (that is, in studying situations where the asymptotically dominant effect has not taken over yet). Perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring.

Smoothed cross-validation

Abstract: For bandwidth selection of a kernel density estimator, a generalization of the widely studied least squares cross-validation method is considered. The essential idea is to do a particular type of “presmoothing” of the data. This is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error. Analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance. The rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions. The method is more appealing than other selectors with this property, because its motivation is not heavily dependent on precise asymptotic analysis, and because its form is simple and intuitive. Theory is also given for choice of the amount of presmoothing, and this is used to derive a data-based method for this choice.

Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data

Abstract: It is shown that the multidimensional signal subspace method, termed weighted subspace fitting (WSF), is asymptotically efficient. This results in a novel, compact matrix expression for the Cramer-Rao bound (CRB) on the estimation error variance. The asymptotic analysis of the maximum likelihood (ML) and WSF methods is extended to deterministic emitter signals. The asymptotic properties of the estimates for this case are shown to be identical to the Gaussian emitter signal case, i.e. independent of the actual signal waveforms. Conclusions concerning the modeling aspect of the sensor array problem are drawn. >

Almost all cubic graphs are Hamiltonian

Abstract: In a previous article the authors showed that at least 98.4% of large labelled cubic graphs are hamiltonian. In the present article, this is improved to 100% in the limit by asymptotic analysis of the variance of the number of Hamilton cycles with respect to populations of cubic graphs with fixed numbers of short odd cycles. © 1992 Wiley Periodicals, Inc.

A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method

Abstract: Moore's asymptotic analysis of vortex-sheet motion predicts that the Kelvin–Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron. Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff–Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.

Subordinacy and spectral theory for infinite matrices

Abstract: The notion of subordinacy, previously used as a tool in the spectral theory of ordinary differential operators, is extended and applied to solutions of three-term recurrence relations. The resulting asymptotic analysis yields a complete description of the spectral behaviour of infinite tridiagonal matrices, and provides a unified approach to the spectral theory of such operators, without detailed special assumptions

Asymptotic theory of thermal convection in rapidly rotating systems

Abstract: An asymptotic theory of marginal thermal convection in rotating systems is constructed for the limit of rapid rotation. Many self-gravitating astronomical bodies, including the major planets, the Sun, and the Earth's liquid core, correspond to this limit. In the laboratory, an analogous system can be constructed with a very rapidly rotating apparatus, in which the centrifugal force plays the role of self-gravitation. The formulation is offered in such a way that both these geophysical systems and laboratory analogues are included as special cases. When the inclination of the outer boundaries relative to the equatorial plane is considered weak, the two types of system are identical at leading order. In this limit, the asymptotic analysis is profoundly simplified, because the system satisfies the Taylor-Proudman theorem to leading order. Nevertheless the system contains a very peculiar property: the mode defined by a conventional WKBJ theory implicitly assuming a locality of convection in the radial direction perpendicular to the axis of rotation cannot be accepted as a correct marginal mode, because a modulation equation gives an exponential growth in the radial direction, which contradicts an implicit initial assumption. The erroneous behaviour is traced to a spatial dispersion of thermal Rossby waves, which governs the marginal mode. The difficulty is resolved by extending the analysis to a complex plane of the radial coordinate of the point where convection amplitude attains its maximum. Such a complex radial distance is defined as the point where the wave dispersion disappears locally. The projection of the solution onto the real axis results in an inclination of the Taylor columns with respect to the radial direction. This is in good agreement with the most recent numerical studies. The isolation of convective Taylor columns in the radial direction weakens and the spiralling gets stronger as the Prandtl number decreases, as a result of the need to displace the critical radial distance further from the real axis.

Asymptotic analysis of a cohesive crack: 1. Theoretical background

Abstract: A method to analyze the evolution of a cohesive crack, particularly appropriate for asymptotic analysis, is presented. Detailed descriptions of the zeroth order and first order asymptotic approaches are given and from these results the far field equivalent elastic crack theorem is derived. An analytically soluble example, the Griffith crack, and a simple model, the Dugdale model, are used to exemplify the results.

On the asymptotic behavior of heterogeneous statistical multiplexer with applications

Abstract: The author combines a heterogeneous statistical multiplexer in heavy traffic with different characteristics in discrete time which is representative of the asynchronous transfer mode (ATM) environment at the cell level. An exact formulation of the queuing model for the multiplexer is presented. Using spectral decomposition method and asymptotic analysis, it is shown that for fixed source average utilization and peak rate, as the burst size of the individual sources increase the tail behavior of the distribution of the number of cells queued in the multiplexer has a simple characterization. This characterization provides a simple approximation of the queuing behavior of the multiplexer, where the impact of each source is quite evident. The accuracy of this approximation is examined. Some applications are considered where both buffer sizing and admission control are discussed. >

The Asymptotic Diffusion Limit of Discretized Transport Problems

Abstract: A well-known asymptotic analysis describes the transition of transport theory to diffusion theory in the limit of optically thick systems with small absorption and sources. Recently, this analysis ...

Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I : The strong fragmentation case

Abstract: The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.

Soliton generation for initial-boundary-value problems.

Abstract: The solution of the initial-boundary-value problem of integrable nonlinear evolution equations, with the spatial variable on a half-infinite line, can be reduced to the solution of a linear integral equation. The asymptotic analysis of this equation for large t shows how the boundary conditions can generate solitons

Finite element asymptotic analysis of slender elastic structures: A simple approach

Abstract: An asymptotic method directly derived from Koiter's theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here. The order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features (theoretical definitions, eigenproblem numerical techniques, suitable FEM implementation) which make such an approach competitive with respect to the classic step-by-step methods. For plane frames and 3D pin-jointed trusses, the performances of the algorithm (numerical accuracy and computational cost) are compared with those of Riks' are-length method.

