Higher Order Asymptotic Expansion Research Articles

Steady advection–diffusion around finite absorbers in two-dimensional potential flows

We perform an exhaustive study of the simplest, nontrivial problem in advection-diffusion -- a finite absorber of arbitrary cross section in a steady two-dimensional potential flow of concentrated fluid. This classical problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. We construct the first accurate numerical solution using an efficient new method, which involves mapping to the interior of the disk and using a spectral method in polar coordinates. Our method also combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive new, high-order asymptotic expansions for high and low P\'eclet numbers ($\Pe$). Remarkably, the `high' $\Pe$ expansion remains accurate even for such low $\Pe$ as $10^{-3}$. The two expansions overlap well near $\Pe = 0.1$, allowing the construction of an analytical connection formula that is uniformly accurate for all $\Pe$ and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($\Nu$) as a function of the P\'eclet number with a maximum relative error of 0.53% for all possible geometries. Because our finite-plate problem can be conformally mapped to other geometries, the general problem of two-dimensional advection-diffusion past an arbitrary finite absorber in a potential flow can be considered effectively solved.

Local likelihood method: a bridge over parametric and nonparametric regression

This paper discusses local likelihood method for estimating a regression function in a setting which includes generalized linear models. The local likelihood function is constructed by first considering a parametric model for the regression function. It is defined as a locally weighted log-likelihood with weights determined by a kernel function and a bandwidth. When a large bandwidth is chosen, the resulting estimator would be close to the fully parametric maximum likelihood estimator, so that a large bandwidth would be a relevant choice in the case where the true regression function is near the parametric family. On the other hand, when a small bandwidth is chosen, the performance of the resulting estimator would not depend much on the assumed parametric model, thus a small bandwidth would be desirable if the parametric model is largely misspecified. In this paper, we detail the way in which the risk of the local likelihood estimator is affected by bandwidth selection and model misspecification. We derive explicit formulas for the bias and variance of the local likelihood estimator for both large and small bandwidths. We look into higher order asymptotic expansions for the risk of the local likelihood estimator in the case where the bandwidth is large, which enables us to determine the optimal size of the bandwidth depending on the degree of model misspecification.

On the diminishing returns of higher-order terms in asymptotic expansions of bias

The relative magnitudes are compared of successive terms in a higher-order asymptotic expansion of the bias of the LSDV estimator in dynamic panels. We find that the leading term accounts for the major part of the actual bias in small samples. This implies that bias correction procedures can be based on relatively simple bias approximation formulas.

Second Order Numerical Methods for First Order Hamilton--Jacobi Equations

We present practical numerical methods which produce provably second order approximations for a class of stationary first order Hamilton--Jacobi partial differential equations. Using probabilistic methods, we derive high order asymptotic expansions for a first order method and then use those results to design second order methods. We prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth. Although we limit our attention to second order schemes, in principle the techniques in this paper can be extended to arbitrarily high order methods. Examples illustrate the rate of convergence as well as global sharp resolution of discontinuities. The Hamilton--Jacobi equations we consider correspond to deterministic optimal control problems, and our rate of convergence results are valid for the value functions and for the optimal feedback controls.

Sharp asymptotics of the small solutions to the nonlinear Schrödinger equations of derivative type

This paper studies the large-time behavior of small solutions to the nonlinear Schrödinger equations in one space dimension. Our relevant equations contain the gauge-invariant cubic nonlinearities of derivative type. Since the nonlinear term is the so-called long-range type, it is well-known that the nonlinear solution tends to the modified linear solution called the first asymptotic term. We present the higher-order asymptotic expansion of the nonlinear solution in weighted $L^2$ and $L^{\infty}$ spaces. The result shows that the nonlinear interaction plays an explicit role in the higher-order asymptotic terms as well as in the phase modification. Our method relies on the nonlinear gauge transformations and the application of $L^{\infty}$ decay estimate by Hayashi--Naumkin [13, 12] for estimating the nonlinear solution in Sobolev spaces.

High-order asymptotic expansions for robust tests

This paper provides high‐order asymptotic expansions for the M‐ratio and the signed‐root M‐ratio robust test statistics, which allow one to compute accurate approximations to significance or critical values using the Edgeworth approximation, the Bartlett correction, the variance correction or the saddlepoint approximation. Specific results are obtained for the linear regression model with the Huber M‐estimator. A Monte Carlo study illustrates the numerical accuracy of these approximations, with respect to the usual first‐order approximations.

