基于微分几何理论的参数失配系统时空同步研究

本文采用状态反馈精确线性化方法,实现了参数失配系统的时空同步控制. 引入微分几何理论,首先采用李导数判定了参数失配系统的仿射模型是否满足状态精确线性化的充要条件,通过非线性坐标变换得到了其线性解耦系统,并结合线性最优控制理论确定了同步控制律,数值仿真证实了该控制方法的有效性,从非线性控制角度探索了混沌系统时空同步的新方法. 关键词： 时空同步 微分几何 精确线性化 参数失配     

Simulation of Stochastic Differential Equations Through the Local Linearization Method. A Comparative Study

A new local linearization (LL) scheme for the numerical integration of nonautonomous multidimensional stochastic differential equations (SDEs) with additive noise is introduced. The numerical scheme is based on the local linearization of the SDE's drift coefficient by means of a truncated Ito–Taylor expansion. A comparative study with the other LL schemes is presented which shows some advantanges of the new scheme over other ones.     

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.     

Piecewise optimal linearization method for nonlinear stochastic differential equations

A method for estimating the dynamical statistical properties of the solutions of nonlinear Langevin-type stochastic differential equations is presented. The non-linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval. The linearization procedure is optimal in the sense of the Chebyshev inequality. Long-term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals. The method is applied to a model nonlinear equation for which the exact time-dependent moments can be obtained by numerical methods. The calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium.Supported in part by the National Science Foundation under Grants CHE77-16307 and PHY76-04761.     

Studies in nonlinear stochastic processes. IV. A comparison of statistical linearization,diagrammatic expansion,and projection operator methods

We compare the methods of statistical linearization, perturbation expansions, and projection operators for the approximate solution of nonlinear multimode stochastic equations. The model equations we choose for this comparison are coupled, nonlinear, first-order, one-dimensional complex mode rate equations. We show that the method of statistical linearization is completely equivalent to the neglect of certain well-defined diagrams in the perturbation expansion resulting in the first Kraichnan-Wyld approximation, and to the retention of only Markovian terms in the projection operator method, i.e., those terms that are local in time.     

Efficient numerical integrators for stochastic models

The efficient simulation of models defined in terms of stochastic differential equations (SDEs) depends critically on an efficient integration scheme. In this article, we investigate under which conditions the integration schemes for general SDEs can be derived using the Trotter expansion. It follows that, in the stochastic case, some care is required in splitting the stochastic generator. We test the Trotter integrators on an energy-conserving Brownian model and derive a new numerical scheme for dissipative particle dynamics. We find that the stochastic Trotter scheme provides a mathematically correct and easy-to-use method which should find wide applicability.     

An L 2 Norm Trajectory-Based Local Linearization for Low Order Systems

Abstract

This paper presents a linear transformation for low order nonlinear autonomous differential equations. The procedure consists of a trajectory-based local linearization, which approximates the nonlinear system in the neighborhood of its equilibria. The approximation is possible even in the non-hyperbolic case which is of a particular interest. The linear system is derived using an L ₂ norm optimization and the method can be used to approximate the derivative at the equilibrium position. Unlike the classical linearization, the L ₂ norm linearization depends on the initial state and has the same order as the nonlinearity. Simulation results show good agreement of the suggested method with the nonlinear system.     

A stochastic Ginzburg–Landau equation with impulsive effects

In this paper we consider a stochastic Ginzburg–Landau equation with impulsive effects. We first prove the existence and uniqueness of the global solution which can be explicitly represented via the solution of a stochastic equation without impulses. Then, based on our obtained result, we study the qualitative properties of the solution, including the boundedness of moments, almost surely exponential convergence and pathwise estimations. Finally, we give a first attempt to study a fractional version of impulsive stochastic Ginzburg–Landau equations.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

EXPLICIT SOLUTION PROCESSES FOR NONLINEAR JUMP-DIFFUSION EQUATIONS

Recent studies have shown that the nonlinear jump-diffusion models give results which are in agreement with financial data. Here we provide linearization criteria together with transformations which linearize the nonlinear jump-diffusion models with compound Poisson processes. Furthermore, we introduce the stochastic integrating factor to solve the linear jump-diffusion equations. Extended Cox–Ingersoll–Ross, Brennan–Schwartz and Epstein models are shown to be linearizable and their explicit solutions are presented.     

A survey of numerical methods for nonlinear filtering problems

The main goal of filtering is to obtain, recursively in time, good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same. In settings where the signal/observation dynamics are significantly nonlinear or the noise intensities are high, an extended Kalman filter (EKF), which is essentially a first order approximation to an infinite dimensional problem, can perform quite poorly: it may require very frequent re-initializations and in some situations may even diverge. The theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations, in the space of probability measures. We survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations; with a special emphasis on an important family of schemes known as the particle filters. A numerical study is presented to illustrate that in settings where the signal/observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended Kalman filter.     

