We consider the meshless approximation for solutions of boundary value problems (BVPs) of elliptic Partial Differential Equations (PDEs) via symmetric kernel collocation. We discuss the importance of the choice of the collocation points, in particular by using greedy kernel methods. We introduce a scale of PDE-greedy selection criteria that generalizes existing techniques, such as the PDE-P -greedy and the PDE-f -greedy rules for collocation point selection. For these greedy selection criteria we provide bounds on the approximation error in terms of the number of greedily selected points and analyze the corresponding convergence rates. This is achieved by a novel analysis of Kolmogorov widths of special sets of BVP point-evaluation functionals. Especially, we prove that target-data dependent algorithms that make use of the right hand side functions of the BVP exhibit faster convergence rates than the target-data independent PDE-P -greedy. The convergence rate of the PDE-f -greedy possesses a dimension independent rate, which makes it amenable to mitigate the curse of dimensionality. The advantages of these greedy algorithms are highlighted by numerical examples.

翻译：我们考虑通过对称核配置方法对椭圆偏微分方程边值问题的解进行无网格逼近。我们讨论了配置点选择的重要性，特别是通过使用贪婪核方法。我们引入了一类PDE-greedy选择准则的尺度，该尺度推广了现有技术，例如用于配置点选择的PDE-P-greedy和PDE-f-greedy规则。针对这些贪婪选择准则，我们给出了以贪婪选择点数为变量的逼近误差界，并分析了相应的收敛速率。这是通过对边值问题点值泛函特殊集合的Kolmogorov宽度进行新颖分析实现的。特别地，我们证明了利用边值问题右端函数的目标数据依赖算法比目标数据独立的PDE-P-greedy具有更快的收敛速率。PDE-f-greedy的收敛速率具有维度无关性，这使其能够缓解维度灾难问题。数值算例凸显了这些贪婪算法的优势。