Guaranteed Convergence Rate Estimates for a Class of Iterative Procedures: Their Comparison

The main difficulty in parallelization and synchronization of computations for speeding up the convergence of iterative procedures is the optimal guaranteed convergence rate estimation. A theorem is formulated concerning the guaranteed convergence rate estimates for certain procedures, in which resources are primarily allotted to the computation of nonlinearity components. Its proof is based on the use of new estimates for the spectral radii of the products of matrix sequences of asynchronous component computations.     

Chebyshev rational spectral method for long-short wave equations

We consider the initial boundary value problem of the long-short wave equations on the whole line. A fully discrete spectral approximation scheme is developed based on Chebyshev rational functions in space and central difference in time. A priori estimates are derived which are crucial to study numerical stability and convergence of the fully discrete scheme. Then, unconditional numerical stability is proved. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented to demonstrate the efficiency and accuracy of the convergence results.     

Convergence of estimates of image interframe deformation parameters in pseudogradient estimation

A local sample of image readings used to find the pseudogradient in estimating the image parameters influences both the convergence of estimates and the computation costs. Problems of estimating parameters of interframe geometrical deformation of images are solved by analysis and selection of values characterizing the speed of convergence of parameter estimates that can be used to optimize the local sample by different criteria.     

Convergence of identification methods based on the instrumental variable approach

A class of identification methods, proposed in [3], are based on the instrumental variable principle. This correspondence contains a continued analysis of convergence of the parameter estimates of these methods. Alternative, sufficient conditions for convergence to correct values are given. It is also shown by construction of counter-examples that the methods do not converge under general conditions.     

Convergence analysis and error estimates of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities

This paper is presented for the convergence analysis of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities arising from the heat-servo control problem. The error estimates illustrate that the convergence order depends not only on the number of basis functions in the moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.     

A family of chebyshev type methods in banach spaces

A family of third order iterative processes, that includes Chebyshev method, is studied in Banach Spaces. Results on convergence and uniqueness of solution are given, as well as error estimates.     

A linear discrete scheme for the Ginzburg–Landau equation

This paper considers a 2D Ginzburg–Landau equation with a periodic initial-value condition. A fully discrete Galerkin–Fourier spectral approximation scheme, which is a linear scheme, is constructed and the dynamical behaviour of the discrete system is then analysed. First, the existence and convergence of global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution without any restriction on the time step, and the convergence of the discrete scheme is then obtained. The numerical stability of the discrete scheme is proved.     

On the convergence of a rule by Monegato for the numerical evaluation of Cauchy principal value integrals

In this paper the authors examine the convergence of an interpolatory type quadrature rule proposed by G. Monegato for the evaluation of Cauchy principal value integrals. A convergence theorem is given for a large class of functions and some estimates of the remainder are established.     

Globally exponentially stable attitude observer with Earth velocity estimation

This paper presents a novel observer for attitude estimation based on a triad of high‐grade rate gyros aided by a body‐fixed vector measurement of a constant inertial vector, departing from the majority of solutions that consider at least two of these vectors. A cascade approach is proposed, where the first block estimates a vector that is related to the angular velocity of the Earth and a second block estimates the attitude. While the topological characteristics are relaxed in the attitude observer so that global exponential stability is achieved, an additional stage is also devised that yields estimates directly on the special orthogonal group, preserving global convergence of the estimation error. Simulation results with realistic sensor noise were performed, including extensive Monte Carlo runs. These results illustrate the convergence of the proposed solution, as well as the achievable performance and robustness to sensor noise.     

A new lifetime distribution

A new two-parameter distribution with decreasing failure rate is introduced. Various properties of the introduced distribution are discussed. The EM algorithm is used to determine the maximum likelihood estimates and the asymptotic variances and covariance of these estimates are obtained. Simulation studies are performed in order to assess the accuracy of the approximation of the variances and covariance of the maximum likelihood estimates and investigate the convergence of the proposed EM scheme. Illustrative examples based on real data are also given.     

