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Doubling the rate -- improved error bounds for orthogonal projection with application to interpolation

Ian H. Sloan, Vesa Kaarnioja

TL;DR

This work establishes a general framework in which an $L_2$-convergence rate for $H$-orthogonal projection of all functions in a Hilbert space $H$ can be doubled for functions in a smoother normed space $B$, provided a compatibility bound $|\langle f,g\rangle_H| \le \|f\|_{L_2}\|g\|_B$ and suitable embeddings hold. Under the assumption $\|f-Pf\|_{L_2} \le M\|f\|_H$ for all $f\in H$, the authors prove that for every $g\in B$ the errors satisfy $\|g-Pg\|_{L_2} \le M^2 \|g\|_B$ and $\|g-Pg\|_H \le M \|g\|_B$, effectively doubling the $L_2$-rate while preserving the $H$-norm rate. The theory is then instantiated to kernel interpolation in RKHS and to radial basis function interpolation: in RKHS, for kernel interpolants, the doubling result yields sharper convergence in $L_2$ and improved rates in the smoother norm; in RBF settings, with native spaces $\mathcal{N}_\Phi$ and $\mathcal{N}_{\Phi*\Phi}$, the same doubling bounds apply. Concrete consequences include sharper $L_2$-rates in Korobov-type spaces and improved high-dimensional PDE approximations, illustrating the practical impact of the doubling phenomenon in kernel- and RBF-based interpolation and approximation.

Abstract

Convergence rates for $L_2$ approximation in a Hilbert space $H$ are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the best rate for functions in the "native space" $H$. Motivated by this, we obtain a general result for $H$-orthogonal projection onto a finite dimensional subspace of $H$: namely, that any known $L_2$ convergence rate for all functions in $H$ translates into a doubled $L_2$ convergence rate for functions in a smoother normed space $B$, along with a similarly improved error bound in the $H$-norm, provided that $L_2$, $H$ and $B$ are suitably related. As a special case we improve the known $L_2$ and $H$-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the $L_2$ convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space $B$.

Doubling the rate -- improved error bounds for orthogonal projection with application to interpolation

TL;DR

This work establishes a general framework in which an -convergence rate for -orthogonal projection of all functions in a Hilbert space can be doubled for functions in a smoother normed space , provided a compatibility bound and suitable embeddings hold. Under the assumption for all , the authors prove that for every the errors satisfy and , effectively doubling the -rate while preserving the -norm rate. The theory is then instantiated to kernel interpolation in RKHS and to radial basis function interpolation: in RKHS, for kernel interpolants, the doubling result yields sharper convergence in and improved rates in the smoother norm; in RBF settings, with native spaces and , the same doubling bounds apply. Concrete consequences include sharper -rates in Korobov-type spaces and improved high-dimensional PDE approximations, illustrating the practical impact of the doubling phenomenon in kernel- and RBF-based interpolation and approximation.

Abstract

Convergence rates for approximation in a Hilbert space are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the best rate for functions in the "native space" . Motivated by this, we obtain a general result for -orthogonal projection onto a finite dimensional subspace of : namely, that any known convergence rate for all functions in translates into a doubled convergence rate for functions in a smoother normed space , along with a similarly improved error bound in the -norm, provided that , and are suitably related. As a special case we improve the known and -norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space .
Paper Structure (7 sections, 6 theorems, 93 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Examples of suitable spaces $H, B$
  3. The main theorem
  4. Kernel interpolation in reproducing kernel Hilbert spaces
  5. A concrete application to kernel interpolation
  6. Radial basis function interpolation
  7. Conclusion

Key Result

Theorem 1

For a domain $D\subseteq \mathbb{R}^d$, let $H$ be a Hilbert space and $B$ a normed space of real-valued functions on $D$ such that $H$ is continuously embedded in $L_2(D)$ and $B$ is continuously embedded in $H$, and such that eq:norms_relation holds. Further, let $P$ be the $H$-orthogonal projecti Then for all $g$ in $B$ we have and

Figures (1)

  • Figure 1: Numerical $L_2$ approximation errors for the functions $f$ and $g$ considered in Example 5.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2
  • Proposition 3
  • proof
  • Corollary 4
  • Definition 5
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 1 more