Asymptotic Correction Research Articles

Height-function-based 4D reference metrics for hyperboloidal evolution

Hyperboloidal slices are spacelike slices that reach future null infinity. Their asymptotic behaviour is different from Cauchy slices, which are traditionally used in numerical relativity simulations. This work uses free evolution of the formally-singular conformally compactified Einstein equations in spherical symmetry. One way to construct gauge conditions suitable for this approach relies on building the gauge source functions from a time-independent background spacetime metric. This background reference metric is set using the height function approach to provide the correct asymptotics of hyperboloidal slices of Minkowski spacetime. The present objective is to study the effect of different choices of height function on hyperboloidal evolutions via the reference metrics used in the gauge conditions. A total of 10 reference metrics for Minkowski are explored, identifying some of their desired features. They include 3 hyperboloidal layer constructions, evolved with the non-linear Einstein equations for the first time. Focus is put on long-term numerical stability of the evolutions, including small initial gauge perturbations. The results will be relevant for future (puncture-type) hyperboloidal evolutions, 3D simulations and the development of coinciding Cauchy and hyperboloidal data, among other applications.

Completely Multipolar Model as a General Framework for Many-Body Interactions as Illustrated for Water.

We introduce a general framework for many-body force fields, the Completely Multipolar Model (CMM), that utilizes multipolar electrical moments modulated by exponential decay of electron density as a common functional form for all terms of an energy decomposition analysis of intermolecular interactions. With this common functional form, the CMM model establishes well-formulated damped tensors that reach the correct asymptotes at both long- and short-range while formally ensuring no short-range catastrophes. CMM describes the separable EDA terms of dispersion, exchange polarization, and Pauli repulsion with short-ranged anisotropy, polarization as intramolecular charge fluctuations and induced dipoles, while charge transfer describes explicit movement of charge between molecules, and naturally describes many-body charge transfer by coupling into the polarization equations. We also utilize a new one-body potential that accounts for intramolecular polarization by including an electric field-dependent correction to the Morse potential to ensure that CMM reproduces all physically relevant monomer properties including the dipole moment, molecular polarizability, and dipole and polarizability derivatives. The quality of CMM is illustrated through agreement of individual terms of the EDA and excellent extrapolation to energies and geometries of an extensive validation set of water cluster data.

Weak long‐range diffusion

AbstractWe study the spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.

Deep learning for derivatives pricing: a comparative study of asymptotic and quasi-process corrections

AbstractIn this study, we compare two methods for using neural networks to efficiently learn the price of derivatives. The first method, proposed by Funahashi (Quant Financ 21(4):575–592, 2021a), involves training the neural networks to learn the difference between the derivative price and its asymptotic expansion, rather than learning the derivative price directly. The target derivative price is then obtained by adding the approximate solution with the predicted value of neural networks. This method reduces the required amount of learning data, often by a factor of one hundred to one thousand, compared to the case where the derivative price is directly learned through neural networks. In the second method, established in this paper, the neural networks learn the difference between the derivative price written on the underlying asset price that follows the target complex stochastic process and the derivative price written on the underlying asset price that has a relatively simple stochastic process that has a closed-form solution for the target derivative prices. This method provides an alternative valuation method when no efficient approximate solution for the derivative value is observed and if one can arbitrarily determine the model parameters of the quasi-process that approximates the original process. We also propose a unified method to determine the model parameters of quasi-processes from underlying asset processes. These methods prove valuable in cases where general analytic solutions are absent, as seen in widely used financial models such as the stochastic volatility models. These cases involve time-consuming numerical calculations to generate learning data, highlighting the value of the two methods, which significantly compress calculation times.

Approximate inferences for Bayesian hierarchical generalised linear regression models

SummaryGeneralised linear mixed regression models are fundamental in statistics. Modelling random effects that are shared by individuals allows for correlation among those individuals. There are many methods and statistical packages available for analysing data using these models. Most require some form of numerical or analytic approximation because the likelihood function generally involves intractable integrals over the latents. The Bayesian approach avoids this issue by iteratively sampling the full conditional distributions for various blocks of parameters and latent random effects. Depending on the choice of the prior, some full conditionals are recognisable while others are not. In this paper we develop a novel normal approximation for the random effects full conditional, establish its asymptotic correctness and evaluate how well it performs. We make the case for hierarchical binomial and Poisson regression models with canonical link functions, for hierarchical gamma regression models with log link and for other cases. We also develop what we term a sufficient reduction (SR) approach to the Markov Chain Monte Carlo algorithm that allows for making inferences about all model parameters by replacing the full conditional for the latent variables with a considerably reduced dimensional function of the latents. We expect that this approximation could be quite useful in situations where there are a very large number of latent effects, which may be occurring in an increasingly ‘Big Data’ world. In the sequel, we compare our methods with INLA, which is a particularly popular method and which has been shown to be excellent in terms of speed and accuracy across a variety of settings. Our methods appear to be comparable to theirs in terms of accuracy, while INLA was faster, for the settings we considered. In addition, we note that our methods and those of others that involve Gibbs sampling trivially handle parameters that are functions of multiple parameters, while INLA approximations do not. Our primary illustration is for a three‐level hierarchical binomial regression model for data on health outcomes for patients who are clustered within physicians who are clustered within particular hospitals or hospital systems.

