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

TL;DR

The paper develops adaptive meshfree methods for linear elliptic PDEs using symmetric kernel collocation and a continuum of PDE-$\beta$-greedy point-selection rules. By analyzing Kolmogorov $n$-widths for the PDE collocation functionals, it derives convergence bounds that interpolate between target-data independent and dependent strategies, with the target-data dependent PDE-$f$-greedy achieving a dimension-independent rate for $\beta\in(0,1]$. The authors prove that the interior and boundary functionals yield decay rates governed by Sobolev regularity $\tau$, yielding $d_n(\Lambda) \lesssim n^{\tfrac{1}{2}-\tfrac{\tau-2}{d}}$, and they demonstrate a fast convergence for PDE-$\beta$-greedy schemes via a generalized power-function analysis. Numerical experiments in 2D and 12D confirm boundary-adaptive collocation and the potential to outperform finite element methods in low dimensions while maintaining robustness in high dimensions, albeit with limitations near singularities. Overall, this work provides a rigorous, scalable, meshfree framework for high-dimensional PDEs with adaptive sampling driven by the PDE RHS and boundary data.

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.

Figures (6)

• Figure 1: Visualization of the scale of the PDE-$\beta$-greedy algorithms on the real line. The important cases for $\beta \in \{0, 1\}$ and $\beta \rightarrow \infty$ are marked.
• Figure 2: Numerical results regarding \ref{['subsec:smooth_case']}: Visualization of the decay of the errors $\sup_{\lambda \in \Lambda} |\lambda(e_i)|$ (left) and $\sup_{x \in \Omega} |e_i(x)|$ (right) for PDE-$P$-greedy (top), PDE-$f \cdot P$-greedy (middle), PDE-$f$-greedy (bottom) for different values of the weighting parameter $w$. For each of the three PDE-greedy methods, the number of selected collocation points $n$ which are selected on the boundary is denoted by $n_{\partial \Omega}$.
• Figure 3: Numerical results regarding \ref{['subsec:smooth_case']}: Visualization of the distribution of 341 collocation points selected by PDE-$f$-greedy (left) and PDE-$P$-greedy (right) for the smooth solution. Collocation points on the boundary are visualized as crosses. On can observe that the PDE-$f$-greedy selected centers cluster adaptively next to the boundary, while the PDE-$P$-greedy centers are rather uniformly distributed.
• Figure 4: Numerical results regarding \ref{['subsec:singular_case']}: Visualization of the decay of the errors $\sup_{\lambda \in \Lambda} |\lambda(u - s_n)|$ (left) and $\sup_{x \in \Omega} |(u-s_n)(x)|$ (right) over the expansion size $n$ ($x$-axis) for PDE-$P$-greedy (top) using $w = 10^6$ and PDE-$f$-greedy (bottom) using $w=10^3$.
• Figure 5: Numerical results regarding \ref{['subsec:singular_case']}: Visualization of the distribution of 1305 collocation points selected by PDE-$f$-greedy (left) and PDE-$P$-greedy (right) for the singular solution. Collocation points on the boundary are visualized as crosses. On can observe that the PDE-$f$-greedy selected centers cluster adaptively next to the singularity, while the PDE-$P$-greedy centers are rather uniformly distributed.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Remark 3: Relation to Kolmogorov $n$-widths for parametric PDEs
• Proposition 4
• proof
• Proposition 5
• Theorem 6: Upper bound on $d_n(\Lambda_L)$
• Theorem 7: Upper bound on $d_n(\Lambda_B)$
• Theorem 8: Upper bound on $d_n(\Lambda)$
• Remark 9