Bernstein Approximation Research Articles

A Combined Taylor–Bernstein Approximation for Solving Non-linear Fitz-Hugh–Nagumo Equation

A Combined Taylor–Bernstein Approximation for Solving Non-linear Fitz-Hugh–Nagumo Equation

Markov and Division Inequalities on Algebraic Sets

A compact set E⊂CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\subset \\mathbb {C}^N$$\\end{document} satisfies the Markov inequality if the supremum norm on E of the gradient of a polynomial p can be estimated from above by the norm of p multiplied by a constant polynomially depending on the degree of p. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^N$$\\end{document}. Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.

Semantic Communication-Based Dynamic Resource Allocation in D2D Vehicular Networks

The semantic communication mechanism enables wireless devices in vehicular networks to communicate more effectively with the semantic meaning. However, in high-dynamic vehicular networks, the transmission of semantic information faces challenges in terms of reliability and stability. To address these challenges, a long-term robust resource allocation scheme is proposed under the Device-to-Device (D2D) vehicular (D2D-V) networks, where multiple performance indicators (user satisfaction, queue stability, and communication delay) are considered. Due to the sophisticated probabilistic form with consideration of channel fluctuations, the Bernstein approximation is introduced to acquire the deterministic constraint more efficiently. The robust resource allocation problem is proposed and separated into two independent subproblems by the Lyapunov optimization method, which includes semantic access control in the application layer and power control in the physical layer. After that, the successive convex approximation method and Karush-Kuhn-Tucher conditions are adopted to solve the subproblems, thereby proposing a robust resource allocation algorithm. The simulations reveal the trade-off relationship between user satisfaction, queue stability, and communication delay, which is on the premise of meeting the user SINR requirement. Moreover, the simulations also prove the necessity of considering channel uncertainty in high-speed mobile vehicular communication scenarios.

An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

On the Randomized Bernstein Approximation Theorem and the Order of Approximation

On the Randomized Bernstein Approximation Theorem and the Order of Approximation

Secure Information Transmission for B5G HetNets: A Robust Game Approach

This article investigates the robust secure transmission problem in two-tier B5G heterogeneous networks with multiple noncollusive eavesdroppers and users, where two types of imperfect channel state information (CSI) scenarios, i.e., instantaneous and statistic CSI scenarios, are considered. Given the two-sidedness of co-channel interference in physical-layer security and the selfishness of femtocell base stations (FBSs), an imperfect-CSI-based noncooperative game framework is proposed to maximize the profits of the macro base station (MBS) and FBSs, while guaranteeing user’s Quality-of-Service (QoS) requirement in terms of outage probability. Specifically, based on the involved two CSI scenarios, the original game where the MBS and FBSs act as players is elaborated as two robust game problems. To address channel uncertainties in the objective function, the worst-case and the mean value of the channel gains are used separately. Besides, the remaining channel uncertainties embodied in the intractable outage probability constraints are treated in a unified way, i.e., the extended Bernstein approximation. The existence and uniqueness of the Nash equilibrium (NE) are analyzed, and the sufficient condition on the uniqueness of the NE is derived. Then, two robust iterative algorithms are given to approach the robust game equilibrium. Finally, numerical results are presented to verify the theoretical analysis and show the robustness and effectiveness of the proposed algorithms.

Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. It is also shown that the Cheney–Sharma operator is the Szász–Mirakyan operator averaged by a certain probability distribution.

Chance-Constrained Optimal Configuration of BESS Considering Uncertain Power Fluctuation and Frequency Deviation Under Contingency

With the accelerating integration of variable renewable energies (VREs), power systems become more vulnerable to active power disturbances, and more drastic frequency dynamics emerge. The battery energy storage system (BESS) is able to handle the uncertainties of VREs, and the decreasing system inertia and frequency regulation capability. This paper proposes a chance-constrained optimal configuration scheme for the BESS to maintain both the uncertain power fluctuations and frequency deviation within predefined limits. First, the required frequency regulation capability of the BESS constrained by the maximum transient frequency deviation (MTFD) and quasi-steady-state frequency deviation (QSSFD) is estimated. Then, the kernel density estimation method is utilized to model the net power fluctuations of VREs and load. A multi-objective chance-constrained programming model accounting for the life cycle cost, energy arbitrage, uncertain power fluctuation, MTFD, and QSSFD is established to optimize the capacity of the BESS. Furthermore, the Bernstein approximation is utilized to process the chance constraint, and transform the optimization model into a deterministic form. Based on the linear weighted method and Benders decomposition, the optimization model is solved through alternating iteration. Case studies were conducted to validate the proposed scheme, showing superior performance in smoothing uncertain power fluctuations, and reducing frequency deviation under contingencies.