The simple pendulum motion by considering the mass of the string: Asymptotic corrections

In this paper the movement of a pendulum is analyzed considering the mass of the string from which the pendulum hangs, the analysis is carried out by perturbations considering that the mass of the string is much less than the mass of the pendulum. The results show that the mass of the string has two effects on the motion of the pendulum, the first is the increase in the inertia of the system and modification of the center of mass of the system, which translates into an increase in the frequency of the pendulum and the second effect is produced by the vibrations of the string itself, modifying the movement of the pendulum.

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

Higher-order asymptotic corrections and their application to the Gamma Variance Model

We present improved methods for calculating confidence intervals and p values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can incorporate uncertainties in parameters that themselves represent uncertainties (informally, “errors on errors”) called the Gamma Variance Model. This model contains fixed parameters, generically denoted by ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, that represent the relative uncertainties in estimates of standard deviations of Gaussian distributed measurements. If the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are small, one can construct confidence intervals and p values using standard asymptotic methods. This is formally similar to the familiar situation of a large data sample, in which estimators for all adjustable parameters have Gaussian distributions. Here we address the important case where the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are not small and as a consequence the first-order asymptotic distributions do not represent a good approximation. We investigate improved test statistics based on the technology of higher-order asymptotics (modified likelihood root and Bartlett correction). The effective application of higher-order corrections removes an important computational barrier to the use of the Gamma Variance Model.

Asymptotic analysis of Timoshenko-like orthotropic beam with elliptical cross-section

This work presents the asymptotically exact analysis of homogeneous-orthotropic prismatic beam with elliptic cross-section. Variational Asymptotic Method (VAM) has been used to systematically reduce the 3D elastic problem of beam analysis into 2D cross-sectional analysis and 1D along the length of beam analysis. This reduction simplifies beam analysis to a great extent. The 2D analysis supplies cross-sectional stiffness constants to the 1D beam problem. This analysis provides closed-form expression of cross-sectional stiffness constants, 3D displacement and strain fields in terms of 1D strain measures. Present work does not make any a priori ad-hoc assumptions about the deformation as it uses VAM and ensures asymptotic correctness of the solution. The results are obtained for Classical and Timoshenko-like beam models loaded with most general loads. These results are validated by comparing them with existing results in the literature and Finite Element Analysis (FEA) results. Current results are in close agreement with FEA results.

Reliability of Extreme Wind Speeds Predicted by Extreme-Value Analysis

The reliability of extreme wind speed predictions at large mean recurrence intervals (MRI) is assessed by bootstrapping samples from representative known distributions. The classical asymptotic generalized extreme value distribution (GEV) and the generalized Pareto (GPD) distribution are compared with a contemporary sub-asymptotic Gumbel distribution that accounts for incomplete convergence to the correct asymptote. The sub-asymptotic model is implemented through a modified Gringorten method for epoch maxima and through the XIMIS method for peak-over-threshold values. The mean bias error is shown to be minimal in all cases, so that the variability expressed by the standard error becomes the principal reliability metric. Peak-over-threshold (POT) methods are shown to always be more reliable than epoch methods due to the additional sub-epoch data. The generalized asymptotic methods are shown to always be less reliable than the sub-asymptotic methods by a factor that increases with MRI. This study reinforces the previously published theory-based arguments that GEV and GPD are unsuitable models for extreme wind speeds by showing that they also provide the least reliable predictions in practice. A new two-step Weibull-XIMIS hybrid method is shown to have superior reliability.

Enumeration of Corner Polyhedra and 3-Connected Schnyder Labelings

We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson.
 Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants : the number $p_n$ of corner polyhedra and $s_n$ of 3-connected Schnyder woods of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and $(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity.
 While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and $-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the associated series are not D-finite.

