On using polynomial chaos for modeling uncertainty in acoustic propagation

The use of polynomial chaos for incorporating environmental variability into propagation models is investigated in the context of a simplified one-dimensional model, which is relevant for acoustic propagation when the random sound speed is independent of depth. Environmental variability is described by a spectral representation of a stochastic process and the chaotic representation of the wave field then consists of an expansion in terms of orthogonal random polynomials. Issues concerning implementation of the relevant equations, the accuracy of the approximation, uniformity of the expansion over the propagation range, and the computational burden necessary to evaluate different field statistics are addressed. When the correlation length of the environmental fluctuations is small, low-order expansions work well, while for large correlation lengths the convergence of the expansion is highly range dependent and requires high-order approximants. These conclusions also apply in higher-dimensional propagation problems.     

Adaptive sparse polynomial chaos expansion based on least angle regression

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e. of Galerkin type) or non intrusive) unaffordable when the deterministic finite element model is expensive to evaluate.To address such problems, the paper describes a non intrusive method that builds a sparse PC expansion. First, an original strategy for truncating the PC expansions, based on hyperbolic index sets, is proposed. Then an adaptive algorithm based on least angle regression (LAR) is devised for automatically detecting the significant coefficients of the PC expansion. Beside the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon. The accuracy of the PC metamodel is checked using an estimate inspired by statistical learning theory, namely the corrected leave-one-out error. As a consequence, a rather small number of PC terms are eventually retained (sparse representation), which may be obtained at a reduced computational cost compared to the classical “full” PC approximation. The convergence of the algorithm is shown on an analytical function. Then the method is illustrated on three stochastic finite element problems. The first model features 10 input random variables, whereas the two others involve an input random field, which is discretized into 38 and 30 ? 500 random variables, respectively.     

随机过程动态自适应小波独立网格多尺度模拟

在随机过程数值仿真中,由多项式混沌展开谱方法得到求解展开系数的确定性偶合方程组。该方程组比相应的确定性仿真时增大许多。并且当多项式展开阶数和随机空间维数提高时,方程维数急剧增加。由于待求未知分量为表征不同尺度波动的混沌展开模,形成节点意义下的的多尺度问题,传统的网格细分自适应逼近不再适用。为此我们采用了小波的多尺度离散,并建立基于空间细化的动态自适应系统,让每个求解点上的多个未知分量有各自独立的小波网格。本文以随机对流扩散方程为例,进行了二个算例的数值实验,论证了此方法的优点。     

随机对流扩散方程的数值仿真

本文针对对流一扩散随机过程在随机输入（即随机输运和源项）,作用下进行数值仿真。我们先将对流扩散随机微分方程中的随机函数采用有限项截断的多项式浑沌展开（Polynomial Chaos Expansion）展开,再由Galerkin映射法得到求解浑沌展开系数的确定性方程组。这是一个在物理空间包含多尺度解的大方程组。为此我...     

Scattering of scalar light wave from a Gaussianben Schell model medium

The scattering of scalar light wave from a random medium with a correlation function of Gaussian–Schell model distribution is studied. It is shown that the properties of the scattered field, i.e., the spectral density and the spectral degree of coherence of the scattered field, are closely related to the properties of the scattering medium, including the scaled effective radius and the scaled correlation length of the correlation function.     

Freak wave events and the wave phase coherence

In this paper we demonstrate the strong correlation in the spectrum area close to the spectral peak in cases when the Benjamin-Feir instability causes intense wave groups of unidirectional deep-water surface waves referred to freak events. A simple phase coherence estimator in the form of an autobicorrelation function is suggested and tested on the basis of the results of numerical simulations within different frameworks, including the primitive Euler equations. The correlation reaches the value of a unity, and, thus, the random phase approximation is definitely violated for these waves.     

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.     

Polynomial chaos representation of spatio-temporal random fields from experimental measurements

Two numerical techniques are proposed to construct a polynomial chaos (PC) representation of an arbitrary second-order random vector. In the first approach, a PC representation is constructed by matching a target joint probability density function (pdf) based on sequential conditioning (a sequence of conditional probability relations) in conjunction with the Rosenblatt transformation. In the second approach, the PC representation is obtained by having recourse to the Rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target Spearman’s rank correlation coefficient (SRCC) matrix. Both techniques are applied to model an experimental spatio-temporal data set, exhibiting strong non-stationary and non-Gaussian features. The data consists of a set of oceanographic temperature records obtained from a shallow-water acoustics transmission experiment [1]. The measurement data, observed over a finite denumerable subset of the indexing set of the random process, is treated as a collection of observed samples of a second-order random vector that can be treated as a finite-dimensional approximation of the original random field. A set of properly ordered conditional pdfs, that uniquely characterizes the target joint pdf, in the first approach and a set of target marginal pdfs and a target SRCC matrix, in the second approach, are estimated from available experimental data. Digital realizations sampled from the constructed PC representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy. The relative advantages and disadvantages of the two proposed techniques are also highlighted.     

