Numerical construction of spherical -designs by Barzilai-Borwein method
A point set on the unit sphere is a spherical -design is equivalent to the nonnegative quantity vanished. We show that if is a stationary point set of and the minimal singular value of basis matrix is positive, then is a spherical -design. Moreover, the numerical construction of spherical -designs is valid by using Barzilai-Borwein method. We obtain numerical spherical -designs with up to at .
Keywords. Spherical -designs, Variational characterization, Barzilai-Borwein method, Singular values.
2010 MSC. 65D99, 65F99.
Distributing finite points on the unit sphere is a challenging problem in the 21st century . Spherical -design is to find the ‘good’ finite sets of points on the unit sphere for spherical polynomial approximations. Spherical -design is very useful in approximation theory, geometry and combinatorics. Recently, it has been applied in quantum mechanics (for quantum -design) and statistics (for rotatable design).
A finite set is a spherical -design if for any polynomial of degree at most such that the average value of on the equals the average value of on , i.e.,
where is the surface of the whole unit sphere , is the space of spherical polynomials on with degree at most and denotes the surface measure on .
The concept of spherical -design was introduced by Delsarte et al.  in 1977. From then on, spherical -designs have been studied extensively [3, 4, 5, 6, 7, 8, 9]. In this paper, we pay attention to 2-dimensional unit sphere .
A lower bound on the number of points to construct a spherical -design for any on was given in :
It is shown that the lower bound can not be achieved, in other words, there is no spherical -design with points for any . Bondarenko et al.  proved spherical -designs exists for points. From the work of Chen et al. , we know that spherical -designs with points exist for all degrees up to 100 on . This encourage us to find higher degrees for spherical -designs.
Extremal points are sets of points on which maximize the determinant of a basis matrix for an arbitrary basis of . For , Chen and Womersley verified a spherical -design exist in a neighborhood of an extremal system . For , An et al.  verified extremal spherical -designs exist for all degrees up to 60 and provided well conditioned spherical -designs for interpolation and numerical integration.
By now, numerical methods have been developed for finding spherical -designs. The problem of finding a spherical -design is expressed as solving nonlinear equations or optimization problems [8, 5]. However, the first order methods for computing spherical -designs are rarely developed. In this paper, we numerically construct spherical -designs by using Barzilai-Borwein method (BB method). The BB method  is a gradient method with modified step sizes, which is motivated by Newton’s method but not involves any Hessian. Further investigations  showed that BB method is locally -linear convergent for general objective functions.
In the next section, we present the required techniques, definitions and first order conditions for spherical -designs. The BB method for computing spherical -designs and its convergence analysis are presented in Section 3. Numerical results for point sets which up to 127 and are included in Section 4. Section 5 ends this paper with a brief conclusion.
2 First order conditions for spherical -design
for degree and order is a complete set of orthonormal real spherical harmonics basis for , where orthogonality with respect to the inner product ,
Note that . It is well known that the addition theorem for spherical harmonics on gives
where is Legendre polynomial and is the inner product in . Sloan and Womersley  introduced a variational characterization of spherical -designs
Theorem 2.1 ().
Let , and . Then
and is a spherical t-design if and only if
It is known that is a spherical -design if and only if vanished. Naturally, one might consider the first order condition to check the global minimizer of .
A point is a stationary point of if , where is the spherical gradient (or surface grident ) of .
Let the basis matrix be where and
A finite set is called a fundamental system for if the zero polynomial is the only element of that vanishes at each point in .
An et al.  described the fundamental system in finding spherical -designs.
Lemma 2.2 ().
A set is a fundamental system for if and only if is of full row rank .
Lemma 2.3 ().
Let and . Assume is a stationary point set of and is a fundamental system for . Then is a spherical -design.
Based on these results, we have the applicable first order condition for spherical -designs as follows.
Let and . Assume is a stationary point set of and the minimal singular value of basis matrix is positive. Then is a spherical -design.
Suppose that the minimal singular value of is positive, then we have all the singular values of are positive immediately. We know that the number of non-zero singular values of equals the rank of , so is of full rank, which means is a fundamental system of by Lemma 2.2. And then suppose that is a stationary point set, then is a spherical -design by Lemma 2.3. Hence, we complete the proof. ∎
Theorem 2.4 is useful in first order optimization method, which provides a simple way to verify the global minimizer to the objective function.
3 Iterative methods for finding spherical -designs
3.1 Algorithm design
Fix and , for objective function , we consider the optimization problem
Apparently, is a non-convex function. For computing conveniently, we assume the first point is the north pole point and the second point is on the primer meridian. Then we can define coordinates convert functions which can convert Cartesian coordinates into spherical coordinates as a vector, and which can convert a vector form spherical coordinates into Cartesian coordinates as a matrix. So for we have
where vector and vector .
We apply BB method in  to construct Algorithm 1 for seeking an efficient way to compute , that achieves the local minimum. Due to the universality of quasi-Newton method , we also apply quasi-Newton method for comparing the efficiency. And then we try to use Theorem 2.4 to prove the local minimum we found is the global minimum, that is, we find the real numerical spherical -design.
To make sure that objective function is sufficient to descend and approximate to which is as near as , we use Armijo-Goldstein rule  and backtracking line search  to lead BB method in a proper way to find local minimum .