Distributed robust power control in two-tier vehicle networks under uncertain channel environments

This paper proposes a novel optimization scheme to support stable and reliable vehicle-to-everything connections in two-tier networks, where the uplink channel of the cellular user is reused by underlay vehicle-to-vehicle communications. However, considering complex channel fading and high-speed vehicle movement, the certainty assumption is impractical and fails to maintain power control strategy in reality in the traditional statical vehicular networks. Rather than the perfect channel state information assumption, the first-order Gauss-Markov process which is a probabilistic model affected by vehicle speed and fading is introduced to describe imperfect channel gains. Moreover, interference management is a major challenge in reusing communications, especially in uncertain channel environments. Power control is an effective way to realize interference management, and optimal power allocation can ensure that interference of the user meets the communication requirements. In this study, the sum-rate-oriented power control scheme and minimum-rate-oriented power control scheme were implemented to manage interference and satisfy different design objectives. Since both of these schemes are non-convex and intractable, the Bernstein approximation and successive convex approximation methods were adopted to transform the original problems into convex ones. Furthermore, a novel distributed robust power control algorithm was developed to determine the optimal solutions. The performance of the algorithm was evaluated through numerical simulations, and the results indicate that the proposed algorithm is effective in vehicular communication networks with uncertain channel environments.

Bernstein operator method for approximate solution of singularly perturbed Volterra integral equations

An approximate solution of integral equations takes an active role in the numerical analysis. This paper presents and tests an algorithm for the approximate solution of singularly perturbed Volterra integral equations via the Bernstein approximation technique. The method of computing the numerical approximation of the solution is properly demonstrated and exemplified in the matrix notation. Besides, the error bound and convergence associated with the numerical scheme are constituted. Finally, particular examples indicate the dependability and numerical capability of the introduced scheme in comparison with other numerical techniques.

Bernstein approximation technique for numerical solution of Volterra integral equations of the third kind

Bernstein approximation technique for numerical solution of Volterra integral equations of the third kind

Efficient Approximation of Two-Terminal Networks Reliability Polynomials Using Cubic Splines

In this article, two new techniques of approximation of the reliability of a two-terminal network are developed based on the constructive theory of functions and related methods. Two methods of generating an approximation cubic spline are used: Lagrange-type interpolation procedures and Bernstein approximation operator. A possibility of minimizing the total error of approximation, based on keeping some properties invariant, is described in case of a large class of pairs of dual two-terminal networks. Simulations are included, showing that the error of approximation is negligible in case of some special initial data.

Maximize Spectrum Efficiency in Underlay Coexistence With Channel Uncertainty

We consider an underlay coexistence scenario where secondary users (SUs) must keep their interference to the primary users (PUs) under control. However, the channel gains from the PUs to the SUs are uncertain due to a lack of cooperation between the PUs and the SUs. Under this circumstance, it is preferable to allow the interference threshold of each PU to be violated occasionally as long as such violation stays below a probability. In this article, we employ Chance-Constrained Programming (CCP) to exploit this idea of occasional interference threshold violation. We assume the uncertain channel gains are only known by their mean and covariance. These quantities are slow-changing and easy to estimate. Our main contribution is to introduce a novel and powerful mathematical tool called Exact Conic Reformulation (ECR), which reformulates the intractable chance constraints into tractable convex constraints. Further, ECR guarantees an equivalent reformulation from linear chance constraints into deterministic conic constraints without the limitations associated with Bernstein Approximation, on which our research community has been fixated on for years. Through extensive simulations, we show that our proposed solution offers a significant improvement over existing approaches in terms of performance and ability to handle channel correlations (where Bernstein Approximation is no longer applicable).

