Showing papers on "Asymptotic analysis published in 1997"

Logarithmic Potentials with External Fields

Abstract: This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions.

Asymptotic statistical theory of overtraining and cross-validation

Abstract: A statistical theory for overtraining is proposed. The analysis treats general realizable stochastic neural networks, trained with Kullback-Leibler divergence in the asymptotic case of a large number of training examples. It is shown that the asymptotic gain in the generalization error is small if we perform early stopping, even if we have access to the optimal stopping time. Based on the cross-validation stopping we consider the ratio the examples should be divided into training and cross-validation sets in order to obtain the optimum performance. Although cross-validated early stopping is useless in the asymptotic region, it surely decreases the generalization error in the nonasymptotic region. Our large scale simulations done on a CM5 are in good agreement with our analytical findings.

Asymptotic performance of multiple description transform codes

Abstract: The multiple description transform coder is introduced for sources with memory and an asymptotic analysis is presented for the squared error distortion. For stationary Gaussian sources, the optimal transform and the optimal bit allocation for the multiple description coder are identical to those for the single description coder.

Random hermitian matrices in an external field

Abstract: In this article, a model of random hermitian matrices is considered, in which the measure $\exp(-S)$ contains a general U(N)-invariant potential and an external source term: $S=N\tr(V(M)+MA)$. The generalization of known determinant formulae leads to compact expressions for the correlation functions of the energy levels. These expressions, exact at finite $N$, are potentially useful for asymptotic analysis.

Sequential linear interpolation of multidimensional functions

Abstract: We introduce a new approach that we call sequential linear interpolation (SLI) for approximating multidimensional nonlinear functions. The SLI is a partially separable grid structure that allows us to allocate more grid points to the regions where the function to be interpolated is more nonlinear. This approach reduces the mean squared error (MSE) between the original and approximated function while retaining much of the computational advantage of the conventional uniform grid interpolation. To obtain the optimal grid point placement for the SLI structure, we appeal to an asymptotic analysis similar to the asymptotic vector quantization (VQ) theory. In the asymptotic analysis, we assume that the number of interpolation grid points is large and the function to be interpolated is smooth. Closed form expressions for the MSE of the interpolation are obtained from the asymptotic analysis. These expressions are used to guide us in designing the optimal SLI structure. For cases where the assumptions underlying the asymptotic theory are not satisfied, we develop a postprocessing technique to improve the MSE performance of the SLI structure. The SLI technique is applied to the problem of color printer characterization where a highly nonlinear multidimensional function must be efficiently approximated. Our experimental results show that the appropriately designed SLI structure can greatly improve the MSE performance over the conventional uniform grid.

The Renormalization-Group Method Applied to Asymptotic Analysis,of Vector Fields

Abstract: The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This formulation actually completes the discussion of the previous work for scalar equations. It is shown in a generic way that the method applied to equations with a bifurcation leads to the Landau-Stuart and the (time-dependent) Ginzburg-Landau equations. It is confirmed that this method is actually a powerful theory for the reduction of the dynamics as the reductive perturbation method is. Some examples for ordinar diferential equations, such as the forced Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this method: The time evolution of the solution of the Lotka-Volterra equation is explicitly given, while the center manifolds of the Lorenz equation are constructed in a simple way in the RG method.

Asymptotic analysis of the Navier-Stokes equations in the domains

Abstract: We are interested in this article with the Navier–Stokes equations of viscous incompressible fluids in three dimensional thin domains Let Ωe be the thin domain Ωe = ω × (0, e), where ω is a suitable domain in R and 0 < e < 1 Our aim is to derive an asymptotic expansion of the strong solution u of the Navier–Stokes equations in the thin domain Ωe when e is small, which is valid uniformly in time This study should give a better understanding of the global existence results in thin domains obtained previously; see [15]–[17] and [23], [22] We consider in this work two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition For the first type of boundary condition we derive an asymptotic expansion of the solution u in terms of the solution of the associated Stokes problem More precisely, we prove that the solution can be written, for e small, as

