Two novel iterative approaches for improved LSPIA convergence

This paper introduces two improved variants of the least squares progressive-iterative approximation (LSPIA) by leveraging momentum techniques. Specifically, based on the Polyak's and Nesterov's momentum techniques, the proposed methods utilize the previous iteration information to update the control points. We name these two methods PmLSPIA and NmLSPIA, respectively. The introduction of momentum enhances the determination of the search directions, leading to a significant improvement in convergence rate. The geometric interpretations of PmLSPIA and NmLSPIA are elucidated, providing insights into the underlying principles of these accelerated algorithms. Rigorous convergence analyses are conducted, revealing that both PmLSPIA and NmLSPIA exhibit faster convergence than LSPIA. Numerical results further validate the efficacy of the proposed algorithms.