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

TL;DR

This paper introduces a numerical method using the Barzilai-Borwein algorithm to construct spherical t-designs on the unit sphere, achieving designs with high degrees up to t+1=127.

Contribution

It demonstrates that stationary points of a specific quantity, combined with basis matrix conditions, can produce spherical t-designs via an efficient numerical approach.

Findings

Successfully constructed spherical t-designs with t+1 up to 127.

Validated the method's effectiveness for designs with N=(t+2)^2 points.

Established theoretical conditions linking stationary points to t-designs.

Abstract

A point set $\mathrm X_N$ on the unit sphere is a spherical $t$-design is equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if $\mathrm X_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular value of basis matrix is positive, then $\mathrm X_N$ is a spherical $t$-design. Moreover, the numerical construction of spherical $t$-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical $t$-designs with $t+1$ up to $127$ at $N=(t+2)^2$.

Tables

Table 1. Table 1: Computing of spherical t 𝑡 t -designs by BB method

$t+1$	$N$	Iteration	$A_{N,t}\left(X_N\right)$	${\parallel {\nabla }^{\ast }{A}_{N,t}\left(X_N\right)\parallel }_{\infty }$	Time	$\min \left({\sigma }_i\right)$
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 2. Table 2: Computing of spherical t 𝑡 t -designs by quasi-Newton method

$t+1$	$N$	Iteration	$A_{N,t}\left(X_N\right)$	${\parallel {\nabla }^{\ast }{A}_{N,t}\left(X_N\right)\parallel }_{\infty }$	Time	$\min \left({\sigma }_i\right)$
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

Equations

$$\frac{1}{N}\sum _{i=1}^{N}p({\mathbf{x}}_i)=\frac{1}{|{\mathbb{S}}^d|}{\int }_{{\mathbb{S}}^d}p(\mathbf{x})d\omega (\mathbf{x})\forall p\in {\Pi }_t,$$

$$N\ge N^{\ast }=\{\begin{array}{ll}\frac{1}{4}(t+1)(t+3),& \text{t is odd,}\\ \frac{1}{4}(t+2{)}^2,& \text{t is even.}\end{array}$$

$$⟨f,g{⟩}_{{\mathbb{S}}^2}:={\int }_{{\mathbb{S}}^2}f(\mathbf{x})g(\mathbf{x})d\omega (\mathbf{x}),f,g\in L_2({\mathbb{S}}^2).$$

$$\sum _{k=1}^{2n+1}{\mathrm{Y}}_n^k(\mathbf{x}){\mathrm{Y}}_n^k(\mathbf{y})=\frac{2n+1}{4\pi }{\mathrm{P}}_n(⟨\mathbf{x},\mathbf{y}⟩)\forall \mathbf{x},\mathbf{y}\in {\mathbb{S}}^2,$$

$$A_{N,t}({\mathrm{X}}_N):=\frac{4\pi }{N^2}\sum _{n=1}^t\sum _{k=1}^{2n+1}(\sum _{i=1}^N{\mathrm{Y}}_n^k({\mathbf{x}}_i){)}^2=\frac{4\pi }{N^2}\sum _{j=1}^N\sum _{i=1}^N\sum _{n=1}^t\frac{2n+1}{4\pi }{\mathrm{P}}_n(⟨{\mathbf{x}}_j,{\mathbf{x}}_i⟩).$$

$$0\le A_{N,t}({\mathrm{X}}_N)\le (t+1{)}^2-1,$$

$$A_{N,t}({\mathrm{X}}_N)=0.$$

$$\underset{{\mathrm{X}}_N\subset {\mathbb{S}}^2}{\min }{A}_{N,t}({\mathrm{X}}_N).$$

$$\eta ({\mathrm{X}}_N)$$

$$\mu (\eta ({\mathrm{X}}_N))$$

$${\mathbf{x}}_1=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],{\mathbf{x}}_2=\left[\begin{array}{c}0.9872\\ 0\\ -0.1595\end{array}\right],{\mathbf{x}}_3=\left[\begin{array}{c}-0.3977\\ 0.6727\\ -0.6239\end{array}\right],{\mathbf{x}}_4=\left[\begin{array}{c}-0.6533\\ -0.7455\\ -0.1318\end{array}\right].$$

$${\mathbf{x}}_1^{\ast }=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],{\mathbf{x}}_2^{\ast }=\left[\begin{array}{c}0.9428\\ 0\\ -0.3333\end{array}\right],{\mathbf{x}}_3^{\ast }=\left[\begin{array}{c}-0.4714\\ 0.8165\\ -0.3333\end{array}\right],{\mathbf{x}}_4^{\ast }=\left[\begin{array}{c}-0.4714\\ -0.8165\\ -0.3333\end{array}\right].$$