On the convergence of least squares estimates in white noise

The problem of convergence of least squares (LS) estimates in a stochastic linear regression model with white noise is considered. It is well known that if the parameter estimates are known to converge, the convergence analysis for many adaptive systems can be rendered considerably less arduous. For an important case where the regression vector is a measurable function of the observations and the noise is Gaussian, it has been shown, by using a Bayesian embedding argument, that the LS estimates converge almost surely for almost all true parameters in the parameter space except for a zero-measure set. However, nothing can be said about a particular given system, which is usually the objective. It has long been conjectured that such a “bad” zero measure set in the parameter space does not actually exist. A conclusive answer to this important question is provided and it is shown that the set can indeed exist. This then shows that to provide conclusive convergence results for stochastic adaptive systems, it is necessary to resort to a sample pathwise analysis instead of the Bayesian embedding approach     

A new information-weighted recursive algorithm for time-varying systems: application to UAV system identification

This paper presents a new recursive identification method which can efficiently estimate time-varying parameters in discrete time systems and has significant advantages over standard recursive least-squares (RLS) method. This new information-weighted recursive algorithm for time-varying systems has three novel features, discounting of inaccurate estimates through weighting by the Information matrix, using the reuse of past data in computing current parameter estimates, a new tuneable damping factor parameter and a precisely designed compensation term to neutralise the estimation error caused by time-varying coefficients. A rigorous proof of convergence is also provided. Simulations show that the new algorithm significantly outperforms standard RLS, exhibiting better tracking performance and faster convergence. Flight tests on a T-REX 800 helicopter Unmanned Aerial Vehicle platform show that it gives system parameter estimates that are accurate enough and converge quickly enough that flight controllers can be designed in real-time based on the online identified model.     

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.     

Almost sure parameter estimation and convergence rates for hidden Markov models

A continuous-time version of Kronecker's Lemma is established and used to give rates of convergence for parameter estimates in hidden Markov models.     

Convergence Rates of Nonparametric Filtering Estimates in Autoregression Dynamic Systems

The Bayes problem of filtration of a random process from observations on an autoregression process with coefficients defined by functions of the useful signal is studied. The main assumption asserts that the conditional observation densities belong to a family of conditional exponential densities with known functions of observations and useful signal, whose distribution is not known in advance. Optimal signal filtering equations and empirical risk estimates are derived. Regularized nonparametric filtering estimates are derived and a theorem on the mean-square convergence and convergence rates of these estimates is formulated.     

Least squares based iterative algorithms for identifying Box–Jenkins models with finite measurement data

A least squares based iterative identification algorithm is developed for Box–Jenkins models (or systems). The proposed iterative algorithm can produce highly accurate parameter estimation compared with recursive approaches. The basic idea of the proposed iterative method is to adopt the interactive estimation theory: the parameter estimates relying on unknown variables are computed by using the estimates of these unknown variables which are obtained from the preceding parameter estimates. The numerical example indicates that the proposed iterative algorithm has fast convergence rates compared with the gradient based iterative algorithm.     

Convergence of discrete-time Kalman filter estimate to continuous time estimate

This article is concerned with the convergence of the state estimate obtained from the discrete-time Kalman filter to the continuous time estimate as the temporal discretisation is refined. The convergence follows from Martingale convergence theorem as demonstrated below; however, surprisingly, no results exist on the rate of convergence. We derive convergence rate estimates for the discrete-time Kalman filter estimate for finite and infinite dimensional systems. The proofs are based on applying the discrete-time Kalman filter on a dense numerable subset of a certain time interval [0, T].     

Finite element approximation of the Cahn–Hilliard equation on surfaces

In this paper, we consider the phase separation on general surfaces by solving the nonlinear Cahn–Hilliard equation using a finite element method. A fully discrete approximation scheme is introduced, and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time. This in turn leads to convergence and optimal error estimates of the discretization scheme. Numerical examples are also provided to substantiate the theoretical results.     

Estimation of convergence rate for multi-regression learning algorithm

In many applications, the pre-information on regression function is always unknown. Therefore, it is necessary to learn regression function by means of some valid tools. In this paper we investigate the regression problem in learning theory, i.e., convergence rate of regression learning algorithm with least square schemes in multi-dimensional polynomial space. Our main aim is to analyze the generalization error for multi-regression problems in learning theory. By using the famous Jackson operators in approximation theory, covering number, entropy number and relative probability inequalities, we obtain the estimates of upper and lower bounds for the convergence rate of learning algorithm. In particular, it is shown that for multi-variable smooth regression functions, the estimates are able to achieve almost optimal rate of convergence except for a logarithmic factor. Our results are significant for the research of convergence, stability and complexity of regression learning algorithm.