On wave propagation in a random micropolar thermoelastic medium,second moments,and associated Green’s tensor

The couple stress theory developed by Eringen comprises granular materials as also composite fibrous materials. As such, micropolar materials present an inclusive model of composite materials. This article endeavors to study aspects of wave propagation in a random weakly thermal micropolar elastic medium. The smooth perturbation technique has been employed. The classical thermoelasticity has been used. Six different types of waves have been observed to propagate in the random interacting medium. Dispersion equations have been derived. The effects due to random variations of micropolar elastic and thermal parameters have been observed. Change of phase speed occurs on account of randomness. Attenuation coefficients for high-frequency waves have been computed. Second moment properties have been discussed with application to wave propagation in the random micropolar elastic medium. 36 + 1 components of the associated Green’s tensor have been computed. Integrals involving correlation functions have been transformed to radial forms. A special type of correlation function has been used to approximately measure effects of random variations of parameters.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

Canonical transformation method in classical electrodynamics

The solutions of Maxwell's equations in the parabolic equation approximation is obtained on the basis of the canonical transformation method. The Hamiltonian form of the equations for the field in an anisotropic stratified medium is also examined. The perturbation theory for the calculation of the wave reflection and transmission coefficients is developed.     

Polynomial chaos expansions for optimal control of nonlinear random oscillators

The polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems. The procedure is demonstrated on a base-excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen-Loeve expansion. An adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement-velocity norm and is applied to the identification of phase orbits of nonlinear oscillators. This approximation is then integrated in the design of an optimal polynomial controller, allowing for the efficient estimation of statistics and probability density functions of quantities of interest. Numerical investigations are carried out employing the polynomial chaos expansions and the Lyapunov asymptotic stability condition based control policy. The results reveal that the performance, as gaged by probabilistic quantities of interest, of the controlled oscillators is greatly improved. A comparative study is also presented against the classical stochastic optimal control, whereby statistical linearization based LQG is employed to design the optimal controller. It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.     

Spreading of atomic wave packets and semiclassical chaos

The correspondence between the statistical properties of the evolution of a quantum system and Lyapunov instability and the chaos of its semiclassical analog has been demonstrated. The results of the analyses of atomic motion in a laser field in the semiclassical approximation (dynamics is described by several nonlinear equations) and without this approximation (dynamics is described by an infinite system of linear equations) are compared. In the ranges of the parameters for which the semiclassical dynamics of point-like atoms is unstable, the fast “spreading” of quantized wave packets in the momentum space is observed. Thus, deterministic chaos “imitates” the statistics of the quantum nondeterministic effects, although the semiclassical and quantum solutions are fundamentally different.     

Statistical characteristics of a wave propagating through a layer with random irregularities

Statistical characteristics of a wave propagating through a layer with random irregularities are investigated by a simulation procedure. The investigation is carried out within the geometrical optics approximation in its validity range. It is shown that when the irregular layer is a long distance from the source and observer, a significant role in the formation of eikonal (phase path) fluctuations is then played by trajectory fluctuations in regions of the propagation medium, free from irregularities before and after the irregular layer. With these variations taken into account, which are neglected in conventional perturbation theory, we obtained approximate expressions for the dispersion and the correlation function of the eikonal. We investigate the behaviour of the eikonal dispersions, the angles and correlation functions of the eikonal and field for different disturbances of the medium, and for different distances of the receiver and transmitter from the layer boundaries.     

Singular perturbation method for short waves in random media

The problem of wave propagation in a randomly inhomogeneous medium is considered on the basis of the parabolic equation approximation. The method of asymptotic expansions construction in powers of the radius of correlation of the random media for the moments of the wave field are proposed.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

弱光场下电子与库仑势散射问题的弱耦合解法

电子与库仑势散射,在圆极化、偶极近似的外光场作用下,其Schr?dinger方程可通过么正变换并引入修饰势来讨论。对修饰势选取恰当的公式展开,并用Floquet分波法,可分离出径向波动方程组,它在弱耦合近似下是可积的,并且近似的波函数,S矩阵和截面可解析表示,其结果与数值迭代解作了比较。 关键词：