$$f(x_k)-f(x_k-{\alpha }_k{g}_k)={\alpha }_k\nabla f(x_k-\kappa {\alpha }_k{g}_k{)}^{\top }{g}_k\le {\alpha }_k(1-\rho )g_k^{\top }{g}_k,$$

$$\rho g_k^{\top }{g}_k\le (\nabla f(x_k)-\nabla f(x_k-\kappa {\alpha }_k{g}_k){)}^{\top }{g}_k.$$

$$\rho g_k^{\top }{g}_k\le (\nabla f(x_k)-\nabla f(x_k-\kappa {\alpha }_k{g}_k){)}^{\top }{g}_k\le \Vert \nabla f(x_k)-\nabla f(x_k-\kappa {\alpha }_k{g}_k)\Vert \Vert g_k\Vert ,$$

$$\Vert \nabla f(x_k)-\nabla f(x_k-\kappa {\alpha }_k{g}_k)\Vert =\Vert {\int }_0^{1}F(x_k-\xi \kappa {\alpha }_k{g}_k)\kappa {\alpha }_k{g}_{k}d\xi \Vert \le M\kappa {\alpha }_k\Vert g_k\Vert .$$

$$\rho g_k^{\top }{g}_k\le M\kappa {\alpha }_k\Vert g_k\Vert ,$$

$${\alpha }_k\Vert g_k\Vert \ge \frac{\rho g_k^{\top }{g}_k}{M\Vert g_k\Vert }.$$

$$f(x_{k+1})\le f(x_k)-{\alpha }_k\rho \Vert g_k\Vert \frac{g_k^{\top }{g}_k}{\Vert g_k\Vert }\le f(x_k)-\frac{{\rho }^2}{M}(\frac{g_k^{\top }{g}_k}{\Vert g_k\Vert }{)}^2,$$

$$\sum _{j=1}^k(\frac{g_j^{\top }{g}_j}{\Vert g_j\Vert }{)}^2\le \frac{M}{{\rho }^2}(f(x_1)-f(x_{k+1})).$$

$$\sum _{j=1}^k(\frac{g_j^{\top }{g}_j}{\Vert g_j\Vert }{)}^2<+\infty ,$$

$$\underset{k\to \infty }{\lim }\frac{g_k^{\top }{g}_k}{\Vert g_k\Vert }=0.$$

Full text

Keywords. Spherical $t$-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 $t$-design is to find the ‘good’ finite sets of points on the unit sphere $\mathbb{S}^{d}:=\{\bm{x}\in\mathbb{R}^{d+1}|\lVert\bm{x}\rVert_{2}=1\}$ for spherical polynomial approximations. Spherical $t$-design is very useful in approximation theory, geometry and combinatorics. Recently, it has been applied in quantum mechanics (for quantum $t$-design) and statistics (for rotatable design).

Definition 1.1**.**

A finite set $\mathrm{X}_{N}:=\{\bm{x}_{1},\ldots,\bm{x}_{N}\}\subset\mathbb{S}^{d}$ is a spherical $t$-design if for any polynomial $p:\mathbb{R}^{d+1}\to\mathbb{R}$ of degree at most $t$ such that the average value of $p$ on the $\mathrm{X}_{N}$ equals the average value of $p$ on $\mathbb{S}^{d}$, i.e.,

[TABLE]

where $|\mathbb{S}^{d}|$ is the surface of the whole unit sphere $\mathbb{S}^{d}$, $\Pi_{t}:=\Pi_{t}(\mathbb{S}^{d})$ is the space of spherical polynomials on $\mathbb{S}^{d}$ with degree at most $t$ and $d\omega(\bm{x})$ denotes the surface measure on $\mathbb{S}^{d}$.

The concept of spherical $t$-design was introduced by Delsarte et al. [2] in 1977. From then on, spherical $t$-designs have been studied extensively [3, 4, 5, 6, 7, 8, 9]. In this paper, we pay attention to 2-dimensional unit sphere $\mathbb{S}^{2}$.

A lower bound on the number of points $N$ to construct a spherical $t$-design for any $t\geq 1$ on $\mathbb{S}^{2}$ was given in [2]:

[TABLE]

It is shown that the lower bound can not be achieved, in other words, there is no spherical $t$-design with $N^{*}$ points for any $t\geq 2$. Bondarenko et al. [3] proved spherical $t$-designs exists for $O(t^{2})$ points. From the work of Chen et al. [7], we know that spherical $t$-designs with $(t+1)^{2}$ points exist for all degrees $t$ up to 100 on $\mathbb{S}^{2}$. This encourage us to find higher degrees $t$ for spherical $t$-designs.

