Stochastic averaging in parametric regions near separatrices of integrability

Stochastic averaging in Hamiltonian framework, as a powerful tool for the dimensional reduction of strongly-nonlinear stochastic dynamical systems, achieves its enormous efficacy by distinguishing the integrability and resonance of associated conservative systems. Its implementation and mathematical expressions of drift and diffusion coefficients are completely different for different categories. For multi-degree-of-freedom systems, a slight changing of stiffness coefficients may induce an immediate conversion of the integrability, and correspondingly, notably different mathematical expressions, which means a sudden changing of dynamical behaviors. This work is devoted to this anti-intuitive phenomenon through a two-degree-of-freedom nonlinear stochastic system with adjustable parameters, by introducing the concept of the degree of integrability, applying stochastic averaging for quasi-integrable/quasi-nonintegrable systems to the parametric region near the separatrix of integrability, and comparing their accuracy of prediction. Numerical results illustrate that there exists a specific band around the separatrix of integrability, as system parameters fall in which, stochastic averaging for quasi-integrable systems achieves higher accuracy than that for quasi-nonintegrable systems even though the system itself is non-integrable. This work uncovers the existence of such specific band, and constitutes a necessary supplement to stochastic averaging in Hamiltonian framework.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Effective implicit finite‐difference method for sensitivity analysis of stiff stochastic discrete biochemical systems

Simulation of cellular processes is achieved through a range of mathematical modelling approaches. Deterministic differential equation models are a commonly used first strategy. However, because many biochemical processes are inherently probabilistic, stochastic models are often called for to capture the random fluctuations observed in these systems. In that context, the Chemical Master Equation (CME) is a widely used stochastic model of biochemical kinetics. Use of these models relies on estimates of kinetic parameters, which are often poorly constrained by experimental observations. Consequently, sensitivity analysis, which quantifies the dependence of systems dynamics on model parameters, is a valuable tool for model analysis and assessment. A number of approaches to sensitivity analysis of biochemical models have been developed. In this study, the authors present a novel method for estimation of sensitivity coefficients for CME models of biochemical reaction systems that span a wide range of time‐scales. They make use of finite‐difference approximations and adaptive implicit tau‐leaping strategies to estimate sensitivities for these stiff models, resulting in significant computational efficiencies in comparison with previously published approaches of similar accuracy, as evidenced by illustrative applications.Inspec keywords: biochemistry, sensitivity analysis, stochastic processes, cellular biophysics, probability, fluctuations, master equation, reaction kinetics, finite difference methodsOther keywords: effective implicit finite‐difference method, sensitivity analysis, stiff stochastic discrete biochemical systems, cellular processes, mathematical modelling, deterministic differential equation models, inherently probabilistic‐stochastic models, random fluctuations, Chemical Master Equation, biochemical kinetics, kinetic parameter estimation, systems dynamics, CME models, biochemical reaction systems, finite‐difference approximations, adaptive implicit tau‐leaping strategies, computational efficiencies     

On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

This paper presents a novel hybrid polynomial dimensional decomposition (PDD) method for stochastic computing in high-dimensional complex systems. When a stochastic response does not possess a strongly additive or a strongly multiplicative structure alone, then the existing additive and multiplicative PDD methods may not provide a sufficiently accurate probabilistic solution of such a system. To circumvent this problem, a new hybrid PDD method was developed that is based on a linear combination of an additive and a multiplicative PDD approximation, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. Two numerical problems involving mathematical functions or uncertain dynamic systems were solved to study how and when a hybrid PDD is more accurate and efficient than the additive or the multiplicative PDD. The results show that the univariate hybrid PDD method is slightly more expensive than the univariate additive or multiplicative PDD approximations, but it yields significantly more accurate stochastic solutions than the latter two methods. Therefore, the univariate truncation of the hybrid PDD is ideally suited to solving stochastic problems that may otherwise mandate expensive bivariate or higher-variate additive or multiplicative PDD approximations. Finally, a coupled acoustic-structural analysis of a pickup truck subjected to 46 random variables was performed, demonstrating the ability of the new method to solve large-scale engineering problems.     

The clock model and its relationship with the Allan and related variances

The clock errors are modeled by stochastic differential equations (SDE) and the relationships between the diffusion coefficients used in SDE and the Allan variance, a typical tool used to estimate clock noise, are derived. This relationship is fundamental when a mathematical clock model is used, for example in Kalman filter, noise estimation, and clock prediction activities.     

Numerical homogenization of N‐component composites including stochastic interface defects

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so defined is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the effective modules method to media random in the micro‐scale. To obtain such an approach the classical mathematical homogenization method is formulated for n‐component composite with random elastic components and implemented in the FEM‐based computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

Maximum likelihood estimation of stochastic chaos representations from experimental data

This paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems (SBVP). The data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing. Starting with a particular mathematical model for the mechanical behaviour of the specimen, the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization. For each set of measurements of the displacement field along the boundary, an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis. Realizations of these coefficients are then used, in conjunction with the maximum likelihood principle, to set‐up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the SBVP. Copyright © 2005 John Wiley & Sons, Ltd.     

