Asymptotic Approximation Method Research Articles

Saddlepoint approximations for the P-values and probability mass functions of some bivariate sign tests

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

Uniform Asymptotic Approximation Method with Pöschl–Teller Potential

In this paper, we study analytical approximate solutions for second-order homogeneous differential equations with the existence of only two turning points (but without poles) by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the Pöschl–Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are ≲0.15∼10% for the first-order approximation of the UAA method. The approximations can be easily extended to high orders, for which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology as well as quasi-normal modes of black holes.

A Cooperative Deception Strategy for Covert Communication in Presence of a Multi-Antenna Adversary

Covert transmission is investigated for a cooperative deception strategy, where a cooperative jammer (Jammer) tries to attract a multi-antenna adversary (Willie) and degrade the adversary’s reception ability for the signal from a transmitter (Alice). For this strategy, we formulate an optimization problem to maximize the covert rate when three different types of channel state information (CSI) are available. The total power is optimally allocated between Alice and Jammer subject to the Kullback-Leibler (KL) divergence constraint, which can be expressed analytically and be widely used as a covertness measurement. Different from the existing literature, in our proposed strategy, we also determine the optimal transmission power at the jammer when Alice is silent, while existing works always assume that the jammer’s power is fixed. Specifically, we apply the S-procedure to convert infinite constraints into linear-matrix-inequalities (LMI) constraints. When statistical CSI at Willie is available, we convert double integration to single integration using asymptotic approximation and substitution method. Finally, our simulation results show that for the proposed strategy, the covert rate is increased with the number of antennas at Willie. Moreover, compared to the benchmark, our proposed strategy is more robust in the presence of imperfect CSI.

Power spectra of slow-roll inflation in the consistent D → 4 Einstein-Gauss-Bonnet gravity

The slow-roll inflation which took place at extremely high energy regimes is in general believed to be sensitive to the high-order curvature corrections to the classical general relativity (GR). In this paper, we study the effects of the high-order curvature term, the Gauss-Bonnet (GB) term, on the primordial scalar and tensor spectra of the slow-roll inflation in the consistent D → 4 Einstein Gauss-Bonnet (4EGB) gravity. The GB term is incorporated into gravitational dynamics via the re-scaling of the GB coupling constant α → α/(D-4) in the limit D → 4. For our purpose, we calculate explicitly the primordial scalar and tensor power spectra with GB corrections accurate to the next-to-leading order in the slow-roll approximation in the slow-roll inflation by using the third-order uniform asymptotic approximation method. The corresponding spectral indices and their runnings of the spectral indices for both the scalar and tensor perturbations as well as the ratio between the scalar and tensor spectra are also calculated up to the next-to-leading order in the slow-roll expansions. These results represent the most accurate results obtained so far in the literature. In addition, by studying the theoretical predictions of the scalar spectral index and the tensor-to-scalar ratio with Planck 2018 constraint in a model with power-law potential, we show that the second-order corrections are important in future measurements.

Statistical Inference of the Class of Nonparametric Tests for the Panel Count and Current Status Data from the Perspective of the Saddlepoint Approximation

Many statisticians resort to using the asymptotic normal approximation method to carry out statistical inference for many statistical tests, especially nonparametric ones. In this article, the saddlepoint approximation method is proposed as an alternative to the asymptotic normal approximation method to carry out statistical inference for a number of nonparametric tests for an important type of data that appears frequently in many clinical studies such as cancer and tumorigenicity studies. In clinical trials, there are many strategies through which treatments are assigned to patients. Equal allocation of both treatments is a largely prevalent approach in clinical trials to eliminate experimental bias and increase power. Accordingly, the statistical analysis is carried out based on the truncated binomial design, which is one of the designs that achieve a perfect balance between the two treatments. To clarify the accuracy of the proposed approximation method, two sets of real data are analyzed, and for the same purpose, a comprehensive simulation study is carried out.

Saddlepoint p-values for a class of nonparametric tests for the current status and panel count data under generalized permuted block design

<abstract><p>Current status and panel count data appear in many applied fields, including medicine, clinical trials, epidemiology, econometrics, demography, engineering and public health. Therefore, in this article, we use the saddlepoint approximation method to approximate the exact p-value of a number of nonparametric tests for the current status and panel count data under a generalized permuted block design. The saddlepoint approximation is referred to as higher-order approximation and it is more accurate than the methods that lead to approximations that are accurate to the first order, such as the asymptotic normal approximation method. To verify the accuracy and efficiency of the saddlepoint approximation method, a simulation study is conducted. The simulation study results confirm that the saddlepoint approximation method is more powerful than the existing approximation method. Furthermore, number of real current status and panel count data sets are analyzed and displayed as illustrative examples.</p></abstract>