Extremal points are sets of $(t+1)^{2}$ points on $\mathbb{S}^{2}$ which maximize the determinant of a basis matrix for an arbitrary basis of $\Pi_{t}$ [4]. For $N=(t+1)^{2}$, Chen and Womersley verified a spherical $t$-design exist in a neighborhood of an extremal system [6]. For $N\geq{(t+1)^{2}}$, An et al. [5] verified extremal spherical $t$-designs exist for all degrees $t$ up to 60 and provided well conditioned spherical $t$-designs for interpolation and numerical integration.

By now, numerical methods have been developed for finding spherical $t$-designs. The problem of finding a spherical $t$-design is expressed as solving nonlinear equations or optimization problems [8, 5]. However, the first order methods for computing spherical $t$-designs are rarely developed. In this paper, we numerically construct spherical $t$-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 $R$-linear convergent for general objective functions.

In the next section, we present the required techniques, definitions and first order conditions for spherical $t$-designs. The BB method for computing spherical $t$-designs and its convergence analysis are presented in Section 3. Numerical results for point sets which $t+1$ up to 127 and $N=(t+2)^{2}=16384$ are included in Section 4. Section 5 ends this paper with a brief conclusion.

2 First order conditions for spherical $t$-design

$\{\mathrm{Y}_{0}^{1},\mathrm{Y}_{1}^{1},\ldots,\mathrm{Y}_{t}^{2t+1}\}$ for degree $n=0,...,t$ and order $k=1,\ldots,2n+1$ is a complete set of orthonormal real spherical harmonics basis for $\Pi_{t}$, where orthogonality with respect to the $L_{2}$ inner product [12],

[TABLE]

Note that $\mathrm{Y}_{0}^{1}=\frac{1}{\sqrt{4\pi}}$. It is well known that the addition theorem for spherical harmonics on $\mathbb{S}^{2}$ gives

[TABLE]

where $\mathrm{P}_{n}:[-1,1]\to\mathbb{R}$ is Legendre polynomial and $\langle\bm{x},\bm{y}\rangle:=\bm{x}^{\top}\bm{y}$ is the inner product in $\mathbb{R}^{3}$. Sloan and Womersley [8] introduced a variational characterization of spherical $t$-designs

[TABLE]

Theorem 2.1** ([8]).**

Let $t\geq 1$, and $\mathrm{X}_{N}\subset\mathbb{S}^{2}$. Then

[TABLE]

and $\mathrm{X}_{N}$ is a spherical t-design if and only if

[TABLE]

It is known that $\mathrm{X}_{N}$ is a spherical $t$-design if and only if $A_{N,t}(\mathrm{X}_{N})$ vanished. Naturally, one might consider the first order condition to check the global minimizer of $A_{N,t}(\mathrm{X}_{N})$.

Definition 2.1**.**

A point $\bm{x}$ is a stationary point of $f\in C^{1}(\mathbb{S}^{2})$ if $\nabla^{*}f(\bm{x})=0$, where $\nabla^{*}:=\nabla^{*}_{\mathbb{S}^{2}}$ is the spherical gradient (or surface grident [12]) of $f$.

Let the basis matrix be $\mathbf{Y}_{t}(\mathrm{X}_{N}):=\begin{bmatrix}\frac{1}{\sqrt{4\pi}}\mathbf{e}^{\top}\\ \\ \mathbf{Y}_{t}^{0}(\mathrm{X}_{N})\end{bmatrix}\in\mathbb{R}^{(t+1)^{2}\times N},$ where $\mathbf{e}=\begin{bmatrix}1\\ \vdots\\ 1\end{bmatrix}\in\mathbb{R}^{N}$ and $\mathbf{Y}_{t}^{0}(\mathrm{X}_{N})=\begin{bmatrix}\mathrm{Y}_{1}^{1}(\bm{x}_{1})&\cdots&\mathrm{Y}_{1}^{1}(\bm{x}_{N})\\ \vdots&\ddots&\vdots\\ \mathrm{Y}_{t}^{2t+1}(\bm{x}_{1})&\cdots&\mathrm{Y}_{t}^{2t+1}(\bm{x}_{N})\end{bmatrix}\in\mathbb{R}^{(t+1)^{2}-1\times N}.$

Definition 2.2**.**

A finite set $\mathrm{X}_{N}:=\{\bm{x}_{1},\ldots,\bm{x}_{N}\}\subset\mathbb{S}^{2}$ is called a fundamental system for $\Pi_{t}$ if the zero polynomial is the only element of $\Pi_{t}$ that vanishes at each point in $\mathrm{X}_{N}$.

