Foundations of Computational Mathematics

, Volume 16, Issue 5, pp 1193–1240

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Article

DOI: 10.1007/s10208-015-9274-8

Cite this article as:
Dũng, D. Found Comput Math (2016) 16: 1193. doi:10.1007/s10208-015-9274-8

Abstract

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.

Keywords

Linear sampling algorithmsOptimal sampling recovery Cubature formulasOptimal cubatureSparse gridsBesov-type spaces of anisotropic smoothnessB-spline quasi-interpolation representations

Mathematics Subject Classification

41A1541A0541A2541A5841A63

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Information Technology InstituteVietnam National University, HanoiHanoiVietnam