Computing on spheres: From spherical designs to scattered, random, and data-driven points

Speaker：Haoning Wu (The University of Georgia)

Time：2026-6-10 10:30

Location：Conference Room C503 at Administration Building at Haiyun Campus

Abstract:

Spherical t-designs provide a foundation for numerical integration and approximation on the sphere, offering spectral accuracy by ensuring exact integration of polynomials up to degree t. However, their computational construction becomes prohibitively difficult for large t, creating a significant practical barrier to computation on spheres. To overcome this limitation, we introduce a framework that relaxes the stringent requirement of quadrature exactness imposed by traditional numerical analysis. Leveraging the Marcinkiewicz--Zygmund inequality, we derive error bounds for hyperinterpolation, a discrete version of the L^2 orthogonal projection, that enable its use with point sets that are not strict spherical designs, thereby relaxing its original requirement for high quadrature exactness. This relaxed approximation scheme maintains provably good convergence rates. We further apply this framework in two settings of numerical analysis. First, we develop a hyperinterpolation-based spectral method for the Allen--Cahn equation posed on the sphere. Second, we introduce a quadrature-based discretization for Fredholm integral equations of the second kind on the sphere. In both cases, the same relaxed sampling assumptions guarantee the stability and convergence of the resulting numerical schemes. These results enable rigorous numerical analysis for approximation, PDEs, and integral equations on the sphere using flexible point sets, including scattered or randomly generated nodes. Numerical experiments validate the theoretical analysis and demonstrate that the proposed methods remain accurate and stable.