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

报告人：吴浩宁（美国佐治亚大学）

时  间：2026年6月10日10:30

地  点：海韵园行政楼C503

内容摘要:

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.

个人简介：

吴浩宁，现为佐治亚大学数学系博士后，主要研究方向包括应用调和分析与逼近论、数值分析、控制与优化。2019年本科毕业于暨南大学数学系，2023年博士毕业于香港大学数学系，师从袁晓明教授。此后先后在香港大学与佐治亚大学从事博士后研究。近年来在Applied and Computational Harmonic Analysis, Inverse Problems, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing等国际期刊发表论文多篇。

联系人：陈黄鑫