High-order approximation of wave propagation constants in a uniformly curved waveguide

New high-accuracy approximations of the angular propagation constants of LEpm and LMpm modes in a continuously curved rectangular waveguide are derived. They are high-order asymptotic expansions that are obtained using analytical computations. The formulas can be used for both propagating modes and an arbitrary number of evanescent ones. The approximations are applicable over a broad range of curvature variation and a wide range of wavenumber values. They are efficient tools for the numerical solution of modal equations.

Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the (4−∈)-Dimensionally Regularized O(n)-Symmetric φ4 Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the O(n)-symmetric φ4 theory are found based on the instanton approach in the minimal subtraction scheme for the (4−∈)-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.

Second-order behavior of M-estimators in linear regression with long-memory errors

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the regression parameter in a one-parameter linear regression model when the errors form a long-memory moving average. Suitably standardized difference between an M-estimator and the least-squares estimator is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter θ . For example, in the case of the symmetric common error distribution, if 1/3< θ <1, then a suitably standardized difference between the least absolute deviation and the least-squares estimators converges weakly to a normal distribution. If 0< θ <1/3 then the corresponding limiting distribution is not normal.

Model Selection Using the Estimative and the Approximate p* Predictive Densities

Model selection procedures, based on a simple cross-validation technique and on suitable predictive densities, are taken into account. In particular, the selection criterion involving the estimative predictive density is recalled and a procedure based on the approximate p* predictive density is defined. This new model selection procedure, compared with some other well-known techniques on the basis of the squared prediction error, gives satisfactory results. Moreover, higher-order asymptotic expansions for the selection statistics based on the estimative and the approximate p* predictive densities are derived, whenever a natural exponential model is assumed. These approximations correspond to meaningful modifications of the Akaike's model selection statistic.

The near tip field expansion of a propagating anti‐plane crack with nonuniform velocity†

In this study, a higher order asymptotic expansion of anti‐plane crack tip fields is investigated for dynamic transient crack growth. A six term and a three term expansion for the shear stresses around the crack tip are obtained for cracks propagating at constant and nonuniform velocity, respectively. In order to compare with the near tip field solution derived in this study, an exact transient solution of propagating cracks with constant velocity is examined. The result shows that the use of the singular field to represent the actual stress field should be carefully considered, especially in the early stages of the dynamic transient field.

Coordinate-space approach to the bound-electron self-energy: Coulomb field calculation

A formulation for calculating the one-photon self-energy of an electron in an external field completely in coordinate space, described in an earlier paper, is applied to the calculation of the self-energy of an electron in a Coulomb field by two different numerical methods. In one numerical approach, a high-order asymptotic expansion of the Coulomb Green function is employed to accelerate the convergence of the summation over angular momentum. In the other, a term-by-term subtraction of the divergent terms in the summation over angular momentum is employed. The latter method is found to be preferable from the point of view of both simplicity and precision. Numerical results are compared to the results of previous calculations.

Pulsed-beam propagation in lossless dispersive media I Theory

This first part of a two-part investigation is concerned with the effects of dispersion on the propagation characteristics of the scalar field associated with a highly localized pulsed-beam (PB) wave packet in a lossless homogeneous medium described by the generic wave-number profile k(ω)=ω/c(ω), where c(ω) is the frequency-dependent wave propagation speed. While comprehensive studies have been performed for the one-dimensional problem of pulsed plane-wave propagation in dispersive media, particularly for specific c(ω) profiles of the Lorentz or Debye type, even relatively crude measures tied to generic k(ω) profiles do not appear to have been obtained for the three-dimensional problem associated with a PB wave packet with complex frequency and wave-number spectral constituents. Such wave packets have been well explored in nondispersive media, and simple asymptotic expressions have been obtained in the paraxial range surrounding the beam axis. These paraxially approximated wave objects are now used to formulate the initial conditions for the lossless generic k(ω) dispersive case. The resulting frequency inversion integral is reduced by simple saddle-point asymptotics to extract the PB phenomenology in the well-developed dispersive regime. The phenomenology of the transient field is parameterized in terms of the space–time evolution of the PB wave-front curvature, spatial and temporal beam width, etc., as well as in terms of the corresponding space–time-dependent frequencies of the signal, which are related to the local geometrical properties of the k(ω) dispersion surface. These individual parameters are then combined to form nondimensional critical parameters that quantify the effect of dispersion within the space–time range of validity of the paraxial PB. One does this by performing higher-order asymptotic expansions beyond the paraxial range and then ascertaining the conditions for which the higher-order terms can be neglected. In Part II [J. Opt. Soc. Am. A15, 1276 (1998)], these studies are extended to include the transitional regime at those early observation times for which dispersion is not yet fully developed. Also included in Part II are analytical and numerical results for a simple Lorentz model that permit assessment of the performance of various nondimensional critical estimators.

