Numerical construction of spherical $t$-designs by Barzilai-Borwein method
Provides a new numerical method for constructing spherical t-designs, which are important for numerical integration and approximation on the sphere.
The paper proves that stationary points of a certain nonnegative quantity correspond to spherical t-designs under a positivity condition, and uses the Barzilai-Borwein method to numerically construct spherical t-designs for t+1 up to 127 with N=(t+2)^2.
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 $N=(t+2)^2$.