Minimizing Energy Cost for Green Data Center by Exploring Heterogeneous Energy Resource

With the deteriorating effects resulting from global warming in many areas, geographically distributed data centers contribute greatly to carbon emissions, because the major energy supply is fossil fuels. Considering this issue, many geographically distributed data centers are attempting to use clean energy as their energy supply, such as fuel cells and renewable energy sources. However, not all workloads can be powered by a single power sources, since different workloads exhibit different characteristics. In this paper, we propose a fine-grained heterogeneous power distribution model with an objective of minimizing the total energy costs and the sum of the energy gap generated by the geographically distributed data centers powered by multiple types of energy resources. In order to achieve these two goals, we design a two-stage online algorithm to leverage the power supply of each energy source. In each time slot, we also consider a chance-constraint problem and use the Bernstein approximation to solve the problem. Finally, simulation results based on real-world traces illustrate that the proposed algorithm can achieve satisfactory performance.

Numerical analysis of fractional Volterra integral equations via Bernstein approximation method

In this study, Bernstein approximation method has been applied along with Riemann–Liouville fractional integral operator to solve both the second and the first kind of fractional Volterra integral equations (2nd FVIEs and 1st FVIEs respectively). In order to show the applicability and efficiency of the proposed technique, some convergence analysis has been provided. Illustrative numerical experiments with comparison are included to indicate the validity and practicability of the method. All of the numerical calculation in this research has been done on a personal computer implementing some programs written in MATLAB.

Energy-Spectral Efficiency Optimization in Vehicular Communications: Joint Clustering and Pricing-Based Robust Power Control Approach

Smart, and green cities impose stringent requirements on spectral efficiency (SE), and energy efficiency (EE) of vehicular networks. For the current vehicular ad-hoc networks (VANETs), vehicle's mobility leads to rapid topology changes, and high channel uncertainty. However, clustering schemes for establishing stable clusters, and robust power control (RPC) combating with channel fluctuation are investigated independently. In this paper, joint clustering, and RPC schemes are proposed to optimize the SE, and EE of the involved VANETs. Via the same fixed-length slot, the synchronized interference constraints of cluster heads (CHs) are formed, and offer conditions for RPC. Due to the random channel fluctuations, all CHs’ synchronized interference constraints are formulated as probability constraints. Besides, a pricing-based utility which avoids the separate optimization between SE, and EE is introduced, and the price's impact on the tradeoff between them is involved. Since the probability constraints are intractable, and the unified utility is nonconvex, the Bernstein approximation, and successive convex approximation (SCA) are used to transform the problem into a tractable convex one. Through dual decomposition, two RPC algorithms are proposed to determine the optimal solutions for the fixed price $C$ , and the optimal price $C^*$ , respectively. Numerical simulations are used to evaluate the algorithmic performances in high-dynamic system, and the results show that the proposed algorithms are effective. The validity of the clustering method, and the proposed RPC scheme is further verified by comparisons.

On New Modification of Bernstein Operators: Theory and Applications

Approximation theory has a significant place in the studies in mathematics, and the researches on it are increasing with each passing day. Accordingly, a number of studies have been presented about Bernstein approximation which is one of the most known linear positive operators. In this study, a new modification of Bernstein operators which fix constant and preserve Korovkin’s other test functions in limit case has been introduced. Then, the approximation properties of the newly defined operators such as asymptotic formulas, weighted approximation and rate of convergence have been presented. Moreover, numerical simulations are included. Finally, discussion and conclusions are presented.

Energy Efficiency Optimization for Distributed Antenna Systems With D2D Communications Under Channel Uncertainty

In this paper, we investigate the energy efficiency (EE) maximization problem in distributed antenna systems (DAS) with underlaid device-to-device (D2D) communications. The EE optimization relies on the channel state information (CSI), and the interference from D2D pairs to the cellular user equipment (CUE) also closely depends on CSI. In this paper, we consider the case that the CSI in the systems is uncertain, which is more realistic and reasonable. We aim to obtain the robust power allocation (PA) solution that can achieve the maximum EE of the D2D systems. The optimization problem can be formulated as a non-convex and non-linear problem with infinite interference constraint. In order to solve it, the interference constraint is treated as chance constraint and handled by Bernstein approximation. Moreover, an equivalent objective function with a sub-tractive form is transformed by exploiting fractional programming theory. After that, the non-convex objective function can be transformed by using the difference of convex (D.C.) programming. The concave convex procedure (CCCP) algorithm is used to tackle the problem and obtain the optimal power allocation for D2D users. Simulation results show the robustness of the system and the effectiveness of the proposed algorithm.

On a Uniqueness Theorem of E. B. Vul

We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved by elementary means.

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.