Structural Interval Reliability Algorithm Based on Bernstein Polynomials and Evidence Theory

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors.Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid.The uncertain parameters mainly exist in the form of intervals.This method requires a lot of calculation and is often difficult to achieve efficiently.In order to solve this problem,this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty.Based on the non-probabilistic reliability index method,the extreme value of the limit state function is obtained using the properties of Bernstein polynomials,thus avoiding the need for a lot of sampling to solve the reliability analysis problem.The method is applied to numerical examples and engineering applications such as experiments,and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method,especially for some reliability problems with higher nonlinearity.Moreover,this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.

A New Subdivision Algorithm for the Bernstein Polynomial Approach to Global Optimization

In this paper, an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed algorithm is based on the Bernstein polynomial approach. Novel features of the proposed algorithm are that it uses a new rule for the selection of the subdivision point, modified rules for the selection of the subdivision direction, and a new acceleration device to avoid some unnecessary subdivisions. The performance of the proposed algorithm is numerically tested on a collection of 16 test problems. The results of the tests show the proposed algorithm to be superior to the existing Bernstein algorithm in terms of the chosen performance metrics.

Analysis of Nonlinear Electrical Circuits Using Bernstein Polynomials

In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.

SATURATION THEOREMS FOR GENERALIZED BERNSTEIN POLYNOMIALS

In the present paper, we give the explicit formula of the principal part of ∑k=0^n（{k}q-[n]qx）^sxk ∏m=0^n-k-1（1-q^mx） with respect to [n]q for any integer s and q ∈ （0, 1]. And, using the expressions, we obtain saturation theorems for Bn （f , qn,x） approximating to f（x） ∈ C[O, 1], 0 〈 qn ≤ 1, qn → 1.

Application of Bernstein polynomials for solving Fredholm integro-differential-difference equations

In this paper,the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions.This method is using a simple computational manner to obtain a quite acceptable approximate solution.The main characteristic behind this method lies in the fact that,on the one hand,the problem will be reduced to a system of algebraic equations.On the other hand,the efficiency and accuracy of the Bernstein polynomials method for solving these equations are high.The existence and uniqueness of the solution have been proved.Moreover,an estimation of the error bound for this method will be shown by preparing some theorems.Finally,some numerical experiments are presented to show the excellent behavior and high accuracy of this algorithm in comparison with some other well-known methods.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

A CLASS OF INEQUALITIES ABOUT BERNSTEIN POLYNOMIAL AND THEIR APPLICATIONS

In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and rational Bézier curves can be improved greatly.

Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems.Systems of ordinary differential equations like systems of second-order boundary value problems(BVPs),Brusselator system and stiff system are significant in science and engineering.One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations.Bernstein polynomials method with residual correction procedure is used to treat those challenges.The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way.In it,we introduce a method called residual correction procedure,to correct some previous approximate solutions for such systems.We study the error analysis of our given method.We first introduce a new result to approximate the absolute solution by using the residual correction procedure.Second,we introduce a new result to get appropriate bound for the absolute error.The collocation method is used and the collocation points can be found by applying Chebyshev roots.Both techniques are explained briefly with illustrative examples to demonstrate the applicability,efficiency and accuracy of the techniques.By using a small number of Bernstein polynomials and correction procedure we achieve some significant results.We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method,continuous genetic algorithm,rational homotopy perturbation method and adomian decomposition method.

LIMIT OF ITERATES FOR BERNSTEIN POLYNOMIALS DEFINED ON A TRIANGLE

Let f ∈ C[0,1] , and Bn(f,x) be the n-th Bernstein polynomial associated with function f. In 1967, the limit of iterates for Bn(f,x) was given by Kelisky and Rivlin. After this, Many mathematicians studied and generalized this result. But anyway, all these discussions are only for univariate case. In this paper, the main contribution is that the limit of iterates for Bernstein polynomial defined on a triangle is given completely.

Algorithms and Identities for(q, h)-Bernstein Polynomials and(q, h)-Bézier Curves–A Non-Blossoming Approach

We establish several fundamental identities, including recurrence relations, degree elevation formulas, partition of unity and Marsden identity, for quantum Bernstein bases and quantum Bezier curves. We also develop two term recurrence relations for quantum Bernstein bases and recursive evaluation algorithms for quantum Bezier curves. Our proofs use standard mathematical induction and other elementary techniques.

THE BEST LIPSCHITZ CONSTANTS OF BERNSTEIN POLYNOMIALS AND BEZIER NETS OVER A GIVEN TRIANGLE

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i. e. f(P)∈Lip4α, then the corresponding Bernstein Bezier net fn∈LipAsecαψα, here ψ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fn∈ LipBα, then its elevation Bezier net Efn∈LipBα; and (3) If f(P)∈Lipαa, then the corresponding Bernstein polynomials Bn(f;P)∈LipAsecαψα, and the constant Asecαψ best in some sense.

UNIFORM APPROXIMATION BY COMBINATIONS OF BERNSTEIN POLYNOMIALS

We prove a pointwise characterization result for combinations of Bernstein polynomials. The main result of this paper includes an equivalence theorem of H. Berens and G. G. Lorentz as a special case.

AN ASYMPTOTIC EXPANSION FORMULA FOR BERNSTEIN POLYNOMIALS ON A TRIANGLE

In this paper, an asymptotic expansion formula for approximation to a continuous function by Bernstein polymomials on a triangle is obtained.

BEST LIPSCHITZ CONSTANTS FOR THE BEZIER NETS AND BERNSTEIN POLYNOMIALS OVER A SIMPLEX

The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.

Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .

On Absolute Convergence of Bernstein Polynomials *

This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely summable by |C,1| method. Moreover, the estimate of the remainders of the |C,1| sum of the sequence of the Bernstein polynomials is obtained.

Random Sampling Scattered Data with Multivariate Bernstein Polynomials

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous fimction and Lp convergence for these operators are studied, respectively.

Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials

We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.

ON COMPUTING ZEROS OF A BIVARIATE BERNSTEIN POLYNOMIAL

In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the function considered. The algorithm works for rectangular as well as triangular domains. The outlined procedure can also be applied for the computation of the intersection of a Bezier patch and a plane as well as in the determination of an algebraic curve restricted to a compact domain. In particular, singular points of the algebraic curve are reliably detected.

THE BERNSTEIN TYPE INEQUALITY AND SIMULTANEOUS APPROXIMATION BY INTERPOLATION POLYNOMIALS IN COMPLEX DOMAIN

In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.