Bernstein Coefficients Research Articles

Bernstein computational algorithm for integro-differential equations

In this study, we introduce a computational algorithm for solving Integro-Differential Equations (IDEs) using Bernstein polynomials as basis functions. The algorithm approximates the solution by expressing it in terms of Bernstein polynomials and substituting this assumed solution into the IDE. Collocating the resulting equation at evenly spaced points yields a system of linear algebraic equations, which is solved via matrix inversion to find the Bernstein coefficients. These coefficients are then used to construct the approximate solution. Numerical examples demonstrate the method's accuracy and efficiency, highlighting its advantages in reducing computational effort.

DISCRETE WEIGHTED RESIDUAL METHODS WHICH ARE TECHNIQUES USED FOR NUMERICAL SOLUTION TO MIXED VOLTERRA-FREDHOLM FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

This study aims to solve the most general form of linear mixed Volterra-Fredholm integro-differential equations of multi-fractional order in the Caputo sense (MV-FIFDEs), which is solved by using orthogonal generalized Bernstein’s polynomial expansion with collocation and moment in the discrete weighted residual method under suitable conditions; then, the Clenshaw-Curtis formula is applied to approximate the integral terms of the equation numerically. In this work, the integro-fractional differential equations are reduced to algebraic linear equations and then to an operational matrix, and the solution of the resultant system yields the unknown Bernstein coefficients of approximation solutions. An algorithm has been created for each technique to handle the MV-FIFDEs using the described methods. Furthermore, numerical examples are presented to demonstrate and compare the techniques’ validity and applicability and comparisons with previous results. The majority of programs are performed on a computer using MATLAB v. 9.7. Keywords: Mixed Fractional Integro-Differential Equations, Caputo Derivative, Bernstein Polynomial, Collocation, Moment, Discrete Weighted Residual, Clenshaw-Curtis Formula, Volterra-Fredholm Equations DOI： https://doi.org/10.35741/issn.0258-2724.58.3.43

Parameter synthesis of polynomial dynamical systems

Parametric dynamical systems emerge as a formalism for modeling natural and engineered systems ranging from biology, epidemiology, and medicine to cyber-physics. Parameter tuning is a complex task usually performed exploiting heavy simulations having high computational complexity and not ensuring the correctness of the synthesized systems. We consider the problem of parameter synthesis for discrete-time polynomial systems. We propose a formal method based on Bernstein coefficients that allows refining the set of parameters according to a temporal specification defined as a Signal Temporal Logic formula. The synthesized system is correct with respect to the specification and we demonstrate the scalability of the approach by implementing it in the C++ library Sapo, also available within a stand-alone application and as a web application. Finally, we illustrate the tool usage and the interface through a simple yet realistic epidemiological model and consider an intriguing application enhancing accuracy of the verification of neural networks.

Numerical optimization and positivity certificates for polynomials and rationals over simplices

Multivariate rational control functions r(x) of high degree d can be expressed in the Bernstein form defined over k-dimensional simplices (triangles). The range of rational functions can be optimized by the enclosure bound (minimum and maximum) of Bernstein coefficients. In this paper, linear and quadratic rate of convergence for the maximum error bound between the enclosure and range bounds are investigated. Furthermore, minimization and numerical identities for certifying the positivity of any given rational function over simplices are given. In order to provide global and local positivity certificates for r, the degree of Bernstein functions and number of simplex subdivision steps are estimated. Subsequently, we establish a bound for the maximum degree of Bernstein polynomials.

Direct Algorithm for Bernstein Enclosure Boundary of Polynomials

Multivariate polynomials of finite degree can be expanded into Bernstein form over a given simplex domain. The minimum and maximum Bernstein control points optimize the polynomial curve over the same domain. In this paper, we address methods for computing these control points in the simplicial case of maximum degree L. To this end, we provide arithmetic operations and properties for obtaining a fast computational method of Bernstein coefficients. Furthermore, we give an algorithm for direct determination of the minimum and maximum Bernstein coefficients (enclosure boundary) in the simplicial multivariate case. Subsequently, the implicit form, monotonicity, and dominance cases are investigated.