Mesoscale Asymptotic Approximations in the Dynamics of Solids with Defects

With reference to the homogenization theory, as well as problems of vibrations in physics and mechanics, we note a growing interest in the analysis of domains whose boundaries are singularly perturbed. We consider solids containing large clusters of defects that may possess unusual dynamic responses to external loads. We review the method of the mesoscale uniform asymptotic approximations, which use the Green kernels and take into account different scales (defined relative to the size of individual inclusions), together with boundary layer fields. The method provides an efficient analytical tool, as well as solvers for hybrid numerical schemes aimed at solutions of boundary value and spectral problems for mesoscale structures, containing large clusters of defects. We discuss examples that include solids with many small inclusions, where a small parameter, the relative size of an inclusion, may compete with a large parameter, representing an overall number of inclusions. We note that in some cases, the approach of mesoscale asymptotic approximations provides a powerful alternative to the conventional homogenization algorithms for solids with large clusters of small inclusions.

Surface waves on a coated incompressible elastic half-space

Abstract Recently, Khajiyeva et al. (2018) studied the dynamics of surface wave propagation in a pre-stressed incompressible half-space when accompanied by both vertical and tangential loads. In this study, however, we extend the work of Khajiyeva et al. to include the presence of a coating layer above the half-space, upon which the loads are acted. Moreover, we prescribe sufficient perfect continuity conditions between the two layers, resulting in an inhomogeneous composite structure. Furthermore, related effective boundary conditions within the long-wave assumption are acquired for the model through the application of the asymptotic approximation method. Finally, approximate uncoupled pseudo-differential equations are derived on the surface, thereby admitting all the results of Khajiyeva et al. as limiting cases of concern.

Meso-scale method of asymptotic analysis of elastic vibrations in periodic and non-periodic multi-structures

Summary The method of meso-scale asymptotic approximations has proved to be very effective for the analysis of models of solids containing large clusters of defects, such as small inclusions or voids. Here, we present a new avenue where the method is extended to elastic multi-structures. Geometrically, a multi-structure makes a step up in the context of overall dimensions, compared to the dimensions of its individual constituents. The main mathematical challenge comes from the analysis of the junction regions assigned to the multi-structure itself. Attention is given to problems of vibration and on the coupling of vibration modes corresponding to displacements of different orientations. The method is demonstrated through the dynamic analysis of infinite or finite multi-scale asymmetric flexural systems consisting of a heavy beam connected to a non-periodic array of massless flexural resonators within some interval. In modelling the interaction between the beam and the resonators, we derive a vectorial system of partial differential equations through which the axial and flexural motions of the heavy beam are coupled. The solution of these equations is written explicitly in terms of Green’s functions having intensities determined from a linear algebraic system. The influence of the resonators on the heavy beam is investigated within the framework of scattering and eigenvalue problems. For large collections of resonators, dynamic homogenization approximations for the medium within the location of the resonant array are derived, leading to (i) the classical Rayleigh beam for symmetric systems and (ii) a generalized Rayleigh beam for asymmetric structures that support flexural–longitudinal wave coupling. Independent numerical simulations are also presented that demonstrate the accuracy of the analytical results.

Parameter identification for portfolio optimization with a slow stochastic factor

Abstract In this paper, we intend to identify two significant parameters – expected return and absolute risk aversion – in the Merton portfolio optimization problem under an exponential utility function where volatility is driven by a slow mean-reverting diffusion process. First, we find the approximate solution of the fully nonlinear Hamilton–Jacobi–Bellman equation for the Merton model by the stochastic asymptotic approximation method. Second, we estimate parameters – expected return and absolute risk aversion – through the approximate solution and prove the uniqueness and stability of the parameter identification problem. Finally, we provide an illustrative example to demonstrate the capacity and efficiency of our method.

New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas

Purpose of the study is to obtain formulas for such a speed of imaginary particles that the circulation of the speed of a (real) fluid along any circuit consisting of these imaginary particles changes (in the process of motion of imaginary particles) according to a given time law. (Until now, only those speeds of imaginary particles were known, at which the mentioned circulation during the motion remained unchanged). Method. Without implementation of asymptotic, numerical and other approximate methods, a rigorous analysis of the dynamic equation of motion (flow) of any continuous fluid medium, from an ideal liquid to a viscous gas, is carried out. Plane-parallel and nonswirling axisymmetric flows are considered. The concept of motion of imaginary particles is used, based on the K. Zoravsky criterion (which is also called A. A. Fridman’s theorem). Results. Formulas for the speed of imaginary particles are proposed. These formulas include the parameters of the (real) flow, their spatial derivatives and the function of time, which determines the law of the change in time of the (real fluid) velocity circulation along the contours moving together with the imaginary particles. In addition, it turned out that for a given function of time (and, as a consequence, for a given law of change in circulation with respect to time), the speed of imaginary particles is determined ambiguously. As a result, a method is proposed to change the speed and direction of motion of imaginary particles in a certain range (while maintaining the selected law of changes in circulation in time). For a viscous incompressible fluid, formulas are proposed that do not include pressure and its derivatives. Conclusion. A new Lagrangian point of view on the vorticity evolution in two-dimensional flows of fluids of all types is proposed. Formulas are obtained for the velocity of such movement of contours, at which the real fluid velocity circulation along any contour changes according to a given time law. This theoretical result can be used in computational fluid dynamics to limit the number of domains when using a gridless method for calculating flows of a viscous incompressible fluid (the method of viscous vortex domains).

