Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

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.     

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

逼近积分点数下限的五阶容积卡尔曼滤波定轨算法

为了在保持滤波定轨精度不变的条件下提高定轨计算的实时性,提出一种新的逼近积分点个数下限的五阶容积卡尔曼滤波定轨算法.首先,采用一种数值容积准则对非线性函数的高斯加权积分进行近似,该准则所需的积分点个数仅比五阶代数精度容积准则积分点个数的理论下限多一个积分点,并在贝叶斯滤波算法框架下推导出本文算法的更新步骤.然后,给出实时定轨所需的状态方程和量测方程,在状态方程中考虑了J2项引力摄动和大气阻力摄动,在量测方程中利用坐标系转换推导了轨道状态与测量元素之间的非线性关系.仿真实验结果表明,本文所提算法在定轨精度方面与已有的五阶滤波算法相当,但所需的积分点个数最少,计算实时性最高,从而验证了本文算法的有效性.     

Coherent ensemble of pseudospins in an external field

The behavior in time is considered for the state vector of an ensemble of coherent pseudospins in external permanent and variable fields. Relaxation of the state vector of the ensemble is described by a Landau-Lifshits equation. The vector equation of motion is reduced to a first-order differential equation. Exact solutions of this equation are obtained for different combinations of both weak and strong fields acting on the ensemble. Conditions for applicability of the solutions obtained are discussed.     

基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程

波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.     

Nonlinear Landau Damping in Spherically Symmetric Vlasov Poisson Systems

The Vlasov Poisson system is a partial differential equation widely used to describe collisionless plasma. It is formulated in a six-dimensional phase space, this prohibits a numerical solution on a complete phase space grid. In some applications, however, spherical symmetry is given, which introduces singularities into the Vlasov Poisson equation. We focus on such problems and propose a stable algorithm using accommodating boundaries. At first, the method is tested in the linear regime, where analytical solutions are available. Thereafter it is applied to large disturbances from equilibrium.     

基于滚动时域估计的飞行器姿态估计及三轴磁强计在线校正

针对飞行器姿态估计以及三轴磁强计在线校正问题, 提出了一种实时滚动时域估计算法. 首先, 为了解决在卡尔曼滤波框架下系统约束不能显式求解的问题, 设计了滚动时域估计滤波算法. 该算法将飞行器姿态估计问题转化为优化问题, 显式求解四元数归一化性质, 缩小搜索空间的同时提高了搜索效率和精度. 其次, 滤波时域窗内应用高斯-牛顿迭代法求解最优状态估计值, 满足了实时性要求. 最后, 在没有增加系统状态维数的情况下, 在线求解了三轴磁强计校正参数, 保证了磁强计量测值以矢量形式输入系统. 仿真结果表明, 由于合理地利用了历史信息, 该方法精度较高, 且对初始误差、系统误差均不敏感, 具有一定鲁棒性.     

自回归状态空间模型下时变声速剖面跟踪方法

讨论了一种适用于浅海的时变声速剖面跟踪方法。该方法将时变水体声速剖面的反演问题建模为由描述声速剖面时变特性的状态方程与包含声压场局部测量信息的测量方程组成的状态-空间模型,提出利用自回归分析拟合方法将声速场扰动建模为高阶自回归演化模型,并通过集合卡尔曼滤波序贯地估计时间演化的海洋声速场。利用2001年亚洲海实验环境与声速测量数据,仿真分析了基于高阶自回归演化模型的时变声速剖面集合卡尔曼滤波估计方法。结果表明,相比于利用传统随机游走状态演化模型的估计方法,该改进方法可有效降低声速的跟踪误差,并且在较低信噪比条件下仍具有较好的跟踪性能。     

基于粒子滤波的一种改进的资料同化方法

针对在粒子数较少时传统的集合卡尔曼滤波和粒子滤波方法不能有效表征后验概率密度函数(PDF)的问题, 提出了一种改进的粒子滤波方法. 主要思想是在预测步之后引入更新步, 并且将观测时刻与非观测时刻的同化分析进行区别处理. 对典型的低维和高维混沌系统的仿真结果表明:改进粒子滤波方法是一种非常有效的估计非线性非高斯随机系统状态的方法.     

On a nonlinear Kalman filter with simplified divided difference approximation

We present a new ensemble-based approach that handles nonlinearity based on a simplified divided difference approximation through Stirling’s interpolation formula, which is hence called the simplified divided difference filter (sDDF). The sDDF uses Stirling’s interpolation formula to evaluate the statistics of the background ensemble during the prediction step, while at the filtering step the sDDF employs the formulae in an ensemble square root filter (EnSRF) to update the background to the analysis. In this sense, the sDDF is a hybrid of Stirling’s interpolation formula and the EnSRF method, while the computational cost of the sDDF is less than that of the EnSRF. Numerical comparison between the sDDF and the EnSRF, with the ensemble transform Kalman filter (ETKF) as the representative, is conducted. The experiment results suggest that the sDDF outperforms the ETKF with a relatively large ensemble size, and thus is a good candidate for data assimilation in systems with moderate dimensions.     