Now we give a small numerical example by using Algorithm 1, which is used to illustrate the numerical construction of spherical -design.
We generate spiral points from ,
By using Algorithm 1, we obtain the termination output: , , and ends with value
In fact, is a set of regular tetrahedron vertices, which is known as a spherical -design. As a result, Algorithm 1 reaches the global minimum , thus numerical solutions for spherical -design found. We can see the explicit change of by using Algorithm 1 from Figure 1 and the behavior of objective function from Figure 2.
3.2 Convergence analysis
From the view of (4), we know for . We assume that
The level set is bounded, and there exists such that , where is a Hessian matrix of .
By using the Armijo rule (mark 8) from Algorithm 1 and mean value theorem, we have
where is a gradient of and , then
According to Cauchy inequality, we obtain
moreover, by using mean value theorem
Since is bounded, we know exists, therefore
Now we assume that . We can find a set of (), when , , and there exist such that . Therefore, (15) can not be hold, which contradicts. Thus, , we complete the proof. ∎
4 Numerical results
Based on the code in [8, 17], we present the feasibility of Algorithm 1 to compute spherical -design with the point set where for up to 127. As an initial point set to solve the optimization problem of minimizing from (4), we use the extremal systems from  without any additional constraints. To make sure BB method is meaningful in spherical -designs, we compare BB method with quasi-Newton method(QN). These methods are implemented in Matlab R2015b and tested on an Intel Core i7 4710MQ CPU with 16 GB DDR3L memory and a 64 Bit Windows 10 Education.
We present the results in Table 1 and Table 2, these numerical spherical -designs can be founded in . We observe that BB method cost less time than quasi-Newton method, especially in large . Furthermore, all point sets are verified to be fundamental systems. In fact, we use singular value decomposition (SVD)  to obtain all singular values of , which are defined as for . As a result, the , then is of full rank, thus is a fundamental system. This is a strong numerical support to Theorem 2.4. Here we set .
In this paper, we employ Barzilai-Borwein method for finding numerical spherical -designs with up to 126 with . This method performs high efficiency and accuracy. Moreover, we check numerical solution as global minimizer with positivity of minimal singular value of basis matrix. Numerical experiments show that Barzilai-Borwein method is better than quasi-Newton method in time efficiency for solving large scale spherical -designs. These numerical results are interesting and inspiring. The numerical construction of higher order spherical -designs are expected in future study.
This work is supported by National Natural Science Foundation of China (Grant No.11301222) and the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (Grant No.2018014).
-  S. Smale, Mathematical problems for the next century, The Mathematical Intelligencer 20 (2) (1998) 7–15.
-  P. Delsarte, J. Goethals, J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (3) (1977) 363–388.
-  A. Bondarenko, D. Radchenko, M. Viazovska, Optimal asymptotic bounds for spherical designs, Annals of mathematics (2013) 443–452.
-  I. H. Sloan, R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Advances in Computational Mathematics 21 (1-2) (2004) 107–125.
-  C. An, X. Chen, I. H. Sloan, R. S. Womersley, Well conditioned spherical designs for integration and interpolation on the two-sphere, SIAM Journal on Numerical Analysis 48 (6) (2010) 2135–2157.
-  X. Chen, R. S. Womersley, Existence of solutions to systems of underdetermined equations and spherical designs, SIAM Journal on Numerical Analysis 44 (6) (2006) 2326–2341.
-  X. Chen, A. Frommer, B. Lang, Computational existence proofs for spherical t-designs, Numerische Mathematik 117 (2) (2011) 289–305.
-  I. H. Sloan, R. S. Womersley, A variational characterisation of spherical designs, Journal of Approximation Theory 159 (2) (2009) 308–318.
-  E. Bannai, E. Bannai, A survey on spherical designs and algebraic combinatorics on spheres, European Journal of Combinatorics 30 (6) (2009) 1392–1425.
-  J. Barzilai, J. M. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis 8 (1) (1988) 141–148.
-  Y. Dai, L. Liao, R-linear convergence of the barzilai and borwein gradient method, IMA Journal of Numerical Analysis 22 (1) (2002) 1–10.
-  W. Freeden, T. Gervens, M. Schreiner, Constructive approximation on the sphere with applications to geomathematics, Oxford University Press on Demand, 1998.
-  W. Sun, Y.-X. Yuan, Optimization theory and methods: nonlinear programming, Vol. 1, Springer Science & Business Media, 2006.
-  R. Bauer, Distribution of points on a sphere with application to star catalogs, Journal of Guidance, Control, and Dynamics 23 (1) (2000) 130–137.
-  Y.-X. Yuan, Problems on convergence of unconstrained optimization algorithms, Numerical Linear Algebra and Optimization (1999) 95–107.
-  M. J. D. Powell, Algorithm for minimization without exact line searches, Nonlinear programming 9 (1976) 53.
-  M. Schmidt, minfunc: unconstrained differentiable multivariate optimization in matlab, http://www.cs.ubc.ca/~schmidtm/Software/minFunc.html (2005).
-  Y. Xiao, C. An, Numerical results of spherical -designs with up to 127, https://github.com/fkxych/Numerical-construction-of-spherical-t-designs-by-Barzilai-Borwein-method.git (2019).