基于核贪婪算法的微分方程数值求解——贪婪算法的微分方程数值求解

Numerical Solution of Differential Equations Based on Kernel Greedy Algorithm—Greedy Algorithm for Numerical Solution of Differential Equations

摘要: 基于再生核Hilbert空间的相关知识，再生核Hilbert空间中的函数可以表示用n个再生点构成的核函数的线性表示进行逼近。为了实现n个再生点的最优选取，文章借助Santin等人提出f-greedy、P-greedy、f/P-greedy等贪婪类算法，实现上述方法在微分方程数值求解的理论推导及算法实验。与传统的均匀取点相比，P-greedy和预正交自适应Fourier分解(POAFD)的贪婪算法逼近误差更优，但f-greedy和f/P-greedy效果还存在改进的空间。

Abstract: Based on the knowledge of real regenerated kernel Hilbert space, the function in the regenerated kernel Hilbert space can be approximated by a linear representation of the kernel function composed of n regenerated points. In order to achieve the optimal selection of n regeneration points, Santin et al. put forward a greedy algorithm such as F-greedy, P-greedy, and f/P-greedy to achieve the theoretical derivation and algorithm experiment of the above method in numerical solution of differential equations. Compared to the traditional method of uniformly taking points, the P-greedy and Pre-Orthogonal Adaptive Fourier Decomposition (POAFD) greedy algorithms provide better approximation errors, while the f-greedy and f/P-greedy algorithms do not perform as well and have room for improvement.