勷勤数学•专家报告

题      目：Spherical configurations and quadrature methods for integral equations of the second kind

报  告  人：安聪沛 教授  (邀请人：叶颀)

                                      贵州大学

时      间： 12月3日  10：30-11:30

地     点：数科院西楼111报告厅

报告人简介:

        安聪沛，本科、硕士毕业于中南大学，博士毕业于香港理工大学，先后工作于暨南大学，西南财经大学，现为贵州大学一流学科特聘教授。研究成果在球 t-设计，振荡积分的近似计算，多项式构造逼近，反问题计算上取得不少同行关注的结果。

摘      要：

        Ｗe propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere . We employ quadrature rules that satisfy the Marcinkiewicz--Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution, in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical  error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution, and the other characterized by the Marcinkiewicz--Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.

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