随机系统的概率密度函数形状调节

针对受高斯白噪声激励的非线性随机系统,提出了使状态响应的概率密度函数形状跟踪期望形状的调节方法.首先,确立了非线性随机系统的多项式反馈机制,同时对系统中的非线性部分进行多项式展开;然后,以Fokker-Planck-Kolmogorov方程为工具,导出了与控制增益相关的各阶矩递推方程,并根据跟踪问题的要求,构造了矩逼近优化问题,用梯度搜索法求解该优化问题,获得了调节函数;再依据特征函数与概率密度函数构成Fourier对的关系,对状态响应的概率密度函数进行重构;最后,通过两个例子仿真,验证了本文方法的有效性.     

On stochastic parameter estimation using data assimilation

Data assimilation-based parameter estimation can be used to deterministically tune forecast models. This work demonstrates that it can also be used to provide parameter distributions for use by stochastic parameterization schemes. While parameter estimation is (theoretically) straightforward to perform, it is not clear how one should physically interpret the parameter values obtained. Structural model inadequacy implies that one should not search for a deterministic “best” set of parameter values, but rather allow the parameter values to change as a function of state; different parameter values will be needed to compensate for the state-dependent variations of realistic model inadequacy. Over time, a distribution of parameter values will be generated and this distribution can be sampled during forecasts. The current work addresses the ability of ensemble-based parameter estimation techniques utilizing a deterministic model to estimate the moments of stochastic parameters. It is shown that when the system of interest is stochastic the expected variability of a stochastic parameter is biased when a deterministic model is employed for parameter estimation. However, this bias is ameliorated through application of the Central Limit Theorem, and good estimates of both the first and second moments of the stochastic parameter can be obtained. It is also shown that the biased variability information can be utilized to construct a hybrid stochastic/deterministic integration scheme that is able to accurately approximate the evolution of the true stochastic system.     

Analysis of a nonlinear stochastic model of cooperative behaviour

A stochastic model of cooperative behaviour is analyzed with regard to its critical properties. A cumulant expansion to fourth order is used to truncate the infinite set of coupled evolution equations for the moments. Linear stability analysis is performed around all the permissible steady states. The method is shown to be incapable of reproducing the critical boundary and the nature of the phase transition. A linearization, which respects the symmetry of the potential, is proposed which reproduces all the basic features associated with the model. The dynamics predicted by this approximation is shown to agree well with the Monte-Carlo simulation of the nonlinear Langevin equation.     

Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations

We study a class of methods for the numerical solution of the system of stochastic differential equations (SDEs) that arises in the modeling of turbulent combustion, specifically in the Monte Carlo particle method for the solution of the model equations for the composition probability density function (PDF) and the filtered density function (FDF). This system consists of an SDE for particle position and a random differential equation for particle composition. The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size. The four primary contributions of the paper are: (i) establishing that the coefficients in the particle equations can be frozen at the mid-time (while preserving second-order accuracy), (ii) examining the performance of three existing schemes for integrating the SDEs, (iii) developing and evaluating different splitting schemes (which treat particle motion, reaction and mixing on different sub-steps), and (iv) developing the method of manufactured solutions (MMS) to assess the convergence of Monte Carlo particle methods. Tests using MMS confirm the second-order accuracy of the schemes. In general, the use of frozen coefficients reduces the numerical errors. Otherwise no significant differences are observed in the performance of the different SDE schemes and splitting schemes.     

Studies in nonlinear stochastic processes. III. Approximate solutions of nonlinear stochastic differential equations excited by Gaussian noise and harmonic disturbances

A fusion of the highly successful methods of harmonic and statistical linearization is used as a first approximation in determining, either iteratively or via a nonlinear integral equation, the effects of higher harmonics and non-Gaussian distortion terms on the second-order statistics of a wide variety of nonlinear stochastic differential equations perturbed by some linear combination of Gaussian noise and a periodic deterministic/stochastic excitation. Physical a posteriori applicability criteria are presented which justify when these higher order effects may be neglected. A simple modification of this statistical-harmonic linearization procedure based upon the Fokker-Planck variance is proposed.This work was supported in part by the National Science Foundation under grant CHE75-20624.     

Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg–Landau, Kuramoto–Sivashinsky and Cahn–Hilliard equations with stochastic forcing.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability

For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X _t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows.     

A statistical theory of the surface of an anharmonic crystal

We perform a linearization of the transcendental equations for the moments of the single-particle atomic distribution functions of an anharmonic crystal with a surface. The transcendental equations can be derived from the nonlinear integral equations of the nonsymmetrized self-consistent field method. With the help of these equations we consider the relaxation of the lattice of an fcc crystal near its three surfaces and the mean square displacement of the atoms, assuming nearest-neighbor interactions. We discuss effects which result when interactions between nonnearest-neighbors are taken into account and also the application of the method to small cyrstalline particles.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 44–49, January, 1986.