Operator-free Equilibrium on the Sphere
This work addresses the challenge of efficient point generation for spherical integration, offering a deterministic approach that is incremental in simplifying computations without pseudodifferential operators.
The authors tackled the problem of generating equidistributed pointsets on the sphere by proposing a generalized minimum discrepancy derived from Legendre's ODE and spherical harmonics, resulting in a method that requires fewer points than Monte Carlo to approximate targets in arbitrary dimensions.
We propose a generalized minimum discrepancy, which derives from Legendre's ODE and spherical harmonic theoretics to provide a new criterion of equidistributed pointsets on the sphere. A continuous and derivative kernel in terms of elementary functions is established to simplify the computation of the generalized minimum discrepancy. We consider the deterministic point generated from Pycke's statistics to integrate a Franke function for the sphere and investigate the discrepancies of points systems embedding with different kernels. Quantitive experiments are conducted and the results are analyzed. Our deduced model can explore latent point systems, that have the minimum discrepancy without the involvement of pseudodifferential operators and Beltrami operators, by the use of derivatives. Compared to the random point generated from the Monte Carlo method, only a few points generated by our method are required to approximate the target in arbitrary dimensions.