Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^\gamma$

TL;DR

This paper studies the convergence rates of sampling operators based on energy-norm sparse grids for Sobolev functions, providing sharp polynomial decay rates and optimal algorithms for various embeddings in high-dimensional spaces.

Contribution

It introduces energy-norm based sparse grids for sampling, achieving sharp decay rates and optimality results for Sobolev space embeddings, extending previous work on linear approximation.

Findings

Sharp polynomial decay rates for sampling numbers when embedding $H^{eta}$ into $H^{eta}$ with $eta< ext{target}$

Optimality of Smolyak's algorithm for certain Sobolev embeddings

Sampling numbers match approximation numbers except in a limiting case with a logarithmic gap

Abstract

We investigate the rate of convergence of linear sampling numbers of the embedding $H^{\alpha,\beta} (\mathbb{T}^d) \hookrightarrow H^\gamma (\mathbb{T}^d)$. Here $\alpha$ governs the mixed smoothness and $\beta$ the isotropic smoothness in the space $H^{\alpha,\beta}(\mathbb{T}^d)$ of hybrid smoothness, whereas $H^{\gamma}(\mathbb{T}^d)$ denotes the isotropic Sobolev space. If $\gamma>\beta$ we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on "energy-norm based sparse grids" for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and D\~ung, Ullrich, where general linear approximations have been considered. In addition, we study the embedding $H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrow H^{\gamma}_{mix}(\mathbb{T}^d)$ and achieve optimality for Smolyak's algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding $H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrow L_q(\mathbb{T}^d)$ for $2<q\leq \infty$ where again Smolyak's algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation $\beta = \gamma$ (including the embedding into $L_2(\mathbb{T}^d)$). The best what we could prove there is a (probably) non-sharp results with a logarithmic gap between lower and upper bound.