A generalized polynomial chaos based ensemble Kalman filter with high accuracy

As one of the most adopted sequential data assimilation methods in many areas, especially those involving complex nonlinear dynamics, the ensemble Kalman filter (EnKF) has been under extensive investigation regarding its properties and efficiency. Compared to other variants of the Kalman filter (KF), EnKF is straightforward to implement, as it employs random ensembles to represent solution states. This, however, introduces sampling errors that affect the accuracy of EnKF in a negative manner. Though sampling errors can be easily reduced by using a large number of samples, in practice this is undesirable as each ensemble member is a solution of the system of state equations and can be time consuming to compute for large-scale problems. In this paper we present an efficient EnKF implementation via generalized polynomial chaos (gPC) expansion. The key ingredients of the proposed approach involve (1) solving the system of stochastic state equations via the gPC methodology to gain efficiency; and (2) sampling the gPC approximation of the stochastic solution with an arbitrarily large number of samples, at virtually no additional computational cost, to drastically reduce the sampling errors. The resulting algorithm thus achieves a high accuracy at reduced computational cost, compared to the classical implementations of EnKF. Numerical examples are provided to verify the convergence property and accuracy improvement of the new algorithm. We also prove that for linear systems with Gaussian noise, the first-order gPC Kalman filter method is equivalent to the exact Kalman filter.     

随机多项式展开多特征向量约束-匹配场 声源定位算法*

多特征向量约束的自适应环境宽容匹配场算法可用于随机和不确知海洋声传播介质中的声源位置定位。该方法需要对拷贝场复声压的互谱矩阵进行估计。为了克服蒙特卡洛统计方法计算互谱矩阵较为耗时的缺点,文章将起伏介质中的随机或不确知声压表示为以正交的随机多项式为基底的级数,随机多项式基底的输入变量为描述环境随机性或不确知性的随机变量。利用随机多项式展开基底的正交性可快速估计拷贝场复声压互谱矩阵。仿真结果表明：在声源频率较低、浅海海水声速存在随机起伏的条件下,在计算效率上使用随机多项式展开方法估计拷贝场复声压互谱矩阵较蒙特卡洛统计方法可提高一个数量级;在高信噪比下,多特征向量约束匹配场声源定位算法在定位准确率和输出峰均比上优于线性匹配场和对角加载的最小方差匹配场声源定位方法。     

多项式混沌展开的浅海水声环境参数敏感度分析

水下声场是海洋环境参数的非线性函数,因此环境参数的不确定性必然会引起水下声场预测的不确定性,从而导致与声场预测相关的声呐探测和水声通信等设备性能的下降.该文将Sobol敏感度指数的计算与多项式混沌展开方法相结合,将敏感度指数表达为环境条件的函数,利用Q范数约束的双曲截断方案来减少多项式项数,有效地分析了设定的"浅海负梯...     

嵌入随机多项式的抛物方程不确定声场快速算法

为了得到准确且高效计算起伏海洋介质中随机声场的算法,本文将随机多项式展开嵌入到宽角抛物方程声场计算模型(简称RAM模型)中,发展了一种不确定声场的快速算法。其计算结果比使用嵌入随机多项式的窄角抛物方程准确,计算时间小于作为参考的蒙特卡洛方法。在仿真算例中,随机多项式展开法对声强均值、方差、概率密度的计算准确;在一定的随机变量维度和随机多项式展开截断幂次内,其计算效率比蒙特卡洛方法至少提高一个数量级。     

Noncommutative cumulants for stochastic differential equations and for generalized Dyson series

For the stochastic equationU=VU, Kubo's ansalze for U in the form of differential and integrodifferential equations is investigated and a newansatz as an integral equation is added. Unique solutions in terms of noncommutative W- and K-cumulants are found by elementary functional differentiation, and expressions of van Kämpen and Terwiel are recovered. For the cumulants we find simple recursion relations and prove the important cluster property. Surprisingly, it is found that the Gaussian approximation in the differential equationansatz leads to positivity problems, while this is not the case with the integral and integrodifferential equation. The cumulant expansion technique is carried over to generalized Dyson series. In a companion paper we apply our results to quantum shot noise.     

Asymptotical p-moment stability of stochastic impulsive differential system and its application to chaos synchronization

In this paper,the asymptotical p-moment stability of stochastic impulsive differential equations is studied and a comparison theory to ensure the asymptotical p-moment stability of the trivial solution is established,which is important for studying the impulsive control and synchronization in stochastic systems.As an application of this theory,we study the problem of chaos synchronization in the Chen system excited by parameter white-noise excitation,by using the impulsive method.Numerical simulations verify the feasibility of this method.     

广义混沌同步中的多稳定同步流形

从构造驱动系统和响应系统之间的函数关系出发,一般性地研究了广义混沌同步中同步流形的多值性问题,并对同步流形的稳定性进行了分析,提出了多稳定同步流形存在的条件.利用该稳定性条件对两个实例进行了分析,一个为Genesio-Rssler耦合系统,另一个为具有二次、三次非线性的耦合Duffing系统,结果表明前者仅有唯一的稳定同步流形,而后者随着耦合强度的增加,从具有两个稳定的同步流形变为只存在一个稳定同步流形. 关键词： 混沌同步 多稳定性 同步流形     

Probing quantum dissipative chaos using purity

In this Letter, the purity of quantum states is applied to probe chaotic dissipative dynamics. To achieve this goal, a comparative analysis of regular and chaotic regimes of nonlinear dissipative oscillator (NDO) are performed on the base of excitation number and the purity of oscillatory states. While the chaotic regime is identified in our semiclassical approach by means of strange attractors in Poincaré section and with the Lyapunov exponent, the state in the quantum regime is treated via the Wigner function. Specifically, interesting quantum purity effects that accompany the chaotic dynamics are elucidated in this Letter for NDO system driven by either: (i) a time-modulated field, or (ii) a sequence of pulses with Gaussian time-dependent envelopes.     

