An identity for multivariate Bernstein polynomials

Polynomial basis conversion made stable by truncated singular value decomposition

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions

Division algorithms for Bernstein polynomials

Robust arithmetic for multivariate Bernstein-form polynomials

This paper presents a unified framework for constructing transfinite finite elements with arbitrary node distributions, using either Lagrange or Bernstein polynomial bases. Three distinct classes of elements are considered. The first includes elements with structured internal node layouts and arbitrarily positioned boundary nodes. The second comprises elements with internal nodes arranged to allow smooth transitions in a single direction. The third class consists of elements defined on structured T-meshes with selectively omitted internal nodes, resulting in sparsely populated or incomplete grids. For all three classes, new macro-element (global interpolation) formulations are introduced, enabling flexible node configurations. Each formulation supports representations based on either Lagrange or Bernstein polynomials. In the latter case, two alternative Bernstein-based models are developed as follows: one that is numerically equivalent to its Lagrange counterpart, and another that offers modest improvements in numerical performance.

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

This manuscript presents a framework for trajectory generation for soft continuum robots using principles from optimal control. The problem is constrained over the partial differential kinematic equations of the Cosserat rod model, capturing all modes of deformation which soft continuum systems can achieve. The derived optimal control problem is transformed to a nonlinear programming problem which can be solved using the Bernstein polynomial basis. Non-unit quaternions are used to discretely apply rotational transformations to values approximated over Bernstein polynomials, allowing individual components of strain to be constrained separately. Included within this manuscript are numerical results as well as validation through experimental results.

A general method for rendering isosurfaces of multivariate rational and polynomial tensor products is described. The method is robust up to degree 15, handling singularities without introducing spurious rendering artifacts. The approach does not solve the problem of singularities in general, but it removes the problem from the rendering domain to the interpolation/approximation domain. It is based on finding real roots of a polynomial in Bernstein form. This makes it particularly suitable for parallel and pipelined processing. It is envisioned that the tensor products will be used as approximants or interpolants for empirical data or scalar fields. An interpolation scheme is given as an example. >

We present an open source toolbox in Scilab for multivariate polynomial optimization based on the Bernstein form. The developed toolbox finds the global minimum of unconstrained polynomial optimization problems. We first describe the method used by the toolbox, and then demonstrate its performance on several standard optimization examples. We also compare the quality of the results obtained using the developed toolbox with those of popular global optimization methods, such as genetic algorithms, simulated annealing, pattern search, and global search interior point methods. The tests show the superior performance of the presented toolbox.

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

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 function and L p convergence for these operators are studied, respectively.

Conversion from NURBS to Bézier representation

In this paper, we develop a Bernstein dual-Petrov-Galerkin method for the numerical simulation of a two-dimensional fractional diffusion equation. A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix based approach for the spatial discretization. It is shown that the method leads to banded linear systems that can be solved efficiently. The stability and convergence of the proposed method is discussed. Finally, some numerical examples are provided to support the theoretical claims and to show the accuracy and efficiency of the method.

An algorithm for smoothing results of three-dimensional (3-D) Monte Carlo ion implantation simulations and translating them from the grid used for the Monte Carlo simulation to an arbitrary unstructured 3-D grid is presented. This algorithm is important for joining various process simulation steps, where data have to be smoothed or transferred from one grid to another. Furthermore, it is important for integrating the ion implantation simulator into a process flow. One reason for using different grids is that for certain Monte Carlo simulation methods, using orthogrids is mandatory because of performance reasons. The algorithm presented sweeps a small rectangular grid over the points of the new tetrahedral grid and uses approximation by generalized Bernstein polynomials. This approach was put on a mathematically sound basis by proving several properties of these polynomials. It does not suffer from the adverse effects of least squares fits of polynomials of fixed degree as known from the response surface method. The most important properties of Bernstein polynomials generalized to cuboid domains are presented, including uniform convergence, an asymptotic formula, and the variation diminishing property. The smoothing algorithm which works very fast is described and, in order to show its applicability, the resulting values of a 3-D real world implantation example are given and compared with those of a least squares fit of a multivariate polynomial of degree two, which yielded unusable results.