Asymptotic Expansion Research Articles

Eigenvalue variations of the Neumann Laplace operator due to perturbed boundary conditions

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold (M,g,∂M) under a singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive an asymptotic expansion of the perturbed eigenvalues as the Dirichlet part shrinks to a point x∗∈∂M in terms of the spectral parameters of the unperturbed system. This asymptotic expansion demonstrates the impact of the geometric properties of the manifold at a specific point x∗. Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic expansion. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green’s function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work.

Stability and related zero viscosity limit of steady plane Poiseuille-Couette flows with no-slip boundary condition

We study the existence and stability of smooth solutions to the steady Navier-Stokes equations near plane Poiseuille-Couette flow in the absence of any external force in two-dimensional domain Ω=(0,L)×(0,2). Under the assumption 0<L≪1, we prove that there exist smooth solutions to the steady Navier-Stokes equations which are stable under infinitesimal perturbations of plane Poiseuille-Couette flow. In particular, if the basic flow is the Couette flow, then through a formal asymptotic expansion including the Euler correctors and weak boundary layer correctors with different scales near the rigid walls y=0 and y=2, we can prove that for any finite disturbance o(1) of the Couette flow in horizontal direction, there still exist stable smooth solutions to the steady Navier-Stokes equations. Finally, based on the same linear estimates, we establish the zero viscosity limit of the solutions obtained above to the solutions of the steady Euler equations.

Chaotic fields out of equilibrium are observable independent

Chaotic dynamics is always characterized by swarms of unstable trajectories, unpredictable individually, and thus generally studied statistically. It is often the case that such phase-space densities relax exponentially fast to a limiting distribution, that rules the long-time average of every observable of interest. Before that asymptotic time scale, the statistics of chaos is generally believed to depend on both the initial conditions and the chosen observable. I show that this is not the case for a widely applicable class of models, that feature a phase-space (‘field’) distribution common to all pushed-forward or integrated observables, while the system is still relaxing towards statistical equilibrium or a stationary state. This universal profile is determined by both leading and first subleading eigenfunctions of the transport operator (Koopman or Perron–Frobenius) that maps phase-space densities forward or backward in time.

Globality of the DPW construction for Smyth potentials in the case of SU1,1

We construct harmonic maps into SU1,1/U1 starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of L−1dL=ξ. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].

Deterministic Gradient-Descent Learning of Linear Regressions: Adaptive Algorithms, Convergence Analysis and Noise Compensation.

Weight learning forms a basis for the machine learning and numerous algorithms have been adopted up to date. Most of the algorithms were either developed in the stochastic framework or aimed at minimization of loss or regret functions. Asymptotic convergence of weight learning, vital for good output prediction, was seldom guaranteed for online applications. Since linear regression is the most fundamental component in machine learning, we focus on this model in this paper. Aiming at online applications, a deterministic analysis method is developed based on LaSalle's invariance principle. Convergence conditions are derived for both the first-order and the second-order learning algorithms, without resorting to any stochastic argument. Moreover, the deterministic approach makes it easy to analyze the noise influence. Specifically, adaptive hyperparameters are derived in this framework and their tuning rules disclosed for the compensation of measurement noise. Comparison with four most popular algorithms validates that this approach has a higher learning capability and is quite promising in enhancing the weight learning performance.

Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials

Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials

Evaluation of heat flux intensity factor for a V-notched structure by the isogeometric boundary element method

The conventional boundary element method using piecewise polynomial interpolation cannot accurately simulate the singular heat flux field around the vertex of the V-notch. Herein, the singularity eigen-analysis combined with the isogeometric boundary element method is proposed to calculate the singular heat flux field. The V-notched structure is divided into two parts, in which one is the heat flux singularity sector near the vertex and the other is the remained structure without heat flux singularity. In the singularity sector, the asymptotic expansion of the heat flux is introduced to transform the heat conduction governing equation into ordinary differential eigen equation, from which the singularity orders and eigen angular functions can be determined, except for the amplitude coefficients in the asymptotic expansion. The boundary integral equations for the heat conduction analysis established on the remained structure are discretized by the non-uniform rational B-spline (NURBS) elements. The amplitude coefficients, which are corresponding to the heat flux intensity factors, can be yielded by coupling the isogeometric boundary integral equations with the singularity asymptotic expansion analysis. A coordinate system transformation method is then proposed to transform the heat conduction governing equations of orthotropic and anisotropic material into the one of the isotropic material, and the heat flux intensity factors are approved to be invariable before and after coordinate transformation. Since the NURBS elements are used, fewer elements are required to evaluate the heat flux intensity factors compared with the conventional boundary element method.