MR fingerprinting reconstruction with Kalman filter

Magnetic resonance fingerprinting (MR fingerprinting or MRF) is a newly introduced quantitative magnetic resonance imaging technique, which enables simultaneous multi-parameter mapping in a single acquisition with improved time efficiency. The current MRF reconstruction method is based on dictionary matching, which may be limited by the discrete and finite nature of the dictionary and the computational cost associated with dictionary construction, storage and matching.In this paper, we describe a reconstruction method based on Kalman filter for MRF, which avoids the use of dictionary to obtain continuous MR parameter measurements. With this Kalman filter framework, the Bloch equation of inversion-recovery balanced steady state free-precession (IR-bSSFP) MRF sequence was derived to predict signal evolution, and acquired signal was entered to update the prediction. The algorithm can gradually estimate the accurate MR parameters during the recursive calculation. Single pixel and numeric brain phantom simulation were implemented with Kalman filter and the results were compared with those from dictionary matching reconstruction algorithm to demonstrate the feasibility and assess the performance of Kalman filter algorithm.The results demonstrated that Kalman filter algorithm is applicable for MRF reconstruction, eliminating the need for a pre-define dictionary and obtaining continuous MR parameter in contrast to the dictionary matching algorithm.     

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.     

A matrix approach for partial differential equations with Riesz space fractional derivatives

Fractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach.     

Trajectory approach to the Schrödinger–Langevin equation with linear dissipation for ground states

The Schrödinger–Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger–Langevin equation yields the complex quantum Hamilton–Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian–Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.     

利用频域β-warping变换的浅海修正纯方位目标运动分析方法

针对纯方位目标运动分析方法收敛时间较长的问题,提出了一种利用频域β-warping变换的浅海修正纯方位目标运动分析方法.该方法利用频域β-warping变换从声强干涉结构中提取与目标距离成线性关系的时延,进而估计距离特征量,并利用距离特征量推导得到的目标状态向量的线性约束修正纯方位扩展卡尔曼滤波算法。数值仿真结果表明,对于浅海匀速直线运动目标,在信噪比不低于8 dB的情况下,与常规纯方位扩展卡尔曼滤波算法相比,改进算法将距离估计的收敛时间由26.5 min缩短至11.5 min.在浅海水平不变波导远场条件下,该方法可以快速稳定地估计距离特征量,并能够对目标进行可靠地跟踪定位。      

非线性动态特性的逆推演方法及其在干涉术中的应用

提出了研究对象非线性动态特性的逆推演新方法,它的特点是从已知对象的非微分方程解析解导出该对象可能的微分方程,从而开拓它的解域或解空间,或者说恢复丢失的解。同时讨论了当对象为法布里－珀罗干涉（F－P）或迈克耳孙干涉时该方法的应用过程,并且对法布里－珀罗定义新的不透过系数（u＝1／τ）,从而给出更简洁和易分析的对象动态特性微分议程,以及由此导出测量法布里－珀罗干涉相位的过程,值得注意的是由新方法导出的微分议程揭示对象更一般的时空演化特征。而且在现有的经验和知识基础上可以进一步唯象地拓展至其他可能的非线性形式,从而使对象的表达方式更接近它的实际情况。     

Ensemble Kalman filter with the unscented transform

A modification scheme to the ensemble Kalman filter (EnKF) is introduced based on the concept of the unscented transform [S. Julier, J. Uhlmann, H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. 45 (2000) 477-482; S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE 92 (2004) 401-422], which therefore will be called the ensemble unscented Kalman filter (EnUKF) in this work. When the error distribution of the analysis is symmetric (not necessarily Gaussian), it can be shown that, compared with the ordinary EnKF, the EnUKF has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional Taylor series expansion term by term. This implies that, the EnUKF may have better performance in state estimation than the ordinary EnKF in the sense that the deviations from the true states are smaller. For verification, some numerical experiments are conducted on a 40-dimensional system due to Lorenz and Emanuel [E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399-414]. Simulation results support our argument.     

Sampling strong tracking nonlinear unscented Kalman filter and its application in eye tracking

The unscented Kalman filter is a developed well-known method for nonlinear motion estimation and tracking. However, the standard unscented Kalman filter has the inherent drawbacks, such as numerical instability and much more time spent on calculation in practical applications. In this paper, we present a novel sampling strong tracking nonlinear unscented Kalman filter, aiming to overcome the difficulty in nonlinear eye tracking. In the above proposed filter, the simplified unscented transform sampling strategy with n+2 sigma points leads to the computational efficiency, and suboptimal fading factor of strong tracking filtering is introduced to improve robustness and accuracy of eye tracking. Compared with the related unscented Kalman filter for eye tracking, the proposed filter has potential advantages in robustness, convergence speed, and tracking accuracy. The final experimental results show the validity of our method for eye tracking under realistic conditions.     

Quantum tomography as a new approach to simulating quantum processes

A new method is proposed for ab initio calculations of nonstationary quantum processes on the basis of a probability representation of quantum mechanics with the help of a positive definite function (quantum tomogram). The essence of the method is that an ensemble of trajectories associated with the characteristics of the evolution equation for the quantum tomogram is considered in the space where the quantum tomogram is defined. The method is applied for detailed analysis of transient tunneling of a wave packet. The results are in good agreement with the exact numerical solution to the Schrödinger equation for this system. The probability density distributions are obtained in the coordinate and momentum spaces at consecutive instances. For transient tunneling of a wave packet, the probability of penetration behind the barrier and the time of tunneling are calculated as functions of the initial energy.