Slow-rotating charged black hole solution in dynamical Chern-Simons modified gravity

Chern-Simons (CS) gravity is a modified theory of Einstein's general relativity. The CS theory arises from the low energy limit of string theory which involves anomaly correction to the Einstein-Hilbert action. The CS term is given by the product of the Pontryagin density with a scalar field. In this study, we derive a charged slowly rotating black hole (BH) solution. The main incentives of this BH solution are axisymmetric and stationary and form distortion of the Kerr-Newman BH solution with a dipole scalar field. Additionally, we investigate the asymptotic correction of the metric with the inverse seventh power of the radial distance to the BH solution, This indicates that it will escape any meaningful constraints from weak field experiments. To find this kind of BH by observations, we investigate the propagation of the photon near the BH and we show that the difference between the left-rotated polarization and the right-handed one could be observed as stronger than the case of the Kerr-Newman BH. Finally, we derived the stability condition using the geodesic deviations.

Stability of variable density rotating flows: Inviscid case and viscous effects in the limit of large Reynolds numbers

The stability of variable density rotating flows is studied analytically and numerically. We show that the dynamics which arises is of the cylindrical Rayleigh-Taylor type when density gradients are opposed to centrifugal acceleration. Asymptotic viscous correction determines the most unstable mode and is in very good agreement with direct numerical simulations.

Asymptotic corrections to the low-frequency theory for a cylindrical elastic shell

The general scaling underlying the asymptotic derivation of 2D theory for thin shells from the original equations of motion in 3D elasticity fails for cylindrical shells due to the cancellation of the leading-order terms in the geometric relations for the mid-surface deformations corresponding to shear and circumferential extension. As a consequence, a cylindrical shell as an elastic waveguide supports a small cut-off frequency for each circumferential mode. The value of this cut-off tends to zero at the thin shell limit. In this case, the near-cut-off behaviour is strongly affected by the presence of two small parameters associated with the relative thickness and wavenumber. It is not obvious whether it can be treated within the 2D theory. For the first time, a novel special scaling is introduced, in order to derive an asymptotically consistent formulation for a cylindrical shell starting from 3D framework. Comparisons with the previous results obtained using the popular 2D Sanders–Koiter shell theory are made. Asymptotic corrections are deduced for the fourth-order equation of low-frequency motion and some of other relations, including the formulae for tangential shear stress resultants.

Deep Learning for Derivatives Pricing: A Comparative Study of Asymptotic and Quasi-process Corrections

Deep Learning for Derivatives Pricing: A Comparative Study of Asymptotic and Quasi-process Corrections

Whitham shocks and resonant dispersive shock waves governed by the higher order Korteweg–de Vries equation

The addition of higher order asymptotic corrections to the Korteweg–de Vries equation results in the extended Korteweg–de Vries (eKdV) equation. These higher order terms destabilize the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the eKdV equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

Численное моделирование магнетиков «атом-в-атом», закон Блоха 3/2 и третье начало термодинамики

The traditional model of a magnet <<atom-to-atom>> considers the magnetic moments of individual atoms connected by an exchange interaction. The evolution of magnetic moments is described on the basis of the Landau-Lifshitz equation, in which a random Langevin source is introduced, determined by the final temperature of the system. In this case, one of the main problems is the violation of the Bloch 3/2 law and the third law of thermodynamics. The recalculation of the intensity of a random source, carried out by Wuu et al., taking into account magnons provides the correct 5/2 asymptotics for the energy, but does not give the correct asymptotics for the magnetization. To solve this problem, we add a random delta-correlated in time but correlated in space source to the Langevin source. This approach, in conjunction with the Wuu approach, ensures the fulfillment of the third law of thermodynamics and Bloch's 3/2 law.

A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime

We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius mathfrak {a}/N, moving in the three-dimensional unit torus Lambda . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N rightarrow infty . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.

α formation probability in Be10 and Be12 within a microscopic cluster model

The alpha clustering in the 10Be and 12Be has been studied within the framework of the real-time evolution method (REM). By using the effective interaction tuned to reproduce the charge radii and the threshold energies, we have evaluated the alpha reduced width amplitude (RWA) and spectroscopic factor (S-factor) for the ground and excited states. With several improvements made in this work, comparing with our previous calculations, larger rate of clustering results are obtained for the 10Be and 12Be with correct asymptotic at large distance.

On the Prandtl-Kolmogorov 1-equation model of turbulence.

We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Let τ denote a selected time scale. Away from walls, interpreting an early suggestion of Prandtl, we set [Formula: see text]In the near-wall region analysis suggests replacing the traditional [Formula: see text] ([Formula: see text] wall normal distance) with [Formula: see text] giving [Formula: see text]This specification of [Formula: see text] results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct [Formula: see text], balancing energy input with energy dissipation. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.