AI 中文总结
针对离散球面数据，提出一种优化方法构建求积规则，通过研究索伯列夫空间再生核积分精度等构建凸优化模型，可推导稳定性界且高效计算权重，经数值结果验证其在基本逼近任务中的性能。
AI 中文摘要
我们引入一种优化方法来构建任意离散数据上的球面求积规则。该新方法不是设计节点布局，而是专注于为固定配置最优地计算权重。受1933年波利亚求积收敛充要条件启发，我们认为追求离散数据逼近的权重正性和高代数精度并非必要。通过研究索伯列夫空间再生核数值积分的精度以及利用马尔钦凯维奇 - 齐格蒙德不等式的超插值性能，构建具有合适目标泛函的凸优化模型。所得优化模型编码了离散点的空间分布和目标函数空间的解析性质。该方法能推导严格理论稳定性界，且现代凸优化技术可高效计算求积权重。还报告了数值结果以展示该优化方法在离散球面数据数值积分和超插值等基本逼近任务中的性能。
英文摘要
We introduce an optimization approach for constructing spherical quadrature rules on arbitrarily scattered data. Rather than designing node placements, the new approach focuses on optimally computing the weights for fixed configurations. Motivated by Pólya's necessary and sufficient conditions for quadrature convergence in 1933, we argue that pursuing weight positivity and high algebraic exactness for scattered data approximation is not necessary. To align the quadrature design with the underlying theory of approximation, we construct convex optimization models with suitable objective functionals by examining the accuracy of numerical integration with reproducing kernels of Sobolev spaces and the performance of hyperinterpolation with Marcinkiewicz-Zygmund (MZ) inequalities. The resulting optimization models encode the spatial distribution of the scattered sites and the analytic properties of the target function spaces. The proposed approach enables the derivation of rigorous theoretical stability bounds, and the resulting quadrature weights are efficiently computable by modern convex optimization techniques. Numerical results are reported to demonstrate the performance of the optimization approach for fundamental approximation tasks such as numerical integration and hyperinterpolation for scattered spherical data.
Comments53 pages, 14 figures