Neutrino-antineutrino asymmetry of CνB on the surface of the round Earth

It has been claimed that the coherent scattering of relic neutrinos with the Earth will result in a neutrino-antineutrino asymmetry of O(10−4) on the Earth surface, which is five orders of magnitude larger than the naive model expectation. In this work we show that this overdensity was overestimated for the perfectly round Earth by solving the exact solution with partial waves. After summing over all the angular modes, the maximal asymmetry is only around 10−8 above the ground. To achieve the proposed asymmetry of O(10−4), a special geography may be needed as the experimental site. The code for a fast evaluation of Bessel functions with large orders through asymptotic expansions has been attached along with this work as supplementary material.

A dynamic event-triggered approach for observer-based formation control of multi-agent systems with designable inter-event time

This paper addresses the leader-following formation control problems for generic linear multi-agent systems under directed topology with designable inter-event time. A synthesis approach combining controller and observer design is developed under a dynamic event-triggered communication and control scheme. The proposed feedback control, state estimation, and event-triggered rules are distributed, and only local information is required for each agent to implement these algorithms. The proposed dynamic event-triggered protocol incorporates model-based estimation and clock-like auxiliary dynamic variables to prolong the inter-event time and economize the network resources. Furthermore, the inter-event time is designable, which allows more flexible tuning of communication frequency with only minor performance degradation. Sufficient conditions for formation control are established by linear matrix inequalities. The proposed method exhibits significant improvement over the dynamic event-triggered control methods described in the existing literature. Compared to the existing static event-triggered strategy, the proposed approach significantly reduces the utilization of communication resources while preserving asymptotic convergence to the desired formation. Comparative simulations demonstrate the validity and effectiveness of the proposed theoretical results.

A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation

We propose a thresholding algorithm for Willmore-type flows in \mathbb{R}^{N} . This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth-order linear parabolic partial differential equation whose initial data is the indicator function on the compact set \Omega_{0} . The main results of this paper demonstrate that the boundary \partial\Omega(t) of the new set \Omega(t) , generated by our algorithm, is included in O(t) -neighborhood of \partial\Omega_{0} for small t&gt;0 and that the normal velocity from \partial\Omega_{0} to \partial\Omega(t) is nearly equal to the L^{2} -gradient of Willmore-type energy for small t&gt;0 . Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.

Richtmyer–Meshkov instability when a shock is reflected for fluids with arbitrary equation of state

First predicted by Richtmyer in 1960 and experimentally confirmed by Meshkov in 1969, the Richtmyer–Meshkov instability (RMI) is crucial in fields such as physics, astrophysics, inertial confinement fusion and high-energy-density physics. These disciplines often deal with strong shocks moving through condensed materials or high-pressure plasmas that exhibit non-ideal equations of state (EoS), thus requiring theoretical models with realistic fluid EoS for accurate RMI simulations. Approximate formulae for asymptotic growth rates, like those proposed by Richtmyer, are helpful but rely on heuristic prescriptions for compressible materials. These prescriptions can sometimes approximate the RMI growth rate well, but their accuracy remains uncertain without exact solutions, as the fully compressible RMI growth rate is influenced by both vorticity deposited during shock refraction and multiple sonic wave refractions. This study advances previous work by presenting an analytic, fully compressible theory of RMI for reflected shocks with arbitrary EoS. It compares theoretical predictions with heuristic prescriptions using ideal gas, van der Waals gas and three-term constitutive equations for simple metals, the latter being analysed with detailed and simplified ideal-gas-like EoS. We additionally offer an alternative explicit approximate formula for the asymptotic growth rate. The comprehensive model also incorporates the effects of constant-amplitude acoustic waves at the interface, associated with the D'yakov–Kontorovich instability in shocks.

Antiplane effective properties of two‐phase micropolar elastic fiber‐reinforced composites with parallelogram‐like unit cells

AbstractIn this contribution, heterogeneous micropolar elastic fiber‐reinforced composites (FRCs) with a periodic structure are analyzed using the two‐scale asymptotic homogenization method (AHM). We focus on predicting the antiplane effective properties of micropolar two‐phase FRCs with parallelogram‐like unit cells. The periodic structure is defined by unidirectional, infinitely long, and concentric cylindrical fibers embedded in a homogeneous matrix. Constituent materials are assumed centro‐symmetric isotropic materials, and perfect interface conditions are considered. The AHM allows us to address the local problems on the periodic cell and determine the corresponding effective properties. This is achieved by employing two‐scale asymptotic expansions for the displacement and microrotation fields, which depend on both macro‐ and micro‐scales. The complex variable theory, combined with the complex‐potential method and doubly periodic Weierstrass elliptic functions, is applied to determine the solution of the antiplane local problems. Simple closed‐form formulas are provided for the antiplane stiffness and torque effective properties of two‐phase micropolar elastic FRCs, which depend on the physical properties and volume fractions of constituents. Finally, numerical examples are reported and discussed. Comparisons with other theoretical models are also presented, and good agreements are obtained.

