Nonlinear Bayesian Estimation： From Kalman Filtering to a Broader Horizon

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date,one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective,which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics(e.g., mean and covariance) conditioned on a system's measurement data.This article offers a systematic introduction to the Bayesian state estimation framework and reviews various Kalman filtering(KF)techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation to more complicated problems such as simultaneous state and parameter/input estimation.

Log-logistic parameters estimation using moving extremes ranked set sampling design

In statistical parameter estimation problems,how well the parameters are estimated largely depends on the sampling design used.In the current paper,a modification of ranked set sampling(RSS)called moving extremes RSS(MERSS)is considered for the estimation of the scale and shape parameters for the log-logistic distribution.Several traditional estimators and ad hoc estimators will be studied under MERSS.The estimators under MERSS are compared to the corresponding ones under SRS.The simulation results show that the estimators under MERSS are significantly more efficient than the ones under SRS.

Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems

We first give a stabilized improved moving least squares （IMLS） approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.

A NEW ESPRIT METHOD FOR BLIND ESTIMATES OF DOA-DOPPLER FREQUENCY WITH UNKNOWN ARRAY MANIFOLD

A novel rotational invariance technique for blind estimates of direction of arrival (I)OA) and Doppler frequency with unknown array manifold due to array sensor uncertainties is proposed, taking Doppler frequency difference between a successive pulses as rotational parameter. The effectiveness of the new method is confirmed by computer simulation. Compared with the existing 2-D DOA-frequeucy estimate techniques, the computation load of the proposed method can be saved greatly.

Maximum Norm Estimates for Finite Volume Element Method for Non-selfadjoint and Indefinite Elliptic Problems

In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method （FVE） based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.

Critical Review on Improved Electrochemical Impedance Spectroscopy-cuckoo Search-Elman Neural Network Modeling Methods for Whole-life-cycle Health State Estimation of Lithium-ion Battery Energy Storage Systems

Efficient and accurate health state estimation is crucial for lithium-ion battery(LIB)performance monitoring and economic evaluation.Effectively estimating the health state of LIBs online is the key but is also the most difficult task for energy storage systems.With high adaptability and applicability advantages,battery health state estimation based on data-driven techniques has attracted extensive attention from researchers around the world.Artificial neural network(ANN)-based methods are often used for state estimations of LIBs.As one of the ANN methods,the Elman neural network(ENN)model has been improved to estimate the battery state more efficiently and accurately.In this paper,an improved ENN estimation method based on electrochemical impedance spectroscopy(EIS)and cuckoo search(CS)is established as the EIS-CS-ENN model to estimate the health state of LIBs.Also,the paper conducts a critical review of various ANN models against the EIS-CS-ENN model.This demonstrates that the EIS-CS-ENN model outperforms other models.The review also proves that,under the same conditions,selecting appropriate health indicators(HIs)according to the mathematical modeling ability and state requirements are the keys in estimating the health state efficiently.In the calculation process,several evaluation indicators are adopted to analyze and compare the modeling accuracy with other existing methods.Through the analysis of the evaluation results and the selection of HIs,conclusions and suggestions are put forward.Also,the robustness of the EIS-CS-ENN model for the health state estimation of LIBs is verified.

A PRIORI ERROR ESTIMATES OF A FINITE ELEMENT METHOD FOR DISTRIBUTED FLUX RECONSTRUCTION＊

This paper is concerned with a priori error estimates of a finite element method for numerical reconstruction of some unknown distributed flux in an inverse heat conduction problem. More precisely, some unknown distributed Neumann data are to be recovered on the interior inaccessible boundary using Dirichlet measurement data on the outer ac- cessible boundary. The main contribution in this work is to establish the some a priori error estimates in terms of the mesh size in the domain and on the accessible/inaccessible boundaries, respectively, for both the temperature u and the adjoint state p under the lowest regularity assumption. It is revealed that the lower bounds of the convergence rates depend on the geometry of the domain. These a priori error estimates are of immense interest by themselves and pave the way for proving the convergence analysis of adaptive techniques applied to a general classes of inverse heat conduction problems. Numerical experiments are presented to verify our theoretical prediction.

Local and global behavior for algorithms of solving equations

The theory of 'point estimate' and the concept of 'general convergence', which were put forward by Smale in order to investigate the complexity of algorithms for solving equations, have been producing a deep impact on the research about the local behavior, the semi-local behavior and the global behavior of iteration methods. The criterion of point estimate introduced by him not only provides a tool for quantitative analysis of the local behavior but also motivates the establishing of the unified determination for the semi-local behavior. Studying the global behavior in the view of discrete dynamical system will lead to many profound research subjects and open up a rich and colorful prospect. In this review, we will make a summarization about the research progress and some applications in nonsmooth optimizations.