Evaluation of failure probability via surrogate models

Evaluation of failure probability of a given system requires sampling of the system response and can be computationally expensive. Therefore it is desirable to construct an accurate surrogate model for the system response and subsequently to sample the surrogate model. In this paper we discuss the properties of this approach. We demonstrate that the straightforward sampling of a surrogate model can lead to erroneous results, no matter how accurate the surrogate model is. We then propose a hybrid approach by sampling both the surrogate model in a “large” portion of the probability space and the original system in a “small” portion. The resulting algorithm is significantly more efficient than the traditional sampling method, and is more accurate and robust than the straightforward surrogate model approach. Rigorous convergence proof is established for the hybrid approach, and practical implementation is discussed. Numerical examples are provided to verify the theoretical findings and demonstrate the efficiency gain of the approach.     

Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

We give a complete classification of at most second-order differential ladder operators preserving finite-dimensional spaces of monomials and closing under the Lie bracket to give a cubic polynomial of the diagonal generators.     

Stochastic differential equation derivation: Comparison of the Markov method versus the additive method

There are several methods of transforming an ordinary differential equation into a stochastic differential equation (SDE). The two most common are adding noise to a system parameter or variable and transforming to a SDE or deriving the SDE by assuming an underlying Markov process. Using simple one- and two-dimensional systems we investigate the differences in dynamics and bifurcations between SDE derived by each method from simple deterministic population models.     

Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.     

脉冲电压微分反馈法控制buck功率变换器中的混沌

通过3种典型的开关逻辑图分析了降压buck变换器中混沌产生的机理，指出开关S导通时，电 容充电过快是电路系统中混沌产生的主要原因.在此基础上提出了利用输出电压的脉冲微分 反馈对这类电路系统中的混沌进行控制的方法.理论分析、数值计算和电路仿真证实了该方 法的正确性和有效性. 关键词： 混沌 混沌控制 脉冲电压微分反馈 buck变换器     

Time-space white noise eliminates global solutions in reaction-diffusion equations

We prove that perturbing the reaction-diffusion equation u_t=u_xx+(u₊)^p (p>1), with time-space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given.     

Research on the chaos recognition method based on differential entropy

Phase space reconstruction is the first step to recognizing the chaos from observed time series. On the basis of differential entropy, this paper introduces an efficient method to estimate the embedding dimension and the time delay simultaneously. The differential entropy is used to characterize the disorder degree of the reconstructed attractor. The minimum value of the differential entropy corresponds to the optimum set of the reconstructed parameters. Simulated experiments show that the original phase space can be effectively reconstructed from time series, and the accuracy of the invariants in phase space reconstruction is greatly improved. It provides a new method for the identification of chaotic signals from time series.     

Model equations from a chaotic time series

We present a method for obtaining a set of dynamical equations for a system that exhibits a chaotic time series. The time series data is first embedded in an appropriate phase space by using the improved time delay technique of Broomhead and King (1986). Next, assuming that the flow in this space is governed by a set of coupled first order nonlinear ordinary differential equations, a least squares fitting method is employed to derive values for the various unknown coefficients. The ability of the resulting model equations to reproduce global properties like the geometry of the attractor and Lyapunov exponents is demonstrated by treating the numerical solution of a single variable of the Lorenz and Rossler systems in the chaotic regime as the test time series. The equations are found to provide good short term prediction (a few cycle times) but display large errors over large prediction time. The source of this shortcoming and some possible improvements are discussed.     

基于双曲线截断混沌多项式展开的地磁感应电流敏感度分析

地磁感应电流(GIC) 可能会引起变压器直流偏磁从而威胁电力设备和电网的安全。鉴于GIC计算所需的很多输入参数是不确定变量, 全面评估电网GIC水平及对电网的威胁有必要研究GIC的不确定度及GIC对输入变量的敏感度。基于混沌多项式展开(PCE) 提出了一种GIC的不确定度量化方法, 利用双曲线截断技术进一步提高了计算效率, 计算了GIC对输入参数的敏感度指标。以新疆750kV规划电网为例, 利用提出的方法对GIC进行了不确定度量化, 得到了GIC的均值和标准差等统计量。根据混沌多项式系数计算了Sobol敏感度指标, 得到了GIC对电场幅值和电网直流电阻等输入参数的敏感度。与蒙特卡罗法(MC) 相比, 此方法在保证精度的前提下大大提高了计算效率。     

Numerical solution of differential equations with colored noise

Using the general theory of numerical integration of stochastic differential equations, a constructive approach to numerical methods for a system with colored noise is proposed. Efficient methods up to the 5/2 strong order and up to the third weak order, including Runge-Kutta and implicit schemes, are presented. The algorithms are tested on the Kubo oscillator.