A New Form of Asymptotic Expansion for Non-smooth Differential Equations with Time-Decaying Forcing Functions

A New Form of Asymptotic Expansion for Non-smooth Differential Equations with Time-Decaying Forcing Functions

Stress boundary layers for the Phan-Thien–Tanner fluid at the static contact line in extrudate swell

The method of matched asymptotic expansions is used to examine the behaviour of the Phan-Thien–Tanner (PTT) fluid in the neighbourhood of the contact line singularity in extrudate swell. The solution structure for this shear-thinning viscoelastic fluid model has solvent stresses that dominate the polymer stresses, but requires stress boundary layers to fully resolve the solution at the die wall and free surface. The sizes and mechanism of the boundary layers at the two surfaces are different. We derive and present a similarity solution for the boundary layer at the die wall and an exact solution for the boundary layer at the free surface. An expression for the local behaviour of the free surface is also derived. The results for the boundary layers completes the preliminary asymptotic results for the PTT model presented in previous work.

Convergent and asymptotic expansions of the displacement elastodynamic integral in terms of known functions

The integral ∫0∞Jμ(rt)Jν(Rt)tα(t−s)dt plays an essential role in the study of several phenomena in the theory of elastodynamics (Ceballos and Prato, 2014). But an exact evaluation of this integral in terms of known functions is not possible. In this paper, we derive an analytic representation of this integral in the form of convergent series of elementary functions and hypergeometric functions. This series have an asymptotic character for either, small values of the variable s, or for small values of the variables r and R. It is derived by using the asymptotic technique designed in Lopez (2008) for Mellin convolution integrals. Some numerical experiments show the accuracy of the approximation supplied by the first few terms of the expansion.

Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity

Abstract Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation and to quadratic polynomials for the Monge–Ampère equation for dimension n ≥ 3 n\geq 3 , with an extra logarithmic term for n = 2 n=2 . Via Kelvin transforms, we characterize remainders in the asymptotic expansions by a single function near the origin. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge–Ampère equation in even dimension, but only C n − 1 , α C^{n-1,\alpha} for the Monge–Ampère equation in odd dimension 𝑛, for any α ∈ ( 0 , 1 ) \alpha\in(0,1) .

Asymptotic behavior of the reflectance of a narrow beam by a plane-parallel slab

We consider the radiative transfer of a finite width collimated beam incident normally on a plane-parallel slab composed of a uniform absorbing and scattering medium. This problem is fundamental for modeling and interpreting non-invasive measurements of light backscattered by a multiple scattering medium. Assuming that the beam width is the smallest length scale in the problem, we introduce a perturbation method to determine the asymptotic expansion for the solution of this problem. Using this asymptotic expansion, we determine the leading asymptotic behavior of the reflectance. This result includes the influence integral, which gives the influence of the phase function on the leading asymptotic behavior of the reflectance. We validate this asymptotic theory using a novel implementation of the Monte Carlo method that is fully vectorized to run efficiently in MATLAB. We evaluate the usefulness of this asymptotic behavior for different phase functions and show that it provides valuable insight into the influence of the phase function on spatially resolved non-invasive measurements of light backscattered by a multiple scattering medium.

Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear Q11 finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order O(h2+τ2) in H1-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the Q11 element, a new suitable fully discrete scheme is developed, and the extrapolation result of order O(h3+τ2) in H1-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to h. Here h is the mesh size and τ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.

Cubic-like Features of I–V Relations via Classical Poisson–Nernst–Planck Systems Under Relaxed Electroneutrality Boundary Conditions

We focus on higher-order matched asymptotic expansions of a one-dimensional classical Poisson–Nernst–Planck system for ionic flow through membrane channels with two oppositely charged ion species under relaxed electroneutrality boundary conditions. Of particular interest are the current–voltage (I–V) relations, which are used to characterize the two most relevant biological properties of ion channels—permeation and selectivity—experimentally. Our result shows that, up to the second order in ε=λ/r, where λ is the Debye length and r is the characteristic radius of the channel, the cubic I–V relation has either three distinct real roots or a unique real root with a multiplicity of three, which sensitively depends on the boundary layers because of the relaxation of the electroneutrality boundary conditions. This indicates more rich dynamics of ionic flows under our more realistic setups and provides a better understanding of the mechanism of ionic flows through membrane channels.

High-frequency asymptotic expansions for multiple scattering problems with Neumann boundary conditions

We consider the two-dimensional high-frequency plane wave scattering problem in the exterior of a finite collection of disjoint, compact, smooth (C∞), strictly convex obstacles with Neumann boundary conditions. Using integral equation formulations, we determine the Hörmander classes and derive Melrose-Taylor type high-frequency asymptotic expansions of the total fields corresponding to multiple scattering iterations on the boundaries of the scattering obstacles. These asymptotic expansions are used to obtain sharp wavenumber dependent estimates on the derivatives of multiple scattering total fields. Numerical experiments supporting the validity of these expansions are presented.