Abstract
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 . 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.
1. Introduction
The enclosure of Bernstein function estimates the range of polynomials over a given simplex. In order to determine the enclosure boundary, all Bernstein coefficients of degree are needed in the traditional approach and their number is large for Bernstein functions with moderately many variables. The problem of minimizing and bounding polynomials for global optimization problems was considered in [1]. The use of Bernstein expansion for a given power form polynomial over a simplex (triangles) was considered in [2–6]. Applications of a similar approach on shape designs and geometric representations in computer-aided geometric design were generalized in [7]. Furthermore, a computational method for reaction diffusion model was addressed in [8]. In [9], the authors published results in degree elevation and subdivision of the underlying simplex of Bernstein basis for solving global optimization and system problems. In [10, 11], the tensorial Bernstein case over boxes was addressed for computing the enclosure range of a given (multivariate) rational polynomial function, which is slow. Generally speaking, minimizing and maximizing of Bernstein coefficients provide bounds for the range of its polynomial function over any given simplex, whereas the complexity of computing these coefficients is high. In this paper, we simplify the computational of Bernstein coefficients in high degree over a multidimensional simplex. Moreover, we provide a fast method for direct determination of the Bernstein enclosure boundary depending on the indices. This method covers the monotonicity of indices, tolerance case, and implicit Bernstein coefficients. Since our results are mathematically proven, we only give simple examples to be followed by readers.
This paper is organized as follows. In the next section, we briefly recall the simplical polynomial Bernstein form. In Section 3, we present the Bernstein expansion and properties over a simplex. The main results about the optimization and fast computation of Bernstein control points are given in Section 4. Conclusions are given in Section 5.
2. Background of Bernstein Expansion
We present the background and fundamental notations of Bernstein basis over a nondegenerate simplex. From the literature, we give the following definition of simplices.
Definition 1. Let be points of . The list defines a simplex of vertices . The convex hull of the vertex points of is the set of defined as . The largest edge of defines the diameter of .
Throughout this work, the points are affinely independent in which case the simplex is nondegenerate. For simplicity, we consider the standard simplex , where is the zero vector in and is the vector of the canonical basis of , . This is because any simplex in can be linearly transformed to . We also recall that can be formulated as an affine combination of with the barycentric coordinates . If then and . For every multi-index and , we write and . Let the entry-wise relation be given, then for with , we define
If is the degree of any polynomial function such that , then we use the notation .
Definition 2. The Bernstein basis of maximum degree over is defined as , where
Note that Bernstein basis takes nonnegative values on and . Let be a power form polynomial of degree ,
Since the Bernstein expansion forms a basis of the vector space of polynomials of maximum degree , see Proposition 1.6 in [7], then can be expanded as where denote the simplicial coefficients of Bernstein polynomial of degree over .
Remark 1. Let , the grid points of degree associated to are the pointswhere the associated control points to are
Proposition 1 (see Proposition 2.7 [9]). For , the following properties hold:(1)Interpolation at the vertices is as follows:(2)Convex hull of control points: the graph of over is convexly optimized by the convex hull of its associated control points, see Figure 1.(3)Enclosure bound property: the following bound for holds:
Theorem 1 (see Theorem 3.3 in [12]). Let be a polynomial in Bernstein basis of degree . Then, its power form iswhere
3. Simplification of Bernstein
First, we provide the method of affine transformation of upon the standard simplex , where the barycentric coordinates can be transformed to Cartesian coordinates. Subsequently, we simply express power form polynomials into Bernstein form over simplices. Let and for all . Therefore, and . Then, we havefrom which we have
In the following, we simplify the method of expanding power form polynomials in Bernstein form over a simplex.
Notation 1. Consider with , and from (3), we get . Any polynomial power form in (3) of degree can be expressed on asFrom Notation 1, we can write . Additionally ,
Proposition 2. For and , the simplicial Bernstein expansion of maximum degree can be given aswhere
Proof. Let be a polynomial power form of maximum degree . For , we have
4. Main Results
The number of Bernstein coefficients of -variable polynomial of maximum degree is equal to . We aim to store and represent the minimum and maximum Bernstein coefficients in a fast determination method.
In the following proposition, we show the linear combination property of Bernstein coefficients.
Proposition 3. Consider be in the Bernstein basis over a given simplex. The coefficients of Bernstein of maximum degree obtain a linear combination of coefficients of lower degree .
Proof. Let the Bernstein form of maximum degree be given on ,We deduce from the Bernstein basis thatfrom which we have
Remark 2. Consider be a given polynomial of degree over a simplex. The Bernstein coefficients of can be calculated by taking linear combinations of Bernstein coefficients of of degree , i.e., for all ,
Example 1. Let be given over the standard simplex . The computed Bernstein coefficients of are as follows:For determining the enclosure boundary in only one dimensional polynomial of degree 2, we need to compute 6 coefficients.
