Higher Order Asymptotic Expansion Research Articles

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Asymptotic profiles of zero points of solutions to the heat equation

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at O ( t ) O(t) . In a one-dimensional spatial case, we show the zero point’s second and third-order asymptotic profiles in a general situation. We also analyze a zero level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

High order asymptotic expansion for Wiener functionals

Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramér condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space–time white noise is discusses.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition

We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.

Comparative asymptotics for discrete semiclassical orthogonal polynomials

We study the ratio [Formula: see text] asymptotically as [Formula: see text], where the polynomials [Formula: see text] are orthogonal with respect to a discrete linear functional and [Formula: see text] denote the falling factorial polynomials. We give recurrences that allow the computation of high order asymptotic expansions of [Formula: see text] and give examples for most discrete semiclassical polynomials of class [Formula: see text]. We show several plots illustrating the accuracy of our results.

Applications of Laplace’s method in Bayesian analysis and related topics

This article reviews some second-order approximations to the posterior mean, and the marginal distribution of observations called a marginal likelihood, where “second-order” means that the approximation has asymptotic errors of order O ( n − 2 ) O(n^{-2}) as the sample size n n goes to infinity. In addition, the Laplace approximations using asymptotic modes instead of the exact mode of integrand, and higher-order asymptotic expansions are presented.

Higher-order asymptotic expansion for abstract linear second-order differential equations with time-dependent coefficients

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u″(t)+Au(t)+b(t)u′(t)=0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b(t)∼(1+t)−β(−1<β<1). In the case b(t)≡1, it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v′(t)+Av(t)=Fn(t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b(t)v′(t)+Av(t)=0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n-th order profile u˜n is given via v′(t)+mn(t)Av(t)=Fn(t) which coefficient mn (time-scale) differs each other.

Extremal properties of the beta-normal distribution

Asymptotic behaviors of the extremes of the beta-normal distribution are derived. The higher-order asymptotic expansions of the probability density and cumulative distribution functions for the maximum are given under an optimal normalizing constants. In particular, the associated rates of convergence are explicitly calculated.

Asymptotics for Bailey-type mock theta functions

We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem. We encounter cancelation in our estimates for one of the mock theta functions due to the auxiliary function theta _{n,p} arising from the splitting of Hickerson and Mortenson. We deal with this by using higher-order asymptotic expansions for the Jacobi theta functions.

Higher-order small time asymptotic expansion of Itô semimartingale characteristic function with application to estimation of leverage from options

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Itô semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

Imaging of small penetrable obstacles based on the topological derivative method

PurposeThe purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.Design/methodology/approachThe method consists in rewriting the inverse reconstruction problem as a topology optimization problem and then to use the concept of topological derivatives to seek a higher-order asymptotic expansion for the topologically perturbed cost functional. Such expansion is truncated and then minimized with respect to the parameters under consideration, which leads to noniterative second-order reconstruction algorithms.FindingsIn this paper, the authors develop two different classes of noniterative second-order reconstruction algorithms that are able to accurately recover the unknown penetrable obstacles from partial measurements of a field generated by incident waves.Originality/valueThe current paper is a pioneer work in developing a reconstruction method entirely based on topological derivatives for solving an inverse scattering problem with penetrable obstacles. Both algorithms proposed here are able to return the number, location and size of multiple hidden and unknown obstacles in just one step. In summary, the main features of these algorithms lie in the fact that they are noniterative and thus, very robust with respect to noisy data as well as independent of initial guesses.

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

High order asymptotic expansions of a good–bad–ugly wave equation

A heuristic method to find asymptotic solutions to a system of non-linear wave equations near null infinity is proposed. The non-linearities in this model, dubbed good–bad–ugly, are known to mimic the ones present in the Einstein field equations and we expect to be able to exploit this method to derive an asymptotic expansion for the metric in general relativity close to null infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leading asymptotics. For the good–bad–ugly model, we derive formal expansions in which terms proportional to the logarithm of the radial coordinate appear at every order in the bad field, from the second order onward in the ugly field but never in the good field. The model is generalized to wave operators built from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptotic solutions. Finally we define stratified null forms, a generalization of standard null forms, which capture the behavior of different types of field, and demonstrate that the addition of such terms to the original system bears no qualitative influence on the type of asymptotic solutions found.

Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is defined as the sum of a Brownian motion and an independent fractional Brownian motion. Our approach is based on the analysis of the cumulants of this sequence. We show that both the Brownian and fractional Brownian parts contribute to the asymptotic expansion.

A high-order three-scale approach for predicting thermo-mechanical properties of porous materials with interior surface radiation

A high-order three-scale approach developed in this work to analyze thermo-mechanical properties of porous materials with interior surface radiation is systematically studied. The microstructures of the porous structures are described by periodical layout of local cells on the microscopic domain and mesoscopic domain, and surface radiation effect at microscale and mesoscale is also investigated. At first, the three-scale formulas based on reiterated homogenization and high-order asymptotic expansion are established, and the local cell solutions in microscale and mesoscale are also defined. Then, two kinds of homogenized parameters are evaluated by upscaling methods, and the homogenized equations are derived on the whole structure. Further, heat flux and strain fields are constructed as the three-scale asymptotic solutions by assembling the higher-order unit cell solutions and homogenized solutions. The significant features of the proposed approach are an asymptotic high-order homogenization that does not require higher order continuity of the macroscale solutions and a new high-order three-scale formula derived for analyzing the coupled problems. Finally, some representative examples are proposed to verify the presented methods. They show that the three-scale asymptotic expansions introduced in this paper are efficient and valid for predicting the thermo-mechanical properties of the porous materials with multiple spatial scales.

Trapped modes in thin and infinite ladder like domains. Part 2: Asymptotic analysis and numerical application

We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a thick graph , namely a thin structure (the thinness being characterized by a small parameter e > 0) whose limit (when e tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter e) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

Larry Brown’s Contributions to Parametric Inference, Decision Theory and Foundations: A Survey

This article gives a panoramic survey of the general area of parametric statistical inference, decision theory and foundations of statistics for the period 1965–2010 through the lens of Larry Brown’s contributions to varied aspects of this massive area. The article goes over sufficiency, shrinkage estimation, admissibility, minimaxity, complete class theorems, estimated confidence, conditional confidence procedures, Edgeworth and higher order asymptotic expansions, variational Bayes, Stein’s SURE, differential inequalities, geometrization of convergence rates, asymptotic equivalence, aspects of empirical process theory, inference after model selection, unified frequentist and Bayesian testing, and Wald’s sequential theory. A reasonably comprehensive bibliography is provided.