[Paper Review] Sampling and cubature on sparse grids based on a B-spline quasi-interpolation

Ðinh Dũng|arXiv (Cornell University)|Nov 19, 2012

Mathematical Approximation and Integration41 references4 citations

TL;DR

This paper constructs asymptotically optimal linear sampling algorithms and cubature formulas on sparse grids using B-spline quasi-interpolation for functions with anisotropic smoothness in Besov-type spaces. It establishes the exact asymptotic order of the optimal recovery and integration errors, achieving optimal convergence rates by leveraging structured sparse grids and B-spline-based reconstruction.

ABSTRACT

Let $X_n = \{x^j\}_{j=1}^n$ be a set of $n$ points in the $d$-cube $[0,1]^d$, and $Φ_n = \{φ_j\}_{j =1}^n$ a family of $n$ functions on $[0,1]^d$. We consider the approximate recovery functions $f$ on $[0,1]^d$ from the sampled values $f(x^1), ..., f(x^n)$, by the linear sampling algorithm \begin{equation} onumber L_n(X_n,Φ_n,f) \ := \ \sum_{j=1}^n f(x^j)φ_j. \end{equation} The error of sampling recovery is measured in the norm of the space $L_q([0,1]^d)$-norm or the energy norm of the isotropic Sobolev sapce $W^γ_q([0,1]^d)$ for $0 < q \le \infty$ and $γ> 0$. Functions $f$ to be recovered are from the unit ball in Besov type spaces of an anisotropic smoothness, in particular, spaces $B^a_{p,θ}$ of a nonuniform mixed smoothness $a \in {\mathbb R}^d_+$, and spaces $B^{α,β}_{p,θ}$ of a "hybrid" of mixed smoothness $α> 0$ and isotropic smoothness $β\in \mathbb R$. We constructed optimal linear sampling algorithms $L_n(X_n^*,Φ_n^*,\cdot)$ on special sparse grids $X_n^*$ and a family $Φ_n^*$ of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic of the error of the optimal recovery. This construction is based on a B-spline quasi-interpolation representations of functions in $B^a_{p,θ}$ and $B^{α,β}_{p,θ}$. As consequences we obtained the asymptotic of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov type spaces.

Motivation & Objective

• To develop linear sampling algorithms and cubature formulas that are asymptotically optimal for functions with anisotropic smoothness in high-dimensional spaces.
• To overcome the curse of dimensionality by employing sparse grids constructed via B-spline quasi-interpolation on Smolyak-type grids.
• To derive the exact asymptotic order of the optimal recovery error in $ L_q $ and Sobolev energy norms for functions in Besov-type spaces with mixed and non-uniform smoothness.
• To establish the asymptotic order of optimal cubature formulas for numerical integration over the unit cube $[0,1]^d$.
• To unify and extend prior results on sampling recovery and integration for periodic and non-periodic functions with anisotropic smoothness.

Proposed method

• Constructs linear sampling algorithms $ L_n(X_n, ilde{oldsymbol{ heta}}_n, f) = sum_{j=1}^n f(x^j) ilde{ heta}_j $ using a family of functions $ ilde{oldsymbol{ heta}}_n $ derived from B-spline quasi-interpolation on sparse grids.
• Employs sparse grids $ X_n^* $ defined via the Smolyak construction $ ilde{oldsymbol{ heta}}_n = ext{span}igackslash\{ \text{tensor products of B-splines} \} $, with grid points $ x^j = 2^{-k}s $ for $ k eq 0 $, $ s eq 0 $.
• Uses B-spline quasi-interpolation representations to express functions in Besov-type spaces $ B^{oldsymbol{eta}}_{p, heta} $ and $ B^a_{p, heta} $, enabling stable and accurate reconstruction.
• Derives optimal error bounds by analyzing the approximation error in $ L_q $ and Sobolev $ W^ u_q $ norms via interpolation and embedding theorems.
• Applies the duality between sampling recovery and cubature to derive optimal cubature formulas from the sampling algorithms via integration of the reconstruction kernels.
• Establishes asymptotic equivalence between the optimal recovery error and the optimal cubature error using the inequality $ i_n(W) \leq r_n(W)_1 $.

Experimental results

Research questions

• RQ1 What is the asymptotic order of the optimal sampling recovery error for functions in Besov-type spaces with anisotropic smoothness?
• RQ2 How can B-spline quasi-interpolation be used to construct asymptotically optimal sampling algorithms on sparse grids?
• RQ3 What is the optimal convergence rate for numerical integration (cubature) of functions in $ B^{oldsymbol{eta}}_{p, heta} $ and $ B^a_{p, heta} $ on sparse grids?
• RQ4 How does the choice of grid structure (sparse vs. full) affect the convergence rate in high-dimensional approximation problems?
• RQ5 What is the relationship between the optimal sampling recovery error and the optimal cubature error in the context of anisotropic smoothness?

Key findings

• The asymptotic order of optimal sampling recovery in $ L_q({f I}^d) $ or $ W^ u_q({f I}^d) $ for $ f o B^{oldsymbol{eta}}_{p, heta} $ is $ r_n(U^{oldsymbol{eta}}_{p, heta}, B^ u_q) \asymp n^{-\alpha - (\beta - \nu)/d + (1/p - 1/q)_+} $ for $ \beta > \nu $, and $ n^{-\alpha - \beta + \nu + (1/p - 1/q)_+} $ for $ \beta < \nu $.
• For $ B^a_{p, heta} $ with $ a \in \mathbb{R}^d_+ $, the optimal sampling recovery error is $ r_n(U^a_{p, heta}, W^ u_q) \asymp n^{-a_1 + (1/p - 1/q)_+} $, showing dependence on the largest smoothness parameter $ a_1 $.
• The optimal cubature error for $ U^{oldsymbol{eta}}_{p, heta} $ is $ i_n(U^{oldsymbol{eta}}_{p, heta}) \asymp n^{-\alpha - \beta/d + (1/p - 1)_+} $ for $ \beta > 0 $, and $ n^{-\alpha - \beta + (1/p - 1)_+} $ for $ \beta < 0 $.
• For $ B^a_{p, heta} $, the optimal cubature error is $ i_n(U^a_{p, heta}) \asymp n^{-a_1 + (1/p - 1)_+} $, demonstrating that the convergence rate depends only on the largest smoothness parameter $ a_1 $.
• The constructed sampling algorithms and cubature formulas are asymptotically optimal, with error bounds matching the theoretical lower bounds derived via extremal functions.
• The results generalize and unify prior work on periodic and non-periodic functions, extending optimal recovery and integration to non-uniform and mixed smoothness spaces using sparse grids and B-spline quasi-interpolation.