Consensus-Based Approximation and Quadrature on the Sphere via Weighted Least Squares Polynomial Models

<ns3:p>The fundamental tasks of function approximation and numerical integration on the sphere from scattered data nodes present significant challenges, including the curse of dimensionality, the instability of high-degree polynomial methods, and the limitations of existing approaches that often require specially structured node sets. This paper introduces a novel framework that seamlessly integrates Weighted Least Squares (WLS) polynomial approximation, using spherical harmonics and Voronoi-based weights, with a consensus-based optimization strategy for decentralized computation, specifically designed for large, distributed datasets. Our results show that this method provides stable and accurate approximation even with high-degree polynomials on arbitrary, unstructured node sets, ensuring a well-conditioned system. It also leads to the construction of a novel quadrature rule with provably positive weights, guaranteeing stability for numerical integration. We derive rigorous theoretical error bounds that explicitly connect the accuracy of the method to the density of the node set and the polynomial degree. Extensive numerical experiments confirm that our framework outperforms standard least squares and classical scattered-data quadrature rules in both stability and accuracy. We conclude that this consensus-based WLS framework offers a robust, scalable, and distributed solution for a wide range of spherical problems, with significant potential impact in scientific computing and data analysis.</ns3:p>