An et al. [5] described the fundamental system in finding spherical $t$-designs.

Lemma 2.2** ([5]).**

A set $\mathrm{X}_{N}\subset\mathbb{S}^{2}$ is a fundamental system for $\Pi_{t}$ if and only if $\mathbf{Y}_{t}(\mathrm{X}_{N})$ is of full row rank $(t+1)^{2}$.

Lemma 2.3** ([5]).**

Let $t\geq 2$ and $N\geq(t+2)^{2}$ . Assume $\mathrm{X}_{N}\subset\mathbb{S}^{2}$ is a stationary point set of $A_{N,t}$ and $\mathrm{X}_{N}$ is a fundamental system for $\Pi_{t+1}$. Then $\mathrm{X}_{N}$ is a spherical $t$-design.

Based on these results, we have the applicable first order condition for spherical $t$-designs as follows.

Theorem 2.4**.**

Let $t\geq 2$ and $N\geq(t+2)^{2}$. Assume $\mathrm{X}_{N}\subset\mathbb{S}^{2}$ is a stationary point set of $A_{N,t}$ and the minimal singular value of basis matrix $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ is positive. Then $\mathrm{X}_{N}$ is a spherical $t$-design.

Proof.

Suppose that the minimal singular value of $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ is positive, then we have all the singular values of $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ are positive immediately. We know that the number of non-zero singular values of $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ equals the rank of $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$, so $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ is of full rank, which means $\mathrm{X}_{N}$ is a fundamental system of $\Pi_{t+1}$ by Lemma 2.2. And then suppose that $\mathrm{X}_{N}$ is a stationary point set, then $\mathrm{X}_{N}$ is a spherical $t$-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 $t$-designs

3.1 Algorithm design

Fix $N$ and $t$, for objective function $A_{N,t}:\mathbb{S}^{2\times N}\to\mathbb{R}$, we consider the optimization problem

[TABLE]

Apparently, $A_{N,t}$ is a non-convex function. For computing $\mathrm{X}_{N}$ conveniently, we assume the first point $\bm{x}_{1}=(0,0,1)^{\top}$ is the north pole point and the second point $\bm{x}_{2}=(x_{2},0,z_{2})^{\top}$ is on the primer meridian. Then we can define coordinates convert functions $\eta:\mathbb{R}^{3\times N}\to\mathbb{R}^{1\times 2N-3}$ which can convert Cartesian coordinates into spherical coordinates as a vector, and $\mu:\mathbb{R}^{1\times 2N-3}\to\mathbb{R}^{3\times N}$ which can convert a vector form spherical coordinates into Cartesian coordinates as a matrix. So for $(\theta,\phi)\in[0,\pi]\times[0,2\pi)$ we have

[TABLE]

where vector $\Theta:=(\theta_{2},\ldots,\theta_{N})\in\mathbb{R}^{1\times N-1}$ and vector $\Phi:=(\phi_{3},\ldots,\phi_{N})\in\mathbb{R}^{1\times N-2}$.

We apply BB method in [10] to construct Algorithm 1 for seeking an efficient way to compute $A_{N,t}$, that $x$ 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 $t$-design.

To make sure that objective function $f(x_{k})$ is sufficient to descend and approximate to $\varepsilon$ which is as near as [math], we use Armijo-Goldstein rule [13] and backtracking line search [13] to lead BB method in a proper way to find local minimum $x^{*}$.

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

Example 3.1**.**

We generate spiral points $\mathrm{X}_{4}=\{\bm{x}_{1},\bm{x}_{2},\bm{x}_{3},\bm{x}_{4}\}\subset\mathbb{S}^{2}$ from [14],

[TABLE]

By using Algorithm 1, we obtain the termination output: $k=25$, $\lvert A_{N,t}(\mathrm{X}_{4}^{*})\rvert=2.775558\times 10^{-17}$, $\lVert\nabla^{*}A_{N,t}(\mathrm{X}_{4}^{*})\rVert=1.0446\times 10^{-8}$ and $\mathrm{X}_{4}^{*}$ ends with value

[TABLE]

In fact, $\mathrm{X}_{4}^{*}$ is a set of regular tetrahedron vertices, which is known as a spherical $2$-design. As a result, Algorithm 1 reaches the global minimum $\mathrm{X}_{4}^{*}$, thus numerical solutions for spherical $2$-design found. We can see the explicit change of $X_{4}$ 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 $A_{N,t}\in C^{t}(\mathbb{S}^{2})$ for $t\geq 2$. We assume that

Assumption 3.1**.**

The level set $D:=\{x\in\mathbb{R}^{n}|f(x)\leq f(x_{l})\}$ is bounded, and there exists $M>0$ such that $\lVert\nabla^{2}f(x)\rVert\leq M$, where $\nabla^{2}f$ is a Hessian matrix of $f(x)$.