On the evolution of phase boundaries

Abstract: This volume emphasizes the interdisciplinary nature of contemporary research in the field of phase transitions, research which involves ideas from nonlinear partial differential equations, asymptotic analysis, numerical computation and experiment. Topics covered include the treatment of scaling laws that describe the coarsening or ripening behaviour observed during the later stages of phase transitions; novel numerical methods for treating interface dynamics; the mathematical description of geometric models of interface dynamics; determination of the governing equations and interfacial boundary conditions in the context of fluid flow and elasticity. This volume should be valuable for any researcher pursuing modern developments in the theory and applications of phase transitions and interface dynamics.

Liquid‐state integral equations at high density: On the mathematical origin of infinite‐range oscillatory solutions

Abstract: Analytic asymptotic analysis and finite element numerical procedures are used to elucidate the mathematical reasons for the appearance of infinite‐range oscillatory solutions to certain integral equation theories of wall–fluid interfacial structure and liquid state radial distribution functions. The results contribute to two issues of recent debate: (i) what physical significance (if any) can be attributed to the apparent ‘‘solidlike’’ structure that is often (but not always) seen in high density solutions to liquid state integral equation theories and (ii) is the same mathematical structure present in density functional theories (i.e., in the presence of a variational condition arising from a free energy functional)?

Analytic Analysis of Algorithms

Abstract: The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial structures. Asymptotic methods based on complex analysis permit to extract directly coefficients of structurally complicated generating functions without a need for explicit coefficient expansions.

Hierarchical controls in stochastic manufacturing systems with machines in tandem

Abstract: This paper presents an asymptotic analysis of hierarchical production planning in a manufacturing system with serial machines that are subject to breakdown and repair, and with convex costs. The machines capacities are modeled as Markov chains. Since the number of parts in the internal buffers between any two machines needs to be non-negative, the problem is inherently a state constrained problem. As the rate of change in machines states approaches infinity, the analysis results in a limiting problem in which the stochastic machines capacity is replaced by the equilibrium mean capacity. A method of “lifting” and “modification” is introduced in order to construct near optimal controls for the original problem by using near optimal controls of the limiting problem. The value function of the original problem is shown to converge to the value function of the limiting problem, and the convergence rate is obtained based on some a priori estimates of the asymptotic behavior of the Markov chains. As a result, an ...

An asymptotic closed-form representation for the grounded double-layer surface Green's function

Abstract: An efficient closed-form asymptotic representation for the grounded double-layer (substrate-superstrate) Green's function is presented. The formulation is valid for both source (a horizontal electric dipole) and observation points anywhere inside the superstate or at the interfaces. The asymptotic expressions are developed via a steepest descent evaluation of the original Sommerfeld-type integral representation of the Green's function, and the large parameter in this asymptotic development is proportional to the lateral separation between source and observation points. The asymptotic solution is shown to agree with the exact Green's function for lateral distances even as small as a few tenths of the free-space wavelengths, thus constituting a very efficient tool for analyzing printed circuits/antennas. Since the asymptotic approximation gives separate contributions pertaining to the different wave phenomena, it provides physical insight into the field behavior, as shown by examples. >

Computing bit-error probabilities for avalanche photodiode receivers by large deviations theory

Abstract: A bit error probability analysis of direct detection optical receivers is presented employing avalanche photodiodes. An asymptotic analysis for large signal intensities is presented. This analysis provides some useful insight into the balance between the Poisson statistics, the avalanche gain statistics, and the Gaussian thermal noise. The conjugate distribution is developed. It is obtained by applying the large-deviation exponential twisting formula. It is demonstrated that this conjugate distribution can be used to obtain numerically efficient Monte Carlo estimates of the bit-error probability via the importance sampling method. >

Pattern formation in first-order phase transitions

Abstract: Recent developments in the theory of pattern formation in diffusional growth are reviewed. Asymptotic analysis and scaling arguments are used to construct a morphology diagram in parameter space. The relevant variables spanning this space are the anisotropy ∈ of the capillary length and the dimensionless undercooling Δ. Structures are classified according to whether they are compact or fractal and whether they possess orientational order or not. Explicit expressions for selected velocities and typical length scales are given in the scaling regime. For compact structures the theory provides a prescription how to go beyond scaling results.

Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya-Lyapunov asymptotic method

Abstract: An example is a system with homogeneous quadratic right-hand members: in it, gl = ... = gn = i. Among others, the Euler-Poincare equations describing geodesics on Lie groups with invariant metrics fall into this class. A popular example from dynamics is Kirchoff's problem on the motion of a rigid body in an unbounded volume of an ideal liquid. Quasihomogeneous systems are also exemplified by the equations of the problem of many gravitating bodies and by the Euler-Poisson equations describing the rotation of a heavy rigid body about a fixed point. These remarks show that it is expedient to consider quasihomogeneous systems from the viewpoint of applications.

The Painlevé Transcendents as Nonlinear Special Functions

Abstract: The scheme of the asymptotic analysis of the Painleve transcendents via the isomon-odromic deformations method (IDM) is presented. Also, it is shown that the classical Laplace’s method, which is the analytical basis of the wide use of the usual “linear” special functions in problems of mathematical physics, can be treated as the degenerational case of IDM.