The issues and challenges of CLIL implementation in higher education: Teachers’ beliefs in the Ukrainian context

The purpose of CLIL in higher education is to increase the level of students’ both professional knowledge and foreign language competence. At the stage of implementing CLIL, it is important to study teachers’ beliefs on a point, since they can either accelerate or inhibit the CLIL effectiveness. In the research, more than 160 teachers from the Ukrainian universities took part. Conjugation tables, asymptotic approximation method, and Pearson’s statistical criterion χ2 were used to analyze the obtained data. The analysis shows there is relationship between the teachers' subject background and their beliefs on the CLIL appropriateness. The study disproves the influence of the teachers’ pedagogical experience on their views on the CLIL implementation. These findings are confirmed by a factor analysis indicating the importance of appropriate learning materials for CLIL, teachers’ specialty, their professional preferences, the importance of flexibility in choosing the CLIL strategy, depending on students’ content knowledge and language skills. The study indicates challenges for the efficient CLIL implementation, and the directions for the change of teachers’ motivation regarding the importance of CLIL for modern Ukrainian students.

Stiffness Modification-Based Bayesian Finite Element Model Updating to Solve Coupling Effect of Structural Parameters: Formulations

The Bayesian model updating approach (BMUA) benefits from identifying the most probable values of structural parameters and providing uncertainty quantification. However, the traditional BMUA is often used to update stiffness only with the assumption of well-known mass, which allows unidentifiable cases induced by the coupling effect of mass and stiffness to be circumvented and may not be optimal for structures experiencing damages in both mass and stiffness. In this paper, the new BMUA tailored to estimating both mass and stiffness is presented by using two measurement states (original and modified systems). A new eigenequation with a stiffness-modified system is formulated to address the coupling effect of mass and stiffness. The posterior function is treated using an asymptotic approximation method, giving the new objective functions with stiffness modification. Analytical formulations of modal parameters and structural parameters are then derived by a linear optimization method. In addition, the covariance matrix of uncertain parameters is determined by the inverse of the Hessian matrix of the objective function. The performance of the proposed BMUA is evaluated through two numerical examples in this study; a probabilistic damage estimation is also implemented. The results show the proposed BMUA is superior to the traditional one in mass and stiffness updating.

The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

Experimental Evaluation of Individualized Treatment Rules

The increasing availability of individual-level data has led to numerous applications of individualized (or personalized) treatment rules (ITRs). Policy makers often wish to empirically evaluate ITRs and compare their relative performance before implementing them in a target population. We propose a new evaluation metric, the population average prescriptive effect (PAPE). The PAPE compares the performance of ITR with that of non-individualized treatment rule, which randomly treats the same proportion of units. Averaging the PAPE over a range of budget constraints yields our second evaluation metric, the area under the prescriptive effect curve (AUPEC). The AUPEC represents an overall performance measure for evaluation, like the area under the receiver and operating characteristic curve (AUROC) does for classification, and is a generalization of the QINI coefficient used in uplift modeling. We use Neyman’s repeated sampling framework to estimate the PAPE and AUPEC and derive their exact finite-sample variances based on random sampling of units and random assignment of treatment. We extend our methodology to a common setting, in which the same experimental data are used to both estimate and evaluate ITRs. In this case, our variance calculation incorporates the additional uncertainty due to random splits of data used for cross-validation. The proposed evaluation metrics can be estimated without requiring modeling assumptions, asymptotic approximation, or resampling methods. As a result, it is applicable to any ITR including those based on complex machine learning algorithms. The open-source software package is available for implementing the proposed methodology. Supplementary materials for this article are available online.

