Bernstein Polynomial Approximation Research Articles

Adaptive asymptotic tracking control of autonomous underwater vehicles based on Bernstein polynomial approximation

In this article, an adaptive asymptotic tracking control approach is first proposed for the autonomous underwater vehicle (AUV) on the horizontal plane by using the Bernstein polynomial approximation. Bernstein polynomials aim to compensate for the synthetical uncertainties including unknown model dynamics and environmental disturbances. However, they cannot be directly used due to the unknown polynomial coefficients. To avoid this trouble, the adaptive backstepping technique is applied to estimation. Considering that the adaptive backstepping can only procure the bounded steady tracking errors and suffer from the redundancy of adaptive parameters, we further integrate the σ-modification and the minimum learning parameter (MLP) technique with the Bernstein polynomial approximation, so as to guarantee the high-precision tracking performance of AUVs with the asymptotic convergent tracking errors and less computation burden. The direct Lyapunov’s method and the Barbalat’s lemma are introduced for proving the stability of the control system. Finally, simulation reveals the advantage of the proposed scheme.

Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

Tracking control based on adaptive Bernstein polynomial approximation for a class of unknown nonlinear dynamic systems

In this article, an adaptive tracking control approach using Bernstein polynomial approximation is firstly proposed for an unknown nonlinear dynamic system. Bernstein polynomial approximation aims to compensate the unknown nonlinear dynamic function. However, if Bernstein theorem is directly used, the Bernstein polynomial's coefficients need to be derived from the system dynamic function. Nevertheless, the dynamic function is presumed to be unknown, hence the polynomial approximation still cannot be used for designing this control. In order to obtain the available function approximation, adaptive strategy is considered to estimate these coefficients. Finally, by learning from the classical adaptive algorithm, the undetermined coefficient problem is addressed, so that the valid tracking control is found for the unknown nonlinear dynamic system. According to Lyapunov stability analysis and simulation experiment, it is concluded that the new adaptive scheme can realize the control objective.

Bernstein polynomial approximation of non-linear stochastic Itô-Volterra integral equations driven by multi-fractional Gaussian noise

In this article, a discussion about stochastic integral equations driven by multi-fractional Gaussian (MFGN) noise has been done which determines the stochastic solutions to fundamental equations based on effective elastic properties of nanomaterials. A Bernstein polynomials (BP) approach has been implemented to find an approximation of the stochastic integral equations under study which is efficient and has less computational cost. An example based on the integral equation has been solved to further more establish the efficiency of the polynomial scheme.

Consistency of Approximation of Bernstein Polynomial-Based Direct Methods for Optimal Control

Bernstein polynomial approximation of continuous function has a slower rate of convergence compared to other approximation methods. “The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation.”—remarked P.J. Davis in his 1963 book, Interpolation and Approximation. This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time-continuous optimal control problems and can be used for costate estimation of the optimal control problems. This result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.

Tail dependence estimation based on smooth estimation of diagonal section

This paper is mainly developed around the diagonal section which is strongly related to tail dependence coefficients as defined in Nelsen [19]. Hence, we propose a flexible method for estimating tail dependence coefficients based on the new smooth estimation of the diagonal section based on the Bernstein polynomial approximation. To assess the performance of the new estimators we conduct the Monte-Carlo simulation study. As a result of the simulation study, both estimators perform satisfactory performance. Also, the estimation methods are illustrated by real data examples.

Joint analysis of informatively interval-censored failure time and panel count data.

Interval-censored failure time and panel count data, which frequently arise in medical studies and social sciences, are two types of important incomplete data. Although methods for their joint analysis have been available in the literature, they did not consider the observation process, which may depend on the failure time and/or panel count of interest. This study considers a three-component joint model to analyze interval-censored failure time, panel counts, and the observation process within a unique framework. Gamma and distribution-free frailties are introduced to jointly model the interdependency among the interval-censored data, panel count data, and the observation process. We propose a sieve maximum likelihood approach coupled with Bernstein polynomial approximation to estimate the unknown parameters and baseline hazard function. The asymptotic properties of the resulting estimators are established. An extensive simulation study suggests that the proposed procedure works well for practical situations. An application of the method to a real-life dataset collected from a cardiac allograft vasculopathy study is presented.

Bernstein Polynomial-Based Method for Solving Optimal Trajectory Generation Problems.

This paper presents a method for the generation of trajectories for autonomous system operations. The proposed method is based on the use of Bernstein polynomial approximations to transcribe infinite dimensional optimization problems into nonlinear programming problems. These, in turn, can be solved using off-the-shelf optimization solvers. The main motivation for this approach is that Bernstein polynomials possess favorable geometric properties and yield computationally efficient algorithms that enable a trajectory planner to efficiently evaluate and enforce constraints along the vehicles’ trajectories, including maximum speed and angular rates as well as minimum distance between trajectories and between the vehicles and obstacles. By virtue of these properties and algorithms, feasibility and safety constraints typically imposed on autonomous vehicle operations can be enforced and guaranteed independently of the order of the polynomials. To support the use of the proposed method we introduce BeBOT (Bernstein/Bézier Optimal Trajectories), an open-source toolbox that implements the operations and algorithms for Bernstein polynomials. We show that BeBOT can be used to efficiently generate feasible and collision-free trajectories for single and multiple vehicles, and can be deployed for real-time safety critical applications in complex environments.

Optimal stochastic Bernstein polynomials in Ditzian–Totik type modulus of smoothness

We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian–Totik type modulus of smoothness on the interval [0,1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust.

A new approach for Volterra functional integral equations with non-vanishing delays and fractional Bagley-Torvik equationss

A numerical technique for Volterra functional integral equations (VFIEs) with non-vanishing delays and fractional Bagley-Torvik equation is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are utilized to evaluate the accurate results. The findings for examples figs and tables show that the technique is accurate and simple to use.

Numerical solution of system of Emden-Fowler type equations by Bernstein collocation method

The system of Emden-Fowler equations occurs frequently in physical and natural sciences. These system of equations are very difficult to handle due to their singularity and the nonlinearity. We provide a numerically robust technique for approximating the solution of the system of Emden-Fowler equations. This algorithm is based on the Bernstein polynomial accompanied by the collocation method. First, we transform the system of Emden-Fowler equations into their integral form. Then we implement the Bernstein polynomial approximation and the collocation technique to acquire a nonlinear system of equations. We apply the Newton-Raphson method to analyze the resulting nonlinear system of equations. We also discuss the error analysis of this method. We provide the numerical results of the $$L_{\infty }$$ error, the $$L_{2}$$ -norm error, and the residual error of some numerical supporting problems to examine the accuracy of the current technique. The advantage of the present method is exhibited by comparing the numerical results obtained by the proposed technique and other known techniques.

Study of nanoparticle as a drug carrier through stenosed arteries using Bernstein polynomials

In this study, Bernstein polynomial approximation method is used to obtain the expressions of velocity, temperature, wall shear stress and stream function, to portray the transport characteristics of Newtonian gold-blood nanofluid flowing through stenosed artery in the presence of magnetic field. The impact of various flow parameters on velocity profile, wall shear stress and stream function is also investigated. The effect of nanoparticle volume fraction in the presence and absence of external magnetic field is studied. The results reveal that the temperature of the nanofluid and wall shearing stress are increased, whereas the velocity is decreased due to increase in the particle concentration in the presence of magnetic field.

Polynomial convergence order of stochastic Bernstein approximation

Recently, authors of Wu et al. (Adv. Comput. Math. 38:187-205, 2013) studied a Bernstein polynomial approximation scheme based on stochastic sampling and obtained a sixth-order moment estimate for the underlying random variable in terms of the modulus of continuity. In the current paper, we employ a new technique and establish estimates for all the even-order moments. Our work gives a strong indication that the probabilistic convergence rate of the stochastic Bernstein approximation is exponential with respect to the modulus of continuity, which we leave as a conjecture at the end of the paper.

Flexible semiparametric generalized Pareto modeling of the entire range of rainfall amount

AbstractPrecipitation amounts at daily or hourly scales are skewed to the right, and heavy rainfall is poorly modeled by a simple gamma distribution. An important yet challenging topic in hydrometeorology is to find a probability distribution that is able to model well low, moderate, and heavy rainfall. To address this issue, we present a semiparametric distribution suitable for modeling the entire range of rainfall amount. This model is based on a recent parametric statistical model called the class of extended generalized Pareto distributions (EGPDs). The EGPD family is in compliance with extreme value theory for both small and large values, while it keeps a smooth transition between these tails and bypasses the hurdle of selecting thresholds to define extremes. In particular, return levels beyond the largest observation can be inferred. To add flexibility to this EGPD class, we propose to model the transition function in a nonparametric fashion. A fast and efficient nonparametric scheme based on Bernstein polynomial approximations is investigated. We perform simulation studies to assess the performance of our approach. It is compared to two parametric models: a parametric EGPD and the classical generalized Pareto distribution (GPD), the latter being only fitted to excesses above a high threshold. We also apply our semiparametric version of EGPD to a large network of 180 precipitation time series over France.

A new approach for solving Bratu’s problem

Abstract A numerical technique for one-dimensional Bratu’s problem is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are exhibited to verify the efficiency and accuracy of the proposed technique. In this sequel, the obtained error was shown between the proposed technique, Chebyshev wavelets, and Legendre wavelets. The results display that this technique is accurate.

Testing independence for Archimedean copula based on Bernstein estimate of Kendall distribution function

ABSTRACTIn this study, we estimate the Kendall distribution function for Archimedean copula family using Bernstein polynomial approximation and we investigate its performance by Monte Carlo simulation. Then, we introduce a nonparametric test of independence which is based on Cramér–von-Mises distance of the new estimate of Kendall distribution function. Also, we examine the power and the size of the test and we compare it with the classical nonparametric test that is based on the empirical Kendall distribution function.

Design of Near Orthogonal Graph Filter Banks

The processing of signal on graphs is becoming an important emerging area that has great potential in a wide variety of applications, e.g. social network. The work by Narang and Ortega (2012) laid the framework for critically sampled orthogonal two-channel filter bank for signal on undirected bipartite graphs. The design method presented by Narang and Ortega (2012) does not allow the tailoring of the filters spectral response and the control of the reconstruction error of the filter bank. This work present a method to design spectral filters that is based on Bernstein polynomial approximation and constrained optimization. The method allows the trade-off between transition band sharpness, ripple magnitude and reconstruction error.

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) point of view. An original method for estimating distribution functions and densities with Bernstein polynomials is proposed, which takes advantage of results about the eigenstructure of the Bernstein operator to refine a convergence acceleration method. Furthermore, an original data-driven method for choosing the degree of the polynomial is worked out. The method is successfully applied to two data-sets which are important benchmarks in the field of Density Estimation.

Functional CLT of eigenvectors for large sample covariance matrices

In order to investigate property of the eigenvector matrix of sample covariance matrix $$\mathbf {S}_n$$ , in this paper, we establish the central limit theorem of linear spectral statistics associated with a new form of empirical spectral distribution $$H^{\mathbf {S}_n}$$ , based on eigenvectors and eigenvalues of sample covariance matrix $$\mathbf {S}_n$$ . Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of $$H^{\mathbf {S}_n}$$ , indexed by a set of functions with continuous third order derivatives over an interval including the support of Marcenko–Pastur law. This result provides further evidences to support the conjecture that the eigenmatrix of sample covariance matrix is asymptotically Haar distributed.

Functional CLT for sample covariance matrices

Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of sample covariance matrices, indexed by a set of functions with continuous fourth order derivatives on an open interval including $[(1-\sqrt{y})^{2},(1+\sqrt{y})^{2}]$, the support of the Marčenko–Pastur law. We also derive the explicit expressions for asymptotic mean and covariance functions.