Showing papers on "Asymptotic analysis published in 1996"
01 Jan 1996
TL;DR: The method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates, whereas previous stochastic analyses typically consider mean and mean square behavior.
Abstract: This paper presents an analysis of stochastic gradient-based adaptive algorithms with general cost functions. The analysis holds under mild assumptions on the inputs and the cost function. The method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates, whereas previous stochastic analyses typically consider mean and mean square behavior. The parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time. Furthermore, almost sure exponential bounds are given for the rate of convergence of the parameter estimates. The asymptotic distribution of the parameter estimates is shown to be Gaussian with mean equal to the optimum value and covariance matrix that depends on the input statistics. Specific adaptive algorithms that fall under the framework of this paper are signed error least mean squre (LMS), dual sign LMS, quantized state LMS, least mean fourth, dead zone algorithms, momentum algorithms, and leaky LMS.
316 citations
TL;DR: A variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the Boussinesq equations of motion, from which nonlinear Euler-Lagrange equations for the optimal background fields are derived.
Abstract: Building on a method of analysis for the Navier-Stokes equations introduced by Hopf [Math. Ann. 117, 764 (1941)], a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the Boussinesq equations of motion. When supplied with appropriate test background fields satisfying a spectral constraint, reminiscent of an energy stability condition, the variational formulation produces rigorous upper bounds on the Nusselt number (Nu) as a function of the Rayleigh number (Ra). For the case of vertical heat convection between parallel plates in the absence of sidewalls, a simplified (but rigorous) formulation of the optimization problem yields the large Rayleigh number bound Nu\ensuremath{\le}0.167 ${\mathrm{Ra}}^{1/2}$-1. Nonlinear Euler-Lagrange equations for the optimal background fields are also derived, which allow us to make contact with the upper bound theory of Howard [J. Fluid Mech. 17, 405 (1963)] for statistically stationary flows. The structure of solutions of the Euler-Lagrange equations are elucidated from the geometry of the variational constraints, which sheds light on Busse's [J. Fluid Mech. 37, 457 (1969)] asymptotic analysis of general solutions to Howard's Euler-Lagrange equations. The results of our analysis are discussed in the context of theory, recent experiments, and direct numerical simulations. \textcopyright{} 1996 The American Physical Society.
265 citations
TL;DR: This article introduces a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers) and builds into the numerical scheme the asymPTotic balances that lead to this behavior.
240 citations
TL;DR: In this paper, the authors considered the Blasius boundary-layer flow of a micropolar fluid over a flat plate, and the resulting nonsimilar equations were solved using the Keller-box method and solutions for a range of parameters were presented.
156 citations
TL;DR: It is shown how effective boundary conditions which can be substituted to the thin shell can be obtained and analyzed in a simple way and essentially based on a suitable handling of the stability of the solution relative to the thickness.
Abstract: A model problem in the scattering of a time-harmonic wave by an obstacle coated with a thin penetrable shell is examined. In previous studies, the contrast coefficients of the thin shell are assumed to tend to infinity in order to compensate for the thickness considered. In this paper, these coefficients are assumed to remain finite. Such a treatment leads to a singular perturbation term that creates a typical difficulty for the asymptotic analysis of the problem with respect to the thickness of the coating. As a result, the asymptotic analysis is essentially based on a suitable handling of the stability of the solution relative to the thickness. As a consequence, it is shown how effective boundary conditions which can be substituted to the thin shell can then be obtained and analyzed in a simple way.
140 citations
114 citations
TL;DR: In this article, it was shown analytically and numerically that the heat-conduction equation is not suitable for describing the temperature field of a gas in the continuum limit around bodies at rest in a closed domain or in an infinite domain without flow at infinity, where the flow vanishes in this limit.
Abstract: It is shown analytically and numerically on the basis of the kinetic theory that the heat‐conduction equation is not suitable for describing the temperature field of a gas in the continuum limit around bodies at rest in a closed domain or in an infinite domain without flow at infinity, where the flow vanishes in this limit. The behavior of the temperature field is first discussed by asymptotic analysis of the time‐independent boundary‐value problem of the Boltzmann equation for small Knudsen numbers. Then, simple examples are studied numerically: as the Knudsen number of the system approaches zero, the temperature field obtained by the kinetic equation approaches that obtained by the asymptotic theory and not that of the heat‐conduction equation, although the velocity of the gas vanishes.
108 citations
94 citations
TL;DR: In this article, a 1D slab-geometry lumped linear-discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation is presented.
87 citations
TL;DR: In this paper, the authors used an asymptotic analysis for large temporal wave number to analyze time harmonic integral forms and provided correction terms to partially account for the missing portion of the integral surface.
