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Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels

Bernard Haasdonk, Gabriele Santin, Tizian Wenzel, Daniel Winkle

TL;DR

This work eliminates a spurious logarithmic factor in the convergence rates for target-data dependent greedy generalized interpolation with Sobolev kernels by leveraging entropy-number estimates. Focusing on PDE-$\beta$-greedy schemes, it derives a bound of the form $\min_{n+1\le j\le 2n} \|\cdot\|_{L_\infty(\mathcal{X})} \le C' n^{-(\tau-\bar{m})/\bar{d} + (1-\beta)/2} \|\cdot\|_{\mathcal{H}_k(\mathcal{X})}$ for $\beta\in[0,1]$, recovering the nonadaptive rate at $\beta=0$ and achieving a dimension- and smoothness-independent improvement of $n^{-\beta/2}$ for $\beta\in(0,1]$ (maximized at $\beta=1$). The results extend via entropy bounds for unions of operator-induced sets and Lipschitz parametrizations, removing the previous log-factor and generalizing to piecewise-smooth domains and, by extension, symmetric PDE collocation. The analysis provides a broadly applicable, theory-driven pathway to faster target-data dependent convergence in high-dimensional settings.

Abstract

Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functionals, including the approximation of solutions of linear PDEs by symmetric collocation. When applied to kernels generating Sobolev spaces as their native Hilbert spaces, some of these greedy methods can provide the same error guarantee of generalized interpolation on quasi-uniform points. More importantly, certain target-data-adaptive methods even give a dimension- and smoothness-independent improvement in the speed of convergence over quasi-uniform points, thus offering advantages for high-dimensional problems. These convergence rates however contain a spurious logarithmic term that limits this beneficial effect. The goal of this note is to remove this factor, and this is possible by using estimates on metric entropy numbers.

Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels

TL;DR

This work eliminates a spurious logarithmic factor in the convergence rates for target-data dependent greedy generalized interpolation with Sobolev kernels by leveraging entropy-number estimates. Focusing on PDE--greedy schemes, it derives a bound of the form for , recovering the nonadaptive rate at and achieving a dimension- and smoothness-independent improvement of for (maximized at ). The results extend via entropy bounds for unions of operator-induced sets and Lipschitz parametrizations, removing the previous log-factor and generalizing to piecewise-smooth domains and, by extension, symmetric PDE collocation. The analysis provides a broadly applicable, theory-driven pathway to faster target-data dependent convergence in high-dimensional settings.

Abstract

Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functionals, including the approximation of solutions of linear PDEs by symmetric collocation. When applied to kernels generating Sobolev spaces as their native Hilbert spaces, some of these greedy methods can provide the same error guarantee of generalized interpolation on quasi-uniform points. More importantly, certain target-data-adaptive methods even give a dimension- and smoothness-independent improvement in the speed of convergence over quasi-uniform points, thus offering advantages for high-dimensional problems. These convergence rates however contain a spurious logarithmic term that limits this beneficial effect. The goal of this note is to remove this factor, and this is possible by using estimates on metric entropy numbers.
Paper Structure (7 sections, 6 theorems, 38 equations)

This paper contains 7 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Generalized interpolation and the PDE-greedy algorithms
  3. Greedy generalized interpolation
  4. Entropy numbers and generalized interpolation
  5. Convex hulls induced by linear and bounded differential operators
  6. Convergence of PDE-greedy algorithms
  7. Discussion and conclusion

Key Result

Lemma 2

Let $n\in\mathbb{N}$ and $\beta\geq 0$. If $\eta_{n,\beta}(\lambda)\neq0$ for some $\lambda\in \Lambda$, then $\lambda_{n+1}$ is linearly independent from $\lambda_1, \dots, \lambda_n$. If instead $\eta_{n,\beta}(\lambda)=0$ for all $\lambda\in \Lambda$, then $\Lambda\subset\mathop{\mathrm{span}}\no

Theorems & Definitions (14)

  • Remark 1
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 4 more