On a critical case of internal solitary waves in a two-layer fluid

This Brief Communication answers an unsettled question pointed out by Peters and Stoker [Commun. Pure Appl. Math 13, 115 (1960)] with regard to a critical case of a two-layer fluid flow with a free surface supported by a horizontal rigid bottom. In the critical case the coefficient of the nonlinear term in the Korteweg–deVries (KdV) equation vanishes and the KdV theory models only a parallel flow. By using higher-order asymptotic expansion, an extended and a modified KdV equation are derived and internal solitary waves and transition waves in the critical case are found.

Asymptotic expansion of M-estimators with long-memory errors

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the one-sample location parameter when the errors form a long-memory moving average. A suitably standardized difference between an M-estimator and the sample mean is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter $\theta$. If, for example, $1/3 < \theta < 1$, then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric. If $0 < \theta < 1/3$, then the corresponding limiting distribution is nonnormal. This paper thus goes beyond that of Beran who observed, in the case of long-memory Gaussian errors, that M-estimators $T_n$ of the one-sample location parameter are asymptotically equivalent to the sample mean in the sense that $\Var(T_n)/\Var(\bar{X}_n) \to 1$ and $T_n = \bar{X}_n + o_P(\sqrt{\Var(\bar{X}_n)}).$

An asymptotic model for compressional and shear wave excitation in plates by ultrasonic transducers

Pulsed beams excited and detected by ultrasonic transducers are routinely used to characterize elastic structures. An efficient model for transducer–structure interactions is presented and applied to transmission through a fluid-loaded plate with a transmitter–receiver pair. Each transducer is modeled as several complex-transducer points (CTPs) [Zeroug etal., ReviewofProgressinQNDE, edited by Thompson and Chimenti (Plenum, New York, 1994), Vol. 13], which behave as reciprocal, electroacoustic quasi-Gaussian transducers. The interaction of each CTP transmitter–receiver pair with the plate is solved by wave-number spectral decomposition and ray expansions, resulting in a sum of multiply reflected beams propagating within the plate. The resulting time-harmonic beam integrals are approximated asymptotically and transformed to the time domain to yield the received voltage as a finite sum of multiply reflected compressional (P), shear (S), and coupled (P–S) arrivals. Comparison with experiments shows that (a) three CTPs accurately model the flat circular transducers, (b) the efficient asymptotic solution is accurate when the observed arrivals are distinct in time, and (c) at normal incidence, the S and P–S arrivals which are excited by the transducers’ finite spectrum necessitate higher-order asymptotic expansions. This approach can be generalized to transducers’ arbitrary orientation, multilayered and cylindrical configurations.

Estimating error correlation in nonparametric regression

Estimates of error correlations in kernel nonparametric regression are obtained using the method of moments. A high order asymptotic expansion of the estimators shows that they are consistent and asymptotically normal at parametric rates, and provides heuristics for the choice of bandwidth and kernel.

Asymptotic expansion of morphological resonance frequencies in Mie scattering

A high-order asymptotic expansion for the size parameters x(n)((l)) of electromagnetic resonances of large spherical dielectric scatterers is reported. The expansion is calculated up to order n(-8/3), where n is the mode number and is valid for small orders l. Its accuracy is studied as a function of n, l, and the refractive index. For the l = 1 mode, the relative accuracy is better than 10(-4) for n > 50.

Buckling and postbuckling of unsymmetrically laminated angle-ply plates in uniaxial and biaxial compression

A deflection-type perturbation technique is proposed to solve the nonlinear Karman equations of rectangular unsymmetrically laminated angle-ply composite plates with simply supported edges in uniaxial and biaxial compression. Higher-order asymptotic expansions for postbuckling behavior of the plates are constructed by this approach. Effects of initial imperfection, lamination angle and number of layers on buckling and postbuckling of the plates are examined. In some specific cases numerical results obtained from the present solution are in fairly good agreement with other theoretical predictions.

Confluent singularities in 3D continuum Φ 4 theory: resolving critical point discrepancies

A new analysis of the three-dimensional ∅ 4 continuum model series for critical exponents and amplitude ratios is presented. By explicitly allowing for a second correction to scaling term with exponent ω 2 and ω 2/ ω 1 ≈ 2.0, a revision is found of previous critical point estimates that are all in the direction of resolving the remaining discrepancies between the 3D and 4 - ϵ analyses. The new estimates are also consistent with those obtained from high temperature series analyses of 3D lattice Ising models. For example, the exponents γ = 1.238, ν = 0.630, η = 0.0355 and correction to scaling amplitude ratio ɐ ξ/ɐ χ = 0.77 are easily accommodated and possibly even preferred. Detailed verification will require further terms both in the low order perturbation theory and the high order asymptotic expansions.