Adaptive meshfree approximation for linear elliptic partial differential equations with PDE-greedy kernel methods

TL;DR

This paper introduces adaptive, meshfree kernel collocation methods for elliptic PDEs using PDE-greedy point selection, providing theoretical error bounds and demonstrating improved convergence, especially in high dimensions.

Contribution

It develops a generalized PDE-greedy selection framework with convergence analysis and shows target-data dependent algorithms outperform target-data independent ones.

Findings

Faster convergence rates for target-data dependent algorithms.

Dimension-independent convergence rate for PDE-f greedy.

Numerical examples demonstrate the effectiveness of the proposed methods.

Abstract

We consider 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.