Fast Computation of Polynomial Data Points Over Simplicial Face Values

Polynomial functions [Formula: see text] of degree [Formula: see text] have a form in the Bernstein basis defined over [Formula: see text]-dimensional simplex [Formula: see text]. The Bernstein coefficients exhibit a number of special properties. The function [Formula: see text] can be optimised by the smallest and largest Bernstein coefficients (enclosure bounds) over [Formula: see text]. By a proper choice of barycentric subdivision steps of [Formula: see text], we prove the inclusion property of Bernstein enclosure bounds. To this end, we provide an algorithm that computes the Bernstein coefficients over subsimplices. These coefficients are collected in an [Formula: see text]-dimensional array in the field of computer-aided geometric design. Such a construct is typically classified as a patch. We show that the Bernstein coefficients of [Formula: see text] over the faces of a simplex coincide with the coefficients contained in the patch.

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.

Control design and Lyapunov Functions via Bernstein Approximations: Exact Results

We study the control problem for polynomial continuous-time dynamical systems. We consider polynomial Lyapunov functions and controllers, both parameterised in Bernstein form. Specifically, we present necessary and sufficient conditions for existence of polynomial controllers and Lyapunov functions of some maximum degree, providing at the same time explicit upper bounds on the degree of the involved Bernstein polynomials. The formulated conditions are a set of algebraic inequalities, in the space of Bernstein coefficients.

A GPU parallel Bernstein algorithm for polynomial global optimization

We come up with a graphics processing unit (GPU) parallel Bernstein algorithm (BA) aimed at global optimization of multi-variate real polynomials (Garloff in Interval Comput 2:164–168, 1993). We first propose parallel algorithms for (a) computing the multi-index set associated with the Bernstein coefficients (BCs), (b) computing the initial set of BCs using the Matrix method (Ray and Nataraj in Reliab Comput 17(1):40–71, 2012), (c) finding the minimum BC from a given set of BCs, and (d) finding the BCs of the child patches from the parent patch. We then incorporate the above components into the proposed parallel Bernstein algorithm. All the parallel algorithms are programmed for GPU accelerating devices through compute unified device architecture. We compared performance of serial and GPU parallel BA using a test suite of 8 multivariate examples. For the test examples, the proposed parallel algorithm is found 30 times faster as compared to serial one, and needs 96% less time.

Computing the roots of sparse high–degree polynomials that arise from the study of random simplicial complexes

The problem of computing the roots of a particular sequence of sparse polynomials pn(t) is considered. Each instance pn(t) incorporates only the n + 1 monomial terms $t,t^{2},t^{4},\ldots ,t^{2^{n}}$ associated with the binomial coefficients of order n and alternating signs. It is shown that pn(t) has (in addition to the obvious roots t = 0 and 1) precisely n − 1 simple roots on the interval (0,1) with no roots greater than 1, and a recursion relating pn(t) and pn+ 1(t) is used to show that they possess interlaced roots. Closed–form expressions for the Bernstein coefficients of pn(t) on [0,1] are derived and employed to compute the roots in double–precision arithmetic. Despite the severe variation of the graph of pn(t), and tight clustering of roots near t = 1, it is shown that for n ≤ 10, the roots on (0,1) are remarkably well–conditioned and can be computed to high accuracy using both the power and Bernstein forms.

A New Computational Method Based on Bernstein Operational Matrices for Solving Two-Dimensional Linear Stochastic Volterra Integral Equations

This paper is concerned with obtaining the approximate numerical solution of two-dimensional linear stochastic Volterra integral equation by using two-dimensional Bernstein polynomials as basis. Properties of these polynomials and operational matrix of integration together with the product operational matrix are utilized to transform the integral equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Bernstein coefficients. Some theorems are included to show the convergence and advantage of the proposed method. The numerical example illustrates the efficiency and accuracy of the method.

Matrix methods for the tensorial Bernstein form

In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family.