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 $x_{1}=\eta(\mathrm{X}_{N})$ be an initial point and $g_{1}=\eta(\nabla^{*}A_{N,t}(\mu(x_{1})))$ and assume Assumption 3.1 holds. Suppose that ${x_{k}}$ is generated by Algorithm 1, then $\lim\limits_{k\to\infty}\inf{\lVert g_{k}\rVert}=0$.

Proof.

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

[TABLE]

where $\nabla f$ is a gradient of $f$ and $\kappa\in(0,1)$, then

[TABLE]

According to Cauchy inequality, we obtain

[TABLE]

moreover, by using mean value theorem

[TABLE]

Combine (9) and (10), we know

[TABLE]

therefore

[TABLE]

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

[TABLE]

thus

[TABLE]

Since $D$ is bounded, we know $\lim\limits_{k\to\infty}f(x_{k+1})$ exists, therefore

[TABLE]

hence

[TABLE]

Now we assume that $\lim\limits_{k\to\infty}\sup\lVert g_{k}\rVert\neq 0$. We can find a set of $\{k_{n}\}$ ($n\in\mathbb{Z}^{+}$), when $n\to\infty$, $k_{n}\to\infty$, and there exist $\epsilon>0$ such that $\lVert g_{k_{n}}\rVert>\epsilon$. Therefore, (15) can not be hold, which contradicts. Thus, $\lim\limits_{k\to\infty}\inf\lVert g_{k}\rVert=\lim\limits_{k\to\infty}\sup\lVert g_{k}\rVert=0$, 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 $x_{k}$ be the starting points set of Algorithm 1, by Theorem 3.1, then there exist $\epsilon>0$ such that $\lim\limits_{k\to\infty}\nabla^{*}A_{N,t}(\mu(x_{k}))=\bm{0}$ when $k>\epsilon$, where $\bm{0}\in\mathbb{R}^{1\times N}$ is a zero vector. Therefore, $\mathrm{X}_{N}$ is a stationary points set of $A_{N,t}$.

Remark 2**.**

Let $x_{k}$ be the starting points set of Algorithm 1, then we have $\lim\limits_{k\to\infty}A_{N,t}(\mu(x_{k}))=0$. If Theorem 2.4 is established in $x^{*}$, then $x^{*}$ is a spherical $t$-design.

4 Numerical results

Based on the code in [8, 17], we present the feasibility of Algorithm 1 to compute spherical $t$-design with the point set $\mathrm{X}_{N}$ where $N=(t+2)^{2}$ for $t+1$ up to 127. As an initial point set $\mathrm{X}_{N}$ to solve the optimization problem of minimizing $A_{N,t}(\mathrm{X}_{N})$ from (4), we use the extremal systems from [4] without any additional constraints. To make sure BB method is meaningful in spherical $t$-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 $t$-designs can be founded in [18]. We observe that BB method cost less time than quasi-Newton method, especially in large $\mathrm{X}_{N}$. Furthermore, all point sets $\mathrm{X}_{N}$ are verified to be fundamental systems. In fact, we use singular value decomposition (SVD) [13] to obtain all singular values of $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$, which are defined as $\{\sigma_{i}\}$ for $i=1,\ldots,(t+2)^{2}$. As a result, the $\min(\sigma_{i})>0$, then $\mathbf{Y}_{t+1}(\mathrm{X}_{N})$ is of full rank, thus $\mathrm{X}_{N}$ is a fundamental system. This is a strong numerical support to Theorem 2.4. Here we set $\varepsilon_{1}=\varepsilon_{2}=10^{-16}$.

Figure 3(a) and Figure 4(a) are well exhibited the locally R-linear convergence [11] of BB method by numerical computation of $A_{N,t}$ with $t=50,N=2601$. We can see that $A_{N,t}$ converges to [math] with iteration increase.

5 Conclusion

In this paper, we employ Barzilai-Borwein method for finding numerical spherical $t$-designs with $t$ up to 126 with $N=(t+1)^{2}$. 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 $t$-designs. These numerical results are interesting and inspiring. The numerical construction of higher order spherical $t$-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).