FLUID INHOMOGENEITY INFLUENCING WAVE MOTION

This article deals with a problem that describes the propagation of surface waves in a layer of an inhomogeneous fluid. The authors present a mathematical model that describes wave motions on the surface of an ideal exponentially stratified fluid. In the equations and boundary conditions, the transition to dimensionless variables and quantities has been completed. Next, a linear version of the problem follows, the solution of which is in the form of progressive waves of a steady-state form with unknown amplitude coefficients. This type of solution is substituted into the equations and boundary conditions of the linear problem, which makes it possible to reduce the determination of unknown quantities to the problem of solving a system of ordinary differential equations. Solving the system has allowed identifying two areas of physical parameters with different nature of wave motion. Expressions are obtained for the unknown components of the fluid velocity, pressure, free surface shape, and wave frequency. This article contains the analysis of the influence of various parameters of the problem on the wave motion: graphs of the dependence of the phase velocity of the wave on the stratification parameter are constructed for different layer depths and wavelengths. For a better understanding of the nature of wave motion, the expressions for the trajectories of liquid particles are determined. This has required writing the equations of motion of particles using the obtained expressions for the components of the velocity vector; these equations are solved with the method of asymptotic approximations. A graphical analysis of the effect of the stratification parameter value on the particle trajectory shape is carried out. The results have revealed that an increase in stratification leads to a compression of the trajectory in the vertical direction.

RESEARCH OF VIBRATION MACHINES DYNAMICS FOR PRODUCT SURFACES PROCESSING BY MATHEMATICAL MODELING

The article describes the methodology for the study of the dynamics of vibrating machines for surface processing of products by mathematical modeling, which is presented in four main stages. The first stage: analysis of classes of vibrating machines for surface treatment of products, choice of basic for solving the technological problem, project of a unified calculation scheme of the machine. The second stage: development of a nonlinear mathematical model for describing the dynamics of the vibration machine working body and its filling, development of elements of automated calculations of the machine. The third stage: the study of the influence of the parameters of the vibrating machine, product sets and tools (with their various combinations) on the factors of the intensity of products surface processing. The fourth stage: recommendations for choosing vibrating machine parameters and machining bodies that will maximize the processing performance of products with the selected intensity criterion. A mathematical model for describing the motion of a vibrating machine for surface treatment of articles by a set of unrelated bodies of small size is created. It has two unbalance units that generate oscillations of its working body and a spring suspension-mounting of the working chamber (container). The model is parametric and nonlinear, incorporating key dynamic, kinematic and geometric parameters of the vibrating machine in symbolic format. It is constructed by: descriptions of the plane-parallel movement of the mechanical system, the rotational motion of the material point and the body; second-order Lagrange equation; asymptotic (approximate) methods of nonlinear mechanics. With the help of the model it is possible: to describe the oscillatory movement of the working chamber (container) of the vibrating machine; to study the influence of the machine parameters on the efficiency of performance of the set technological task, the conditions of occurrence of non-stationary modes of operation of the vibrating machine and the ways of their regulation.

Efficient Statistical Analysis for Correlated Rare Failure Events via Asymptotic Probability Approximation

In this article, a novel asymptotic probability approximation (APA) method is proposed to estimate the overall rare probability of correlated failure events for complex circuits containing a large number of replicated cells (e.g., SRAM bit-cells). The key idea of APA is to approximate the overall circuit failure rate based on a set of carefully defined failure events. An efficient hierarchical subset simulation (H-SUS) method is developed to calculate the aforementioned failure rate and a statistical methodology is further proposed to estimate the confidence interval of APA. Our numerical experiments demonstrate that APA can accurately and reliably estimate the overall failure rate of correlated rare failure events involving more than 20 000 independent random variables.

Hierarchical Bayesian operational modal analysis: Theory and computations

This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the variability prompted due to modeling errors. Such developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyper probability model, whose mean and covariance matrix are unknown, encapsulating the uncertainty of the modal parameters. Detailed computation of this hierarchical model is addressed under two major algorithms using Markov chain Monte Carlo (MCMC) sampling and Laplace asymptotic approximation methods. Since for a small number of data sets the hyper covariance matrix is often unidentifiable, a practical remedy is suggested through the eigenbasis transformation of the covariance matrix, which effectively reduces the number of unknown hyper-parameters. It is also proved that under some conditions the maximum a posteriori (MAP) estimation of the hyper mean and covariance coincide with the ensemble mean and covariance computed using the optimal estimations corresponding to multiple data sets. This interesting finding addresses relevant concerns related to the outcome of the mainstream Bayesian methods in capturing the stochastic variability from dissimilar data sets. Finally, the dynamical response of a prototype structure tested on a shaking table subjected to Gaussian white noise base excitation and the ambient vibration measurement of a cable footbridge are employed to demonstrate the proposed framework.

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

A full analysis of all regimes for optical dispersive shock wave (DSW) propagation in nematic liquid crystals is undertaken. These dispersive shock waves are generated from step initial conditions for the optical field and are resonant in that linear diffractive waves are in resonance with the DSW, resulting in a resonant linear wavetrain propagating ahead of it. It is found that there are six regimes, which are distinct and require different solution methods. In previous studies, the same solution method was used for all regimes, which does not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham modulation theory are used to find solutions for these resonant dispersive shock waves in a given regime. The solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all regimes. It is found that for small initial jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure.