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

Authors: Dinh Dũng

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

Title: Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation
Authors: Dinh Dũng
Keywords: Linear sampling algorithms, Optimal sampling recovery, Cubature formulas, Optimal cubature, Sparse grids, Besov-type spaces of