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Sparse-grid sampling recovery and numerical integration of functions having mixed smoothness

Dinh Dũng

TL;DR

The paper surveys sparse-grid linear methods for recovering and integrating high-dimensional functions with mixed smoothness, focusing on unweighted Sobolev spaces on ${\mathbb T}^d$ and weighted spaces $W^r_p({\mathbb R}^d;\mu)$ with Freud-type and Gaussian measures. It develops B-spline quasi-interpolation representations and analyzes Smolyak sparse-grid algorithms, establishing asymptotic rates for linear sampling widths and identifying cases where Smolyak constructions are optimal. The text extends results to reproducing-kernel Hilbert spaces and discusses multiple notions of optimality in sampling recovery, revealing how width orders behave in different regimes of $p$, $q$, and dimension $d$. It then advances numerical weighted integration in these weighted Sobolev spaces, presenting assembling-based quadratures that achieve the right asymptotic orders for Freud-type weights and Gaussian measures, and addresses the particularly delicate $p=1$ case via sparse-grid and hyperbolic-cross strategies, including extensions to Markov–Sonin weights. Overall, the work provides a cohesive framework linking sparse-grid recovery and quadrature to the mixed-smoothness landscape, with concrete constructions and asymptotic guarantees across unweighted and weighted settings.

Abstract

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

Sparse-grid sampling recovery and numerical integration of functions having mixed smoothness

TL;DR

The paper surveys sparse-grid linear methods for recovering and integrating high-dimensional functions with mixed smoothness, focusing on unweighted Sobolev spaces on and weighted spaces with Freud-type and Gaussian measures. It develops B-spline quasi-interpolation representations and analyzes Smolyak sparse-grid algorithms, establishing asymptotic rates for linear sampling widths and identifying cases where Smolyak constructions are optimal. The text extends results to reproducing-kernel Hilbert spaces and discusses multiple notions of optimality in sampling recovery, revealing how width orders behave in different regimes of , , and dimension . It then advances numerical weighted integration in these weighted Sobolev spaces, presenting assembling-based quadratures that achieve the right asymptotic orders for Freud-type weights and Gaussian measures, and addresses the particularly delicate case via sparse-grid and hyperbolic-cross strategies, including extensions to Markov–Sonin weights. Overall, the work provides a cohesive framework linking sparse-grid recovery and quadrature to the mixed-smoothness landscape, with concrete constructions and asymptotic guarantees across unweighted and weighted settings.

Abstract

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.
Paper Structure (19 sections, 41 theorems, 298 equations, 3 figures)

This paper contains 19 sections, 41 theorems, 298 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Unweighted sampling recovery
  3. Introducing remarks
  4. B-spline quasi-interpolation representations
  5. Sampling recovery by Smolyak sparse-grid algorithms
  6. Sampling recovery in reproducing kernel Hilbert spaces
  7. Different concepts of optimality in sampling recovery
  8. Weighted sampling recovery
  9. Introducing remarks
  10. Linear sampling recovery and approximation
  11. Numerical weighted integration in the space $W^r_p({\mathbb R}^d;\mu)$
  12. Introducing remarks
  13. Assembling quadratures
  14. Optimal quadrature
  15. Numerical weighted integration in the space $W^r_1(\mathbb{R}^d;\mu)$
  16. ...and 4 more sections

Key Result

Lemma 2.1

Every continuous function $f$ on ${\mathbb T}^d$ is represented as B-spline series converging in the norm of $C({\mathbb T}^d)$, where the coefficient functionals $c_{{\boldsymbol{k}},{\boldsymbol{s}}}(f)$ are explicitly constructed as linear combinations of at most $m_0$ of function values of $f$ for some $m_0 \in {\mathbb N}$ which is independent of ${\boldsymbol{k}},{\boldsymbo

Figures (3)

  • Figure 1: Pictures of integration nodes from DK2023
  • Figure 2: Pictures of step hyperbolic crosses in ${\mathbb R}^2$ from DD2023
  • Figure 3: A picture of Smolyak grid in $[-1,1]^2$ from DD2023

Theorems & Definitions (58)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 48 more