Asymptotic theory of liquid–solid impact

Abstract: The liquid–solid impact problem is analysed with the help of the method of matched asymptotic expansions. This method allows us to estimate the roles of different effects (viscosity of the liquid, surface tension, compressibility, nonlinearity, geometry) on the impact, to distinguish the regions of the flow and the stages of the impact, where and when each of these effects is of major significance, to present a complete picture of the flow, and describe approximately such phenomena as jetting, escape of the shock onto the liquid–free surface and cavitation. Five stages of the impact are distinguished: supersonic stage, transonic stage, subsonic stage, inertia stage and the stage of developed liquid flow. The asymptotic analysis of each stage is based on general principles of hydrodynamics and will be helpful to design experiments on liquid impact and to develop an adequate computational algorithm, as well as to understand the dynamics of the process.

On summability of distributions and spectral geometry

Abstract: Modulo the moment asymptotic expansion, the Cesaro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic pseudodifferential operators. We show how Cesaro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesaro asymptotic development.

An algorithm for the three-dimensional packing problem with asymptotic performance analysis

Abstract: The three-dimensional packing problem can be stated as follows. Given a list of boxes, each with a given length, width, and height, the problem is to pack these boxes into a rectangular box of fixed-size bottom and unbounded height, so that the height of this packing is minimized. The boxes have to be packed orthogonally and oriented in all three dimensions. We present an approximation algorithm for this problem and show that its asymptotic performance bound is between 2.5 and 2.67. This result answers a question raised by Li and Cheng [5] about the existence of an algorithm for this problem with an asymptotic performance bound less than 2.89.

Infinite systems for a biharmonic problem in a rectangle

Abstract: This paper addresses a general analytical method based upon Mellin's transforms technique of investigating asymptotic behaviour of infinite systems of linear algebraic equations occurring in some basic two–dimensional biharmonic problems in a rectangular domain. The object of this paper is to prove the advantages of the asymptotic analysis when studying the concrete problems of an equilibrium of an elastic rectangle, creeping flow of viscous fluid set up in a rectangular cavity by tangential velocities applied along its walls, and bending of a clamped thin rectangular plate by a normal load at one surface. The method is illustrated by several numerical examples; the rapidity of convergence and the accuracy of results are studied.

Impulsive stretching of a surface in a viscous fluid

Abstract: The induced transient viscous flow due to a suddenly stretched surface is studied. After a similarity transform, the unsteady Navier--Stokes equation is solved by several methods, including perturbation for small times, numerical integration, and asymptotic analysis for large times. It is found that the validity of the small-time series can be greatly extended and the approach to steady state is exponential.

Global linear stability analysis of thin aerofoil wakes

Abstract: We investigate the global linear stability properties of the quasi-parallel flow in the neighbourhood of the trailing edge of a thin aerofoil, using a WKBJ/multiple scales formulation in the limit of large Reynolds number, as originally developed by Monkewitz, Huerre & Chomaz. We show that the wake is globally linearly unstable to second order in the asymptotic expansion parameter at all Reynolds numbers provided the effective adverse pressure gradient at the trailing edge, which is related to the aerofoil thickness distribution, is sufficiently large. For smaller adverse pressure gradients, there exists a critical Reynolds number above which the flow is globally linearly unstable, but below which it is globally stable. An asymptotic analysis for large wavenumber indicates that the double Blasius profile, corresponding to a zero adverse pressure gradient, may be absolutely unstable.

Heterogeneous Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

Abstract: A heterogeneous domain-decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems. The method, which is parallelizable at various levels, uses several ideas of asymptotic analysis. The subdomains match the domains of validity of the local (inner and outer) asymptotic expansions, and cut-off functions are used to match solutions in neighboring subdomains. The positions of the interfaces, as well as the mesh widths, depend on the small parameter $\varepsilon$. On the subdomains, iterative solution techniques are used, which may vary from one subdomain to another. The global convergence rate depends on $\varepsilon$; it generally increases like some power of $(\log (\varepsilon ^{-1}))^{-1}$ as $\varepsilon \downarrow 0$. The method is illustrated on several two-dimensional singular perturbation problems.

Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass

Abstract: We consider a hybrid, one-dimensional, linear system con- sisting on two flexible strings connected by a point mass It is known that this system presents two interesting features First, it is well posed in an asymmetric space in which solutions have one more degree of reg- ularity to one side of the point mass Second, that the spectral gap vanishes asymptotically We prove that the first property is a conse- quence of the second one We also consider a system in which the point mass is replaced by a string of length 2" and density 1=2" We show that, as " ! 0, the solutions of this system converge to those of the original one We also analyze the convergence of the spectrum and ob- tain the well-posedness of the limit system in the asymmetric space as a consequence of non-standard uniform bounds of solutions of the approx- imate problems Finally we consider the controllability problem It is well known that the limit system with L 2 -controls on one end is exactly controllable in an asymmetric space We show how this result can be obtained as the limit when " ! 0 of partial controllability results for the approximate systems in which the number of controlled frequencies converges to infinity as " ! 0 This is done by means of some new results on non-harmonic Fourier series

Finite group actions and asymptotic expansion of e P(z)

Abstract: We establish an asymptotic expansion for the number |Hom (G,S n )| of actions of a finite groupG on ann-set in terms of the order |G|=m and the numbers G (d) of subgroups of indexd inG ford|m This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficientsC ν(α;z) in Perron's asymptotic expansion for Laguerre polynomials in the cases α=±1/2

Asymptotic analysis of the linearized Navier-Stokes equations in a general 2D domain

Abstract: We continue our study of the asymptotic behavior of the Navier-Stokes equations linearized around the rest state as viscostiy " approaches zero. We study the convergence as "! 0 to the inviscid type equations. Suitable correctors are obtained which resolve the boundary layer and we obtain convergence results valid up to the boundary. Explicit asymptotic expansion formulas are given which display the boundary layer phenomena. We improve our previous by treating here the general smooth bounded domain in R 2 instead of two-dimensional channels. Curvilinear coordinates are used to resolve the complex geometry.

Asymptotic behavior of solutions to Maxwell's system in bounded domains with absorbing Silver--Müller's condition on the exterior boundary

Abstract: We develop an asymptotic study of Maxwell's system involving an absorbing boundary condition. First- and second-order systems are considered. In both cases, we prove that the solution converges towards a steady state depending on the geometrical properties of the obstacle. The asymptotic state involves the data of the problem.

The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations

Abstract: We study the asymptotic limit of the reaction-diffusion equation E , E 1 r/ EX 1 / E. * e ut = Au 2 / ( u ) + g ( u ) \ t as £ tends to zero in a radially symmetric domain in R subject to the constraint J h(u) dx = const. The energy estimates and the signed distance function approach Q are used to show that a limiting solution can be characterized by moving interfaces . The interfaces evolve by nonlocal (volume preserving) mean curvature flow. Possible interactions between the interfaces are discussed as well.

Asymptotic analysis of first-passage problems

Abstract: The response of stochastically-forced dynamical systems is analyzed in the limit of vanishing noise strength e. We predict asymptotic expressions for the stationary response probability density function (p.d.f.) and for the probability of first-passage of the response to the boundary of a domain in state space. The analysis is limited to Gaussian white noise type perturbations and to domains D in the phase plane “attracted” to an equilibrium point O of the system: all unperturbed trajectories enter D and converge asymptotically to O . In the first stage, the p.d.f. is expressed in terms of an asymptotic WKB form wexp( −Ψ e ) where the “quasi-potential” Ψ can be readily determined numerically by a method of “rays”. A domain of reliability D may then be defined as one bounded by a given contour of quasi-potential, since the latter is a Lyapunov function of the deterministic system. In a second stage, the probability of first-passage is determined in terms of the mean first-passage time to the boundary ∂ D . The latter is found in a singular perturbation solution devised by Matkowsky and Schuss [ SIAM. Appl. Math. 33 , 365 (1977)] in terms of the values reached on ∂ D by Ψ, w and by the deterministic force vector. Several examples demonstrate the validity and usefulness of this approach.