Higher Order Asymptotic Expansion Research Articles

Multiscale nonlocal effective medium model for in-plane elastic wave dispersion and attenuation in periodic composites

This manuscript proposes a multiscale nonlocal homogenization and a nonlocal effective medium model for in-plane wave propagation in periodic composites accounting for dispersion and attenuation due to Bragg scattering. The nonlocal effective medium model is developed based on the spatial-temporal nonlocal homogenization model that is formulated to capture dispersion within the first Brillouin zone with particularly high accuracy along high symmetry directions. The homogenization model is derived by employing high order asymptotic expansions, extending the applicability of asymptotic homogenization to short wavelength regime, to capture wave dispersion and attenuation. The effective medium model is in the form of a second order PDE with a nonlocal effective moduli tensor that contains the nonlocal features of the homogenization model. The proposed models are derived and numerically verified for in-plane elastic wave propagation in two-dimensional periodic composites. It is shown that the dispersion curves of the spatial-temporal nonlocal homogenization model capture the acoustic branch, the first stop band and the optical branch of longitudinal and shear wave modes. The nonlocal effective medium model predicts transient elastic wave propagation in periodic composites and captures wave dispersion and attenuation within the limits of separation of scales.

The second-order bias of quantile estimators

The finite sample theory using higher-order asymptotics provides better approximations of the bias for a class of estimators. Phillips (1991) demonstrated the higher-order asymptotic expansions for LAD estimators. Rilstone et al. (1996) provided the second-order bias results of conditional mean regression estimators. This paper develops new analytical results on the second-order bias of the conditional quantile regression estimators, which enables an improved bias correction and thus to obtain improved quantile estimation. In particular, we show that the second-order bias is larger towards the tails of the conditional density than near the median, and therefore the benefit of the second-order bias correction is greater when we are interested in the deeper tail quantiles, e.g., for the study of income distribution and financial risk management. The Monte Carlo simulation confirms the theory that the bias is larger at the tail quantiles, and the second-order bias correction improves the behavior of the quantile estimators.

Higher order asymptotic behaviour of partial maxima of random sample from generalized Maxwell distribution under power normalization

In this article, the higher order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribution and density of maximum are obtained under nonlinear normalization.

Asymptotic model of size-exclusion grouting

In the construction of foundations of buildings and structures on fragile ground, various technologies of soil grouting are used. When pouring fine-grained concrete into the porous soil, the concrete grains are filtered in the pores of the soil. The filtration process depends on the ratio of the pore sizes of the soil and the solid particles of the injected concrete mortar.Injection of a carrier fluid with small solid particles in a porous medium forms a dynamic concentrations front of suspended and retained particles, separating the suspended particles and the hollow part of the porous frame. The purpose of the study is to construct and calculate an asymptotic model near the concentrations front for the filtration of monodisperse suspension in a porous medium with size-exclusion mechanism of particles retention.The classical mathematical model for one-dimensional filtration of suspensions and colloids in a porous medium is based on the geometric ratio of the particles and pores sizes: the particles freely pass through the large pores and get stuck in the pore throats with sizes smaller than the particles diameter. The model is determined by a system of two quasilinear first-order partial differential equations with the gap between boundary and initial conditions. To construct an asymptotic expansion in the vicinity of the concentrations front, a special small parameter is used that specifies the distance to the front. This parameter provides direct determination of the asymptotic terms from the recurrent system of ordinary linear differential equations of the first order.Near the concentrations front of the suspended and retained particles, a nonlinear high-order asymptotics is constructed for the filtration problem of solid particles transported by a carrier fluid in a porous medium. The obtained solution is zero before the front and nonzero behind the front. Approbation of the asymptotic expansion is carried out. It is shown that, for a linear blocking filtration coefficient, the asymptotics coincides with the exact solution.The asymptotic model of deep bed filtration makes it possible to obtain exact formulas for the high-order asymptotic expansion near the dynamic concentrations front of suspended and retained particles. The asymptotics improves the ability to fine-tune the filtration model depending on the properties of the porous soil and the grout.

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM) [29], whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly capture the singularity of the solution in the near-field. The method requires a fixed number of grid points per wavelength to accurately represent the wave field with an asymptotic convergence rate of O(ω−1/2), where ω is the frequency parameter in the Helmholtz equation. In addition, a fast sweeping-type preconditioner is used to solve the resulting linear system.We present numerical examples in 2D to show both accuracy and efficiency of our method as the frequency increases. In particular, we provide numerical evidence of the convergence rate, and we show empirically that the overall complexity is O(ω2) up to a poly-logarithmic factor.

Corrigendum to “On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters”

A Dirichlet problem for a nonlinear wave equation is investigated. Under suitable assumptions, we prove the solvability and the uniqueness of a weak solution of the above problem. On the other hand, a high-order asymptotic expansion of a weak solution in many small parameters is studied. Our approach is based on the Faedo-Galerkin method, the compact imbedding theorems, and the Taylor expansion of a function.

Higher order expansion for moments of extreme for generalized Maxwell distribution

ABSTRACTIn this paper, the higher order asymptotic expansion of the moments of extreme from generalized Maxwell distribution is gained, by which one establishes the rate of convergence of the moment of the normalized partial maximum to the moment of the associated Gumbel extreme value distribution.

Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg–Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the $${\Gamma}$$ -limit of a sequence of scaled Ginzburg–Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg–Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg–Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg–Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

On steady non-breaking downstream waves and the wave resistance – Stokes’ method

In this work, we have obtained explicit analytical formulae expressing the wave resistance of a two-dimensional body in terms of geometric parameters of nonlinear downstream waves. The formulae have been constructed in the form of high-order asymptotic expansions in powers of the wave amplitude with coefficients depending on the mean depth. To obtain these expansions, the second Stokes method has been used. The analysis represents the next step of the research carried out in Maklakov & Petrov (J. Fluid Mech., vol. 776, 2015, pp. 290–315), where the properties of the waves have been computed by a numerical method of integral equations. In the present work, we have derived a quadratic system of equations with respect to the coefficients of the second Stokes method and developed an effective computer algorithm for solving the system. Comparison with previous numerical results obtained by the method of integral equations has been made.

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Levy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.

A Higher Order Asymptotic Expansion of the Krawtchouk Polynomials

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles.

On the homogenization of thin perforated walls of finite length

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.

Higher-order asymptotics for scoring rules

In this paper we discuss higher-order asymptotic expansions for proper scoring rules generalizing results for likelihood quantities, but meanwhile bring in the difficulty caused by the failure of the information identity. In particular, we derive higher-order approximations to the distribution of the scoring rule estimator, of the scoring rule ratio test statistic and, for a scalar parameter of interest, of the signed scoring rule root statistic. From these expansions, a modified signed scoring rule root statistic is proposed. Examples are given illustrating the accuracy of the modified signed scoring rule root statistic with respect to first-order methods.

Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

Advances in Statistical Inference: Bayesian and likelihood interplay

This volume has been inspired by the aim of gathering a number of theoretical papers that, though from different schools of inference—frequentist andBayesian, share the same interest towards studying complex statistical models. Indeed, both frequentist and Bayesian inference can entail various drawbacks, due to the complexity or misspecification of the model, or to the presence of many nuisance parameters. These difficulties may be overcome from a theoretical point of view through approximate likelihoods and scoring rules, and from a computational point of view through higher-order asymptotic expansions or computational tools from Bayesian inference. Specifically, the volume focuses on the interplay between Bayesian and frequentist inference in addressing these problems. The goal is to highlight and to disseminate some of the topics emerged in the international Workshop Recent advances in Statistical Inference: Theory and case studies, held in Padova, Italy, March 21–23, 2013. The Workshop has been a forum for the exchange of new ideas and thoughts on recent and ongoing research in both Bayesian methodology and modern likelihood inference. Although the topics covered in this volume are diverse, similar themes recur, as research ismostly fueled by the need to deal with complex models, for which traditional methods do not provide viable solutions. Among the topics presented we have Bayesian/frequentist challenges for categorical data analysis, Bayesian-frequentist sample size determination, integrated likelihood inference in semi parametric regression models, approximate Bayesian computation with modified log-likelihood ratios, theory and applications of proper scoring rules, quasi likelihood approximation of posterior distributions for likelihood-intractable complex models, mitigating multicollinearity with spike-and-slab priors, empirical Bayes methods in classical and Bayesian inference. While not exhaustive, this list should give a feeling of the issues discussed at the Workshop.

High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls

We derive a complete asymptotic expansion for the singularly perturbed problem of acoustic wave propagation inside gases with small viscosity. This derivation is for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity exhibits a boundary layer of size [Formula: see text] where η is the dynamic viscosity. The asymptotic expansion, which is based on the technique of multiscale expansion is expressed in powers of [Formula: see text] and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.

Higher-order expansions of distributions of maxima in a Hüsler-Reiss model

The max-stable H\"usler-Reiss distribution which arises as the limit distribution of maxima of bivariate Gaussian triangular arrays has been shown to be useful in various extreme value models. For such triangular arrays, this paper establishes higher-order asymptotic expansions of the joint distribution of maxima under refined H\"{u}sler-Reiss conditions. In particular, the rate of convergence of normalized maxima to the H\"usler-Reiss distribution is explicitly calculated.

Higher-order asymptotics and critical indexes in the [formula omitted] theory

Instanton analysis is applied to the scalar ϕ3 theory. Higher-order asymptotic expansions of the renormalization constants in the minimal subtraction scheme and 6−ϵ dimensional regularization and of the index η are found. Results of the Borel resummation for the presently known terms of the ϵ-expansion of the index η up to the order ϵ4 are presented.

Higher-order Asymptotic Solution for the Fatigue Crack Growth Problem based on Continuum Damage Mechanics

In the paper an analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. In order to evaluate the mechanical behavior in the vicinity of a growing fatigue crack for plane strain and plane stress conditions of mode I the asymptotic governing equations are derived and analyzed by the light of Continuum Damage Mechanics. It is shown that the growing fatigue crack problem can be reduced to the nonlinear eigenvalue problem. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. The eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. The higher-order asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem. The higher order stress, strain and continuity asymptotic fields in the vicinity of the fatigue growing crack are either obtained, in which analytical expressions of the higher order exponents and angular distribution functions (eigenfunctions) of the near tip stress and continuity fields are derived.

Panel data models with nonadditive unobserved heterogeneity: Estimation and inference

This paper considers fixed effects estimation and inference in linear and nonlinear panel data models with random coefficients and endogenous regressors. The quantities of interest—means, variances, and other moments of the random coefficients—are estimated by cross sectional sample moments of generalized method of moments (GMM) estimators applied separately to the time series of each individual. To deal with the incidental parameter problem introduced by the noise of the within-individual estimators in short panels, we develop bias corrections. These corrections are based on higher-order asymptotic expansions of the GMM estimators and produce improved point and interval estimates in moderately long panels. Under asymptotic sequences where the cross sectional and time series dimensions of the panel pass to infinity at the same rate, the uncorrected estimators have asymptotic biases of the same order as their asymptotic standard deviations. The bias corrections remove the bias without increasing variance. An empirical example on cigarette demand based on Becker, Grossman, and Murphy (1994) shows significant heterogeneity in the price effect across U.S. states.