Remaining useful life estimation for deteriorating systems with time-varying operational conditions and condition-specific failure zones

Dynamic time-varying operational conditions pose great challenge to the estimation of system remaining useful life （RUL） for the deteriorating systems. This paper presents a method based on probabilistic and stochastic approaches to estimate system RUL for periodically moni- tored degradation processes with dynamic time-varying operational conditions and condition- specific failure zones. The method assumes that the degradation rate is influenced by specific oper- ational condition and moreover, the transition between different operational conditions plays the most important role in affecting the degradation process. These operational conditioqs are assumed to evolve as a discrete-time Markov chain （DTMC）. The failure thresholds are also determined by specific operational conditions and described as different failure zones. The 2008 PHM Conference Challenge Data is utilized to illustrate our method, which contains mass sensory signals related to the degradation process of a commercial turbofan engine. The RUE estimation method using the sensor measurements of a single sensor was first developed, and then multiple vital sensors were selected through a particular optimization procedure in order to increase the prediction accuracy. The effectiveness and advantages of the proposed method are presented in a comparison with exist- ing methods for the same dataset.

GLOBAL SOLUTION AND ITS LONG TIME BEHAVIOR FOR THE GENERALIZED LONG-SHORT WAVE EQUATIONS

The long time behavior of the solutions of the generalized long-short wave equations with dissipation term is studied. The existence of global attractor of the initial periodic boundary value is proved by means of a uniform a priori estimate for time. And also the dimensions of the global attractor are estimated.

Identification of the nonlinear properties of rubber-bearings in base-isolated buildings with limited seismic response data

Seismic behaviors of base-isolated structures are highly affected by the nonlinear characteristics of the isolated systems. Most of the currently available methods for the identification of nonlinear properties of isolator require either the measurements of all structural responses or the assumptions of the proper mathematic models for the rubber-bearings. In this paper, two algorithms are proposed to identify the nonlinear properties of rubber-bearings in base-isolated buildings using only partial measurements of structural dynamic responses. The first algorithm is applicable to the case that proper mathematical models are available for the base isolators. It is based on the extended Kalman filter for the parametric identification of nonlinear models of rubber-bearing isolators and buildings. For the general case where it is difficult to establish a proper mathematical model to describe the nonlinear behavior of a rubber-bearing isolator, another algorithm is proposed to identify the model-tYee nonlinear property of rubber-bearing isolated system. Nonlinear effect of rubber-bearing is treated as ＇fictitious loading＇ on the linear building under severe earthquake. The algorithm is based on the sequential Kalman estimator for the structural responses and the least-squares estimation of the ＇fictitious loading＇ to identify the nonlinear force of rubber-bearing isolator. Simulation results demonstrate that the proposed two algorithms are capable of identifying the nonlinear properties of rubber-bearing isolated systems with good accuracy.

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL.NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.

ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.

A Posteriori Error Estimates for hp Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints.The control problem is approxi-mated by the hp spectral element method with high accuracy and geometricflexibil-ity.Optimality conditions of the continuous and discrete optimal control problems are presented,respectively.The a posteriori error estimates both for the control and state variables are established in detail.In addition,illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

Convergence of the Three-Dimensional Compressible Navier-Stokes-Poisson- Korteweg Equation to the Incompressible Euler Equation

We study the combination of quasi-neutral limit and viscosity limit of smooth solution for the three-dimensional compressible viscous Navier-Stokes-Poisson-Korteweg equation for plasmas and semiconductors.When the Debye length and viscosity coefficients are sufficiently small,the initial value problem of the model has a unique smooth solution in the time interval where the corresponding incompressible Euler equation has a smooth solution.We also establish a sharp convergence rate of smooth solutions for three-dimensional compressible viscous Navier-Stokes-Poisson-Kortewe equation towards those for the incompressible Euler equation in combining quasi-neutral limit and viscosity limit.Moreover,if the incompressible Euler equation has a global smooth solution,the maximal existence time of three-dimensional compressible Navier-Stokes-Poisson-Korteweg equation tends to infinity as the Debye length and viscosity coefficients goes to zero.

Asymptotic Theory for Relative-Risk Models with Missing Time-Dependent Covariates

Relative-risk models are often used to characterize the relationship between survival time and time-dependent covariates. When the covariates are observed, the estimation and asymptotic theory for parameters of interest are available; challenges remain when missingness occurs. A popular approach at hand is to jointly model survival data and longitudinal data. This seems efficient, in making use of more information, but the rigorous theoretical studies have long been ignored. For both additive risk models and relative-risk models, we consider the missing data nonignorable. Under general regularity conditions, we prove asymptotic normality for the nonparametric maximum likelihood estimators.