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

Soliton solutions of the resonant nonlinear Schrödinger’s equation in optical fibers with time-dependent coefficients by simplest equation approach

In this paper, the resonant nonlinear Schrödinger’s equation is studied with four forms of nonlinearity. This equation is also considered with time-dependent coefficients. The simplest equation method is applied to solve the governing equations and then exact 1-soliton solutions are obtained. It is shown that this method provides us with a powerful mathematical tool for solving nonlinear evolution equations with time-dependent coefficients in mathematical physics.     

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed-form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross-section column, a 7-story frame, and a Kiewitt single-layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.     

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.     

Global sensitivity analysis using polynomial chaos expansions

Global sensitivity analysis (SA) aims at quantifying the respective effects of input random variables (or combinations thereof) onto the variance of the response of a physical or mathematical model. Among the abundant literature on sensitivity measures, the Sobol’ indices have received much attention since they provide accurate information for most models. The paper introduces generalized polynomial chaos expansions (PCE) to build surrogate models that allow one to compute the Sobol’ indices analytically as a post-processing of the PCE coefficients. Thus the computational cost of the sensitivity indices practically reduces to that of estimating the PCE coefficients. An original non intrusive regression-based approach is proposed, together with an experimental design of minimal size. Various application examples illustrate the approach, both from the field of global SA (i.e. well-known benchmark problems) and from the field of stochastic mechanics. The proposed method gives accurate results for various examples that involve up to eight input random variables, at a computational cost which is 2–3 orders of magnitude smaller than the traditional Monte Carlo-based evaluation of the Sobol’ indices.     

Method of analysis of stochastic nonstationary heat conduction equations

A method is proposed for determination of nonstationary fields of mathematical expectation, second moments, and dispersion of a stochastic nonstationary temperature field for inhomogeneous bodies of an arbitrary form and dimension with arbitrary stochastic boundary conditions.Institute for Cybernetics Problems, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 2, pp. 244–251, February, 1993.     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.     

Generation of non-Gaussian stochastic processes using nonlinear filters

Non-Gaussian stochastic processes are generated using nonlinear filters in terms of Itô differential equations. In generating the stochastic processes, two most important characteristics, the spectral density and the probability density, are taken into consideration. The drift coefficients in the Itô differential equations can be adjusted to match the spectral density, while the diffusion coefficients are chosen according to the probability density. The method is capable to generate a stochastic process with a spectral density of one peak or multiple peaks. The locations of the peaks and the band widths can be tuned by adjusting model parameters. For a low-pass process with the spectrum peak at zero frequency, the nonlinear filter can match any probability distribution, defined either in an infinite interval, a semi-infinite interval, or a finite interval. For a process with a spectrum peak at a non-zero frequency or with multiple peaks, the nonlinear filter model also offers a variety of profiles for probability distributions. The non-Gaussian stochastic processes generated by the nonlinear filters can be used for analysis, as well as Monte Carlo simulation.     

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.     

基于小波变换的非平稳脉动风时变功率谱估计方法研究

为揭示非平稳随机脉动风的时频特性,基于小波变换原理推导了时变功率谱的时间、频率和幅值与小波变换系数的关系,建立了非平稳随机脉动风时变功率谱估计的小波函数加权和法,并采用模拟非平稳脉动风和实测台风过程对理论推导结果进行了验证。研究结果表明:非平稳随机过程在某一时刻的不同尺度小波变换系数是一个以此非平稳随机过程的调制函数与小波函数的乘积为调制函数的非平稳随机过程的傅里叶变换,非平稳随机过程的时变功率谱等于不同尺度和不同时移的小波函数模平方的加权和,小波函数加权和法计算的非平稳随机脉动风的时变功率谱与理论结果具有良好的一致性。小波函数加权和法可有效地估计非平稳随机脉动风的时变功率谱,估计的时变功率谱可为进一步理解强(台)风的随机脉动特性奠定基础。     

Least squares stochastic Boundary Element Method

The main issue in this paper is mathematical formulation and computational implementation of the stochastic Boundary Element Method based on the generalized stochastic perturbation technique. The key feature is a replacement of the given order polynomial response function with the least squares method leading to a numerical determination of this response function. This new approach minimizes the approximation error during the recovery of the structural response indexed with the random input parameter, which is a decisive factor for the entire stochastic method accuracy; contrary to some lower order techniques, numerical implementation of up to the fourth order probabilistic moments is displayed. Computational experiments obey both analyses for the homogeneous and heterogeneous structures with Gaussian random material parameters and also some comparison against the Monte-Carlo simulation and analytical results.     

Identification of composite uncertain material parameters from experimental modal data

Stochastic analysis of structures using probability methods requires the statistical knowledge of uncertain material parameters. This is often quite easier to identify these statistics indirectly from structure response by solving an inverse stochastic problem. In this paper, a robust and efficient inverse stochastic method based on the non-sampling generalized polynomial chaos method is presented for identifying uncertain elastic parameters from experimental modal data. A data set on natural frequencies is collected from experimental modal analysis for sample orthotropic plates. The Pearson model is used to identify the distribution functions of the measured natural frequencies. This realization is then employed to construct the random orthogonal basis for each vibration mode. The uncertain parameters are represented by polynomial chaos expansions with unknown coefficients and the same random orthogonal basis as the vibration modes. The coefficients are identified via a stochastic inverse problem. The results show good agreement with experimental data.