Abstract: Means of improving the accuracy of Kirchhoff integral solutions for sound fields in cases where the surface may not be completely closed are investigated. Asymptotic analysis for large temporal wave number is used to analyze time harmonic integral forms. Extension to time dependent equations is discussed briefly. Applicability to the "moderate" temporal wave numbers of real problems is discussed. Stationary phase arguments are used to show geometrically where good results are expected from a Kirchhoff integral . on an open surface. A similar asymptotic analysis is used to provide correction terms to partially account for the missing portion of the integral surface. The present study is restricted to the case where the mean flow is parallel to the open surface, but results are applicable to arbitrary flow situations in any number of dimensions. Two dimensional numerical examples are given to demonstrate and evaluate the method. It is found that the correction terms can dramatically reduce the error in an open surface calculation of the radiated sound field.
87 citations
TL;DR: For a given U(1)-bundle E over M = \{x1,..., xn }, where the xi are n distinct points of, the authors in this paper proved that the curvature of the magnetic field converges to a limiting curvature that is singular along line vortices which connect the x i.
Abstract: For a given U(1)-bundle E over M = \{x1 , ..., xn }, where the xi are n distinct points of , we minimise the U(1)-Higgs action and we make an asymptotic analysis of the minimizers when the coupling constant tends to infinity. We prove that the curvature (= magnetic field) converges to a limiting curvature that we give explicitely and which is singular along line vortices which connect the xi . This work is the three dimensional equivalent of previous works in dimension two (see [3] and [4]). The results presented here were announced in [12].
TL;DR: In this article, a comprehensive analysis of the asymptotic behavior of analytic solutions of the initial value problem is given. But the analysis is restricted to the case where A, Bi and Ci are d × d complex matrices, and y0 is a column vector in ℂd.
Abstract: This paper discusses the initial value problemwhere A, Bi and Ci are d × d complex matrices, pi, qi ∈ (0, 1), i = 1, 2, …, and y0 is a column vector in ℂd. By using ideas from the theory of ordinary differential equations and the theory of functional equations, we give a comprehensive analysis of the asymptotic behaviour of analytic solutions of this initial value problem.
TL;DR: In this paper, an output least-squares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity is introduced, based on network approximation results from an asymptotic analysis and its recovery is based on this model.
Abstract: We introduce an output least-squares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modelled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate the performance of the method.
Book•
25 Apr 1996
TL;DR: Asymptotic expansion and linear extrapolation methods have been studied in this article for philosophy error bounds, stopping rules and monotonicity generalizations, with a focus on linear expansion.
Abstract: Part 1 Asymptotic expansion: asymptotic systems and expansions geometric asymptotic expansions logarithmic asymptotic expansions. Part 2 Linear extrapolation methods: fundamental concepts and general philosophy error bounds, stopping rules and monotonicity generalizations and final remarks.
TL;DR: It is shown that with a generally selected preconditioning matrix the actual performance of the PCG scheme may not be superior to an accelerated inverse power method.
Abstract: SUMMARY In this paper, a detailed description of CG for evaluating eigenvalue problems by minimizing the Rayleigh quotient is presented from both theoretical and computational viewpoints. Three variants of CG together with their asymptotic behaviours and restarted schemes are discussed. In addition, it is shown that with a generally selected preconditioning matrix the actual performance of the PCG scheme may not be superior to an accelerated inverse power method. Finally, some test problems in the finite element simulation of 2-D and 3-D large scale structural models with up to 20200 unknowns are performed to examine and demonstrate the performances.
TL;DR: In this article, sufficient conditions for the asymptotic stability of the trivial solution of an impulsive delay differential equation were obtained by investigating respectively the nonoscillatory solutions and oscillatory solutions of the equation.
TL;DR: The concept of flow-normal hyperbolicity was introduced in this article to guarantee asymptotic completeness for inertial manifolds, which is more natural in this case than the traditional linearized flow near the manifold.
Abstract: An investigation of the asymptotic completeness property for inertial manifolds leads to the concept of `flow-normal hyperbolicity', which is more natural in this case than the traditional form of normal hyperbolicity derived from the linearized flow near the manifold. An example shows that without flow-normal hyperbolicity asymptotic completeness cannot be guaranteed. The analysis also yields a new result on the asymptotic equivalence of ordinary differential equations.
Book•
18 Jun 1996TL;DR: In this paper, the propagation of the truncated second-order moment (i.e., integrated over a finite interval enclosing a constant fraction of the total power) of a diffracted beam is analyzed.
TL;DR: In this article, a Hartmann number asymptotic analysis of the motion of an electrically conducting fluid in the presence of a steady magnetic field is presented. But the analysis is restricted to the case of planar symmetries.
Abstract: The motion of an electrically conducting fluid in the presence of a steady magnetic field is analyzed. For any non‐uniform magnetic field and any non‐electromagnetic driving force, a high Hartmann number asymptotic analysis is developed using curvilinear coordinates based on the magnetic field. This analysis yields the structure of the electric current density and velocity fields. In a second step, orthogonal planar symmetries lead to a significant simplification of the asymptotic structure, depending on the nature of the symmetry. The asymptotic solution is applied to some configurations, some of them corresponding to crystal growth from a melt. In the case of electrically insulating boundaries, the nature of the symmetry is found to govern the magnitude and structure of the damped velocity.
TL;DR: Analysis of stochastic gradient based adaptive algorithms with general cost functions is carried out and almost sure behavior is considered, which means the parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time.
Abstract: This paper presents an analysis of stochastic gradient-based adaptive algorithms with general cost functions. The analysis holds under mild assumptions on the inputs and the cost function. The method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates, whereas previous stochastic analyses typically considered mean and mean square behavior. The parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time. Furthermore, almost sure exponential bounds are given for the rate of convergence of the parameter estimates. The asymptotic distribution of the parameter estimates is shown to be Gaussian with mean equal to the optimum value and covariance matrix that depends on the input statistics. Specific adaptive algorithms that fall under the framework of this paper are signed error least mean square (LMS), dual sign LMS, quantized state LMS, least mean fourth, dead zone algorithms, momentum algorithms, and leaky LMS.
TL;DR: In this paper, the bending and stretching problem of doubly curved laminated shells is formulated on the basis of three-dimensional elasticity, and the formulation is made amenable to asymptotic analysis, which otherwise would be too complicated to deal with.
Abstract: Analysis of the bending and stretching problem of doubly curved laminated shells is formulated on the basis of three-dimensional elasticity. The basic idea underlying the approach is to make the formulation amenable to asymptotic analysis, which otherwise would be too complicated to deal with. In the formulation, the basic field equations are first rearranged into equations in terms of displacements and transverse stresses, and then they are made dimensionless by proper scaling of the field variables. By means of asymptotic expansion the recast equations can be decomposed into recurrent sets of differential equations at various levels. It turns out that the asymptotic equations can be integrated in succession, leading to the two-dimensional equations in the classical laminated shell theory (CST) at each level. Higher-order corrections as well as the first-order solution can be determined by treating the CST equations at multiple levels in a systematic and consistent way. The essential feature of the prese...
TL;DR: In this article, an asymptotic analysis of free vibrations of a beam having varying curvature and cross-section is carried out for the equations of free vibration of curved beams.
Book•
01 Jan 1996
TL;DR: A review of the current state of the theory of pattern formation by a liquid-solid interface during crystal growth is given in this paper, including experimental results, mathematical modeling and linear stability analysis.
Abstract: This book reviews the current state of the theory of pattern formation by a liquid-solid interface during crystal growth. It gives a pedagogical introduction to the subject, including experimental results, mathematical modeling and linear stability analysis. After highlighting the success of the theory in resolving the selection problem of dendritic growth, various new directions of research are presented in which progress has been made recently. These are the formation of nondendritic seaweed-like structures, growth of lamellar eutectics and rapid solidification. The interplay between analytic methods on the one hand (scaling arguments, asymptotic analysis, similarity equation, Sivashinsky singular expansion) and numerical calculations on the other (Newton method, dynamical schemes) is emphasized.
TL;DR: In this paper, the asymptotic behavior of a solute plume undergoing reversible sorption governed by a Freundlich isotherm after single-pulse injection is discussed.
Abstract: We discuss the asymptotic behavior of a solute plume undergoing reversible sorption governed by a Freundlich isotherm after single-pulse injection. Our analysis predicts that the concentration at a fixed position decays asymptotically like a power law with an exponent α = 1/(1 − n) where n is the Freundlich exponent. Correspondingly, the shape of tail is time invariant. The results are checked by comparison with numerical solutions for one-dimensional transport in a homogeneous medium. Some further asymptotic results for this case are derived. The power law behavior provides an alternative way to derive the Freundlich n parameter from breakthrough curves in comparison to the use of inverse estimation methods. This is especially the case when evaluating breakthrough curves obtained in two- or three-dimensional flow domains, for which indirect estimation of parameters becomes very difficult.
TL;DR: In this article, a spike-layer solution for the reaction-diiusion equation 2 4u + Q(u) = 0; x 2 D R N ; @ n u + bu = 0 ; x 2 @D ; where b > 0 and D is a bounded convex domain.
Abstract: In the limit ! 0, a spike-layer solution is constructed for the reaction-diiusion equation 2 4u + Q(u) = 0 ; x 2 D R N ; @ n u + bu = 0 ; x 2 @D ; where b > 0 and D is a bounded convex domain. Here Q(u) is such that there exists a unique radially symmetric function u c (?1 r) satisfying 2 4u c +Q(u c) = 0 in all of R N , with u c () decaying exponentially at innnity. The spike-layer solution has the form u u c ?1 jx ? x 0 j], where the spike-layer location x 0 2 D is to be found subject to the condition that dist(x 0 ; @D) = O(1) as ! 0. The determination of x 0 is shown to be exponentially ill-conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x 0. For a strictly convex domain, it is shown that there is an x 0 that is located at an O() distance away from the point in D which is furthest from @D. Finally, hot-spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.