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

Authors: Dũng, D.

"Let Xn={xj}j=1 n be a set of n points in the d-cube Id: = [0 , 1] d, and Φn={φj}j=1 n a family of n functions on Id. We consider the approximate recovery of functions f on Id from the sampled values f(x1) , … , f(xn) , by the linear sampling algorithm Ln(Xn,Φn,f):=∑j=1 nf(xj)φj. The error of sampling recovery is measured in the norm of the space Lq(Id) -norm or the energy quasi-norm of the isotropic Sobolev space Wq γ(Id) for 1 < q< ∞ and γ> 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces Bp,θ α,β of a “hybrid” of mixed smoothness α> 0 and isotropic smoothness β∈ R, and spaces Bp,θ a of a nonuniform mixed smoothness a∈R+ d. We constructed asymptotically optimal linear sampling algorithms Ln(Xn ∗,Φn ∗,·) on special sparse grids Xn ∗ and a family Φn ∗ 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 Bp,θ α,β and Bp,θ a. 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...".

Link: http://link.springer.com/article/10.1007%2Fs10208-015-9274-8
http://repository.vnu.edu.vn/handle/VNU_123/33849