[Paper Review] High-dimensional periodic sampling on Smolyak grids based on B-spline quasi-interpolation

Ðinh Dũng|arXiv (Cornell University)|Feb 5, 2015

Mathematical Approximation and Integration26 references3 citations

TL;DR

This paper proposes linear sampling recovery and cubature formulas on Smolyak grids for high-dimensional periodic functions with Lipschitz-Hölder mixed smoothness using B-spline quasi-interpolation. It establishes optimal error bounds that are explicit in dimension $d$, refinement level $m$, and number of active variables $ u$, showing that sparse Smolyak grids achieve the same convergence rates as full grids with significantly fewer nodes.

ABSTRACT

We constructed linear algorithms of sampling recovery and cubature formulas on Smolyak grids parametrized by $m \in \mathbb{N}$ of periodic $d$-variate functions having Lipschitz-Hölder mixed smoothness $α> 0$ based on B-spline quasi-interpolation, and studied their optimality. We established lower estimates (for $α\le 2$) and upper bounds of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in $d$, $m$ and the number $ν$ of active variables of functions when $d$ and $m$ may be large.

Motivation & Objective

• To develop linear sampling recovery and cubature algorithms on Smolyak grids for periodic $d$-variate functions with mixed smoothness $\alpha>0$.
• To analyze the optimality of these algorithms in terms of error bounds that are explicit in $d$, $m$, and $\nu$.
• To establish lower bounds for $\alpha \leq 2$ and upper bounds for the error of optimal sampling recovery and integration on Smolyak grids.
• To demonstrate that sparse Smolyak grids achieve the same convergence rates as full grids while using far fewer sampling points.

Proposed method

• Constructs linear sampling recovery and cubature formulas on Smolyak grids $G^d(m)$ using B-spline quasi-interpolation operators.
• Employs univariate symmetric Laurent polynomials $P_\Lambda$ to define quasi-interpolation operators $Q$ with explicit control over approximation error.
• Uses mixed difference operators $\Delta_h^{r,u}$ to define the Lipschitz-Hölder semi-norms in the space $H^\alpha_\infty(\mathbb{T}^d)$.
• Derives error bounds via analysis of $\|P_{\text{even}}^*\|$, $\|P_{\text{odd}}^*\|$, and $\|L_\Lambda\|_{C(\mathbb{T})}$, which control the stability and approximation quality.
• Applies the Faber series representation through piecewise linear B-splines ($r=1$) and extends to cubic ($r=2$) and quintic ($r=3$) B-spline quasi-interpolation.
• Uses hierarchical decomposition and sparse grid construction with $|k|_1 = m$ to ensure sparsity while maintaining optimal convergence rates.

Experimental results

Research questions

• RQ1 Can B-spline quasi-interpolation on Smolyak grids achieve optimal convergence rates for high-dimensional periodic functions with mixed smoothness $\alpha>0$?
• RQ2 What are the explicit error bounds for sampling recovery and integration on Smolyak grids in terms of $d$, $m$, and $\nu$?
• RQ3 How do the approximation properties of B-spline quasi-interpolation compare to standard full grids in high dimensions?
• RQ4 What is the role of the Laurent polynomial $P_\Lambda$ in controlling the stability and accuracy of the quasi-interpolation operator?
• RQ5 Are the error bounds for $\alpha \leq 2$ sharp, and how do they compare to lower bounds?

Key findings

• The error of optimal sampling recovery on Smolyak grids is bounded by $C \cdot 2^{-m\alpha}$, with $C$ depending explicitly on $d$, $\nu$, and $\alpha$, and this bound is optimal for $\alpha \leq 2$.
• For piecewise linear B-spline quasi-interpolation ($r=1$), $\|P_{\text{even}}^*\| = 1/2$, $\|P_{\text{odd}}^*\| = 0$, and $\|L_\Lambda\| = 1$, ensuring stability and optimal convergence.
• For cubic B-spline quasi-interpolation ($r=2$), $\|P_{\text{even}}^*\| = 3/8$, $\|P_{\text{odd}}^*\| = 1/2$, and $\|L_\Lambda\| = 11/9$, yielding high approximation accuracy.
• For quintic B-spline quasi-interpolation ($r=3$), $\|P_{\text{even}}^*\| \approx 0.97$, $\|P_{\text{odd}}^*\| \approx 1.00$, and $\|P_\Lambda\| \approx 2.63$, showing near-optimal stability and high-order accuracy.
• The number of grid points in $G^d(m)$ is smaller than $\frac{1}{(d-1)!}2^m m^{d-1}$, significantly less than $2^{dm}$, confirming the sparsity of Smolyak grids.
• The proposed algorithms achieve the same convergence rates as those on full grids, proving that sparse grids are optimal for high-dimensional periodic functions with mixed smoothness.