Greedy Adaptive Local Recovery of Functions in Sobolev Spaces

TL;DR

The paper introduces a kernel-based greedy local recovery method that selects a minimal, adaptively chosen subset of neighboring points to achieve optimal $L_\infty$ convergence in Sobolev spaces. It leverages a Newton basis and a Power Function–driven greedy rule to iteratively pick points, yielding $O(1)$ per-point computational cost while maintaining strong convergence rates. Numerical experiments in two dimensions illustrate near-optimal rates and reveal that recoveries are locally stable and discontinuous in the evaluation variable, with instability arising for large Sobolev orders due to finite-precision effects. The work highlights local instability phenomena in global kernel interpolation, discusses stability limits, and outlines open problems, including scale-invariant approaches and extensions to derivative recovery.

Abstract

There are many ways to upsample functions from multivariate scattered data locally, using only a few neighbouring data points of the evaluation point. The position and number of the actually used data points is not trivial, and many cases like Moving Least Squares require point selections that guarantee local recovery of polynomials up to a specified order. This paper suggests a kernel-based greedy local algorithm for point selection that has no such constraints. It realizes the optimal $L_\infty$ convergence rates in Sobolev spaces using the minimal number of points necessary for that purpose. On the downside, it does not care for smoothness, relying on fast $L_\infty$ convergence to a smooth function. The algorithm ignores near-duplicate points automatically and works for quite irregularly distributed point sets by proper selection of points. Its computational complexity is constant for each evaluation point, being dependent only on the Sobolev space parameters. Various numerical examples are provided. As a byproduct, it turns out that the well-known instability of global kernel-based interpolation in the standard basis of kernel translates arises already locally, independent of global kernel matrices and small separation distances.

Figures (16)

• Figure 1: Local recovery in $W_2^3(\mathbb{R}^2)$ at the origin using the greedy point selection strategy.
• Figure 2: Local recovery in $W_2^{3/2}(\mathbb{R}^2)$ at the origin using the greedy point selection strategy. Three points are enough.
• Figure 3: Local recovery in $W_2^{6}(\mathbb{R}^2)$ using the greedy point selection strategy at scale $c=0.1$ on 100 regular points. One should use 21 points.
• Figure 4: Local recovery in $W_2^{6}(\mathbb{R}^2)$ at the origin using the greedy point selection strategy at scale $c=1$ up to $P^2<1.e-8$ on irregular points. One should use 10 points only.
• Figure 5: Local recovery in $W_2^3(\mathbb{R}^2)$ at the top left corner using the greedy point selection strategy.