Remark 3. The minimum value of the enclosure bound optimizes the minimum range of its original polynomial (see Figure 2). For positivity analysis of polynomial systems, if the minimum enclosure is positive, then we certify the global positivity of its polynomial over the given domain.
4.1. The Implicit Bernstein Form
In this subsection, we provide a method of computing the implicit Bernstein coefficients over . Letbe a given polynomial power form comprising the monomials . Let , , then can be written asfor some .
For avoiding the constant terms, we assume that for all .
If consists terms of monomials, then each Bernstein coefficient of degree can be added to the corresponding Bernstein coefficient of the next term:where are the coefficients of the th term of Bernstein .
Remark 4. Consider and be in Bernstein form of the same degree . Then, we have
Example 2. Let of degree be given over the standard simplex . Adding Bernstein coefficients to the corresponding coefficients of each term gives the total number of Bernstein coefficients for :The minimum and maximum values of approximate , and the number of computed coefficients is .
Remark 5. Let be of degree and be of degree with . Then, we have
4.2. Monotonicity of Monomial Coefficients
In this subsection, we show the monotonicity of Bernstein coefficients for high dimensional monomials over a simplex. We assume and is an variable monomial.
Lemma 1. Let , , be a monomial and denote the canonical basis of . Then, the Bernstein coefficients are monotone with respect to , ,
Proof. Let , , be a multivariate power form monomial of degree on . We can express in the monomial Bernstein form of degree . Assume without loss the generality that . Then, we conclude thatfor.
4.3. Polynomials under Dominance
In this subsection, we consider polynomials (of two terms) of orders , respectively, and . Assume for simplicity that . We provide a direct determination of and that occur at some , respectively.
Proposition 4. If ,then and obtains with .
If then and obtains with .
Proof. We give the proof of (32) and the proof of (33) is analogous. For all and , we havewhere are decreasing with respect to and the proof follows.
4.4. Monotonicity of Polynomial Coefficients
The determination of enclosure boundary for multivariate polynomials in Bernstein form required number of coefficients. The minimum Bernstein coefficient of multivariate polynomials approximates the minimum range over the same domain (see Figure 2). We dramatically reduce the search space of coefficients by obtaining the monotonicity of Bernstein.
Remark 6. Consider and be polynomials in Bernstein form of degree and , respectively. From [13], we have
Example 3. Let and be given over . The computed Bernstein coefficients of of degree are as follows:The number of computed coefficients of degree 4 is .
In the following theorem, we generalize the method of fast determination of Bernstein enclosure boundary to the case of monotonicity.
Theorem 2. The multi-indices and of the minimum and maximum coefficients of a Bernstein polynomial over a simplex with power form coefficients for all satisfy
Proof. We give the proof of (37), and the proof of (38) is entirely analogous. Assume that . We proceed by contradiction and assume there is some , , from whichIt follows thatTherefore, we deduce that . Then, the coefficients of Bernstein are decreasing with respect to and for all . Additionally, the coefficients are decreasing with respect to and so , which is a contradiction of (40).
Example 4. Let be a polynomial power form given over . By application of Theorem 2 to , we directly find that and . Subsequently, the respective maximum and minimum Bernstein values appear at the corresponding values of and , which can be computed using (17).
Corollary 1. Let be a (multivariate) power form polynomial of terms with coefficients . Then, for all , the multi-indices of minimum and maximum Bernstein coefficients are appeared as and . If the coefficients , then we have , and .
Corollary 2. By Lemma 1 and Theorem 2, the number of coefficients that are needed to determine the enclosure boundary of dimensional polynomials over a simplex does not exceed .
5. Conclusions
In this work, we considered computing the minimum and maximum Bernstein coefficients (enclosure boundary) that optimize the range of polynomials over a simplex. We reduced the high complexity of computing all -dimensional Bernstein coefficients of degree to . Therefore, we covered the cases of monotonicity, dominance, and multivariate monotonicity of Bernstein coefficients. Finally, the index of enclosure boundary was directly determined and only the enclosure value computed. In the future work, we consider generalizing this method to new classes of functions such as rational polynomial functions over triangles.
Data Availability
The authors confirm that the data supporting the findings of this study are included within the thesis https://kops.uni-konstanz.de/handle/123456789/20898.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the financial support from Al Zaytoonah University of Jordan. The figures and Bernstein Matlab software were implemented by Eng. Qutayba Al-Talaq.