Numerical construction of spherical t-designs by Barzilai-Borwein method

Numerical construction of spherical -designs by Barzilai-Borwein method

Yuchen Xiao E-mail address: fkxych@stu.jnu.edu.cn Department of Mathematics, Jinan University, Guangzhou, China Congpei An Corresponding author: andbachcp@gmail.com, cpan@cuhk.edu.hk
Abstract

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.

1 Introduction

Distributing finite points on the unit sphere is a challenging problem in the 21st century [1]. 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).

Definition 1.1.

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.,

(1)

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. [2] 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 [2]:

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. [3] proved spherical -designs exists for points. From the work of Chen et al. [7], 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 [4]. For , Chen and Womersley verified a spherical -design exist in a neighborhood of an extremal system [6]. For , An et al. [5] 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 [10] is a gradient method with modified step sizes, which is motivated by Newton’s method but not involves any Hessian. Further investigations [11] 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 [12],

(2)

Note that . It is well known that the addition theorem for spherical harmonics on gives

(3)

where is Legendre polynomial and is the inner product in . Sloan and Womersley [8] introduced a variational characterization of spherical -designs

(4)
Theorem 2.1 ([8]).

Let , and . Then

(5)

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 .

Definition 2.1.

A point is a stationary point of if , where is the spherical gradient (or surface grident [12]) of .

Let the basis matrix be where and

Definition 2.2.

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. [5] described the fundamental system in finding spherical -designs.

Lemma 2.2 ([5]).

A set is a fundamental system for if and only if is of full row rank .

Lemma 2.3 ([5]).

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.

Theorem 2.4.

Let and . Assume is a stationary point set of and the minimal singular value of basis matrix is positive. Then is a spherical -design.

Proof.

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

(6)

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 [10] to construct Algorithm 1 for seeking an efficient way to compute , that achieves the local minimum. Due to the universality of quasi-Newton method [8], 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 [13] and backtracking line search [13] to lead BB method in a proper way to find local minimum .

1:: spherical polynomial degree; : number of points; : distributing points on unit sphere ; : maximum iterations; : termination tolerance on the first-order optimality; : termination tolerance on progress in terms of function or parameter changes. Initialize , , , and .
2:while  and , or  do
3:     ,
4:     compute step size
5:     if  or  then
6:         
7:     end if
8:     if  and
9:  ,  then
10:          (Armijo-Goldstein rule)
11:     else 
12:         , (backtracking line search)
13:     end if
14:     
15:     compute and search direction
16:end while
17:numerical spherical -designs .
Algorithm 1 Barzilai-Borwein method for computing spherical -designs

Now we give a small numerical example by using Algorithm 1, which is used to illustrate the numerical construction of spherical -design.

Example 3.1.

We generate spiral points from [14],

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.

(a) Initial vertices
(b) Final vertices
Figure 1: Numerical simulation of regular tetrahedron vertices on by using Algorithm 1
(a) The behavior of
(b) The behavior of
Figure 2: Numerical behavior of and with by using Algorithm 1

3.2 Convergence analysis

From the view of (4), we know for . We assume that

Assumption 3.1.

The level set is bounded, and there exists such that , where is a Hessian matrix of .

Now we present the convergence result on Algorithm 1. We shall mention that the idea of proof originated in [15, 16].

Theorem 3.1.

Let be an initial point and and assume Assumption 3.1 holds. Suppose that is generated by Algorithm 1, then .

Proof.

By using the Armijo rule (mark 8) from Algorithm 1 and mean value theorem, we have

(7)

where is a gradient of and , then

(8)

According to Cauchy inequality, we obtain

(9)

moreover, by using mean value theorem

(10)

Combine (9) and (10), we know

(11)

therefore

(12)

By Armijo rule (mark 7) from Algorithm 1 and (12), we have

(13)

thus

(14)

Since is bounded, we know exists, therefore

(15)

hence

(16)

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. ∎

Theorem 3.1 shows the convergence of Algorithm 1. Based on the above theorems, we summarize the following results.

Remark 1.

Let be the starting points set of Algorithm 1, by Theorem 3.1, then there exist such that when , where is a zero vector. Therefore, is a stationary points set of .

Remark 2.

Let be the starting points set of Algorithm 1, then we have . If Theorem 2.4 is established in , then is a spherical -design.

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 [4] 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 [18]. 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) [13] 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 .

Figure 3(a) and Figure 4(a) are well exhibited the locally R-linear convergence [11] of BB method by numerical computation of with . We can see that converges to with iteration increase.

Iteration Time
10 121 100 7.796661e-16 1.8478e-09 1.049909s 1.3270
50 2601 335 1.879594e-12 6.3307e-09 336.663545s 2.3394
96 9409 715 8.237123e-10 7.3212e-08 22724.767736s 2.1647
127 16384 803 8.229142e-10 2.7122e-08 84579.358811s 1.9673
Table 1: Computing of spherical -designs by BB method
Iteration Time
10 121 81 1.054716e-15 2.1856e-09 1.062004s 1.3232
50 2601 278 8.455657e-15 2.4398e-09 397.480675s 2.3380
96 9409 465 1.418436e-14 1.4049e-09 40101.809512s 2.1637
127 16384 543 3.709079e-13 2.0536e-09 143631.176093s 1.9656
Table 2: Computing of spherical -designs by quasi-Newton method
(a) Barzilai-Borwein method
(b) Quasi-Newton method
Figure 3: The behavior of for on in each iteration
(a) Barzilai-Borwein method
(b) Quasi-Newton method
Figure 4: The behavior of for on in each iteration
(a) Barzilai-Borwein method
(b) Quasi-Newton method
Figure 5: Numerical simulation for on by using different methods

5 Conclusion

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.

Acknowledgements

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).

References

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