Reduced difference polynomials and self-intersection computations

A reduced difference polynomialf(u,v)=(p(u)−p(v))/(u−v) may be associated with any given univariate polynomial p(t), t ∈ [ 0, 1 ] such that the locus f(u,v)=0 identifies the pairs of distinct values u and v satisfying p(u)=p(v). The Bernstein coefficients of f(u, v) on [ 0, 1 ]2 can be determined from those of p(t) through a simple algorithm, and can be restricted to any subdomain of [ 0, 1 ]2 by standard subdivision methods. By constructing the reduced difference polynomials f(u, v) and g(u, v) associated with the coordinate components of a polynomial curve r(t)=(x(t),y(t)), a quadtree decomposition of [ 0, 1 ]2 guided by the variation-diminishing property yields a numerically stable scheme for isolating real solutions of the system f(u,v)=g(u,v)=0, which identify self-intersections of the curve r(t). Through the Kantorovich theorem for guaranteed convergence of Newton–Raphson iterations to a unique solution, the self-intersections can be efficiently computed to machine precision. By generalizing the reduced difference polynomial to encompass products of univariate polynomials, the method can be readily extended to compute the self-intersections of rational curves, and of the rational offsets to Pythagorean–hodograph curves.

The Bernstein algorithm using the modified implicit Bernstein form and its GPU parallelization using CUDA

In this paper, we propose a Modified Implicit Bernstein Form (MIBF) for computing the Bernstein coefficients of a polynomial. The MIBF avoids several redundant computations in Smith’s Implicit Bernstein Form (IBF) (Smith in J Glob Optim 43:445–458, 2009). Based on the MIBF, we then propose a serial (or CPU based) Bernstein Algorithm for polynomial global optimization. On a set of test problems, the proposed Bernstein Algorithm is about 1.8 times faster than the one based on IBF in Dhabe and Nataraj (Int J Syst Assur Eng Manag, 2017). To obtain further speedups, we next propose a GPU parallel optimization algorithm based on the MIBF, and obtain speedups of up to 44 times with 97% reductions in computations over the serial version. Thus, we recommend the GPU parallel Bernstein algorithm based on the proposed MIBF form for solving polynomial global optimization problems.

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

A Computational Method Based on Bernstein Polynomials for Solving FredholmIntegro-Differential Equations under Mixed Conditions

In this study, a computational method for solving linear FredholmIntegro-Differential Equation (FIDE) of the first order under the mixed conditions using the Bernstein polynomials. First, we present some properties of these polynomials and the method is explained. These properties are then used to convert the integro-differential equation to a system of linear algebraic equations with unknown Bernstein coefficients. Using Galerkin method, we give an approximate solution. This method seems very attractive and simple to use. Illustrative examples show the efficiency and validity of the method we discuss the results using error analysis, the results are discussed.

Numerical solution of linear integral equations system using the Bernstein collocation method

Abstract Since in some application mathematical problems finding the analytical solution is too complicated, in recent years a lot of attention has been devoted by researchers to find the numerical solution of this equations. In this paper, an application of the Bernstein polynomials expansion method is applied to solve linear second kind Fredholm and Volterra integral equations systems. This work reduces the integral equations system to a linear system in generalized case such that the solution of the resulting system yields the unknown Bernstein coefficients of the solutions. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

Bounds on the Range of Multivariate Rational Functions

AbstractBy utilising the expansion of a polynomial into Bernstein polynomials, we construct bounds for the range of a multivariate rational function over a box. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adjoint Optimization of a Wing Using the Class-Shape-Refinement-Transformation Method

This paper will demonstrate the potential of the class-shape-refinement-transformation (CSRT) method for aerodynamically optimizing three-dimensional surfaces. The CSRTmethodwas coupled to an in-house Euler solver and this combination was used in an optimization framework to optimize the ONERA M6 wing in transonic conditions. The gradients of the flow variables with respect to the design parameters were computed using an adjoint solver integrated in theEuler code.The optimizationwasperformedbya trust region reflective algorithm.A two-step approachwas used to optimize the wing. First, a general optimizationwas done using the Bernstein coefficients of the shape function. Second, a regional refinement was performed using the B-spline coefficients of the refinement function. It was shown that using this strategy a considerable improvement of the lift-to-drag ratio of 22% could be achieved. The work presented in this paper proves that the CSRT method is a very intuitive and effective way of parametrizing aircraft shapes, both in two aswell as in three dimensions. Themethod allows for a two-step approach which has the potential to significantly increase the lift-to-drag ratio of various aircraft shapes. It was also shown that using an adjoint algorithm provides the computational efficiency necessary to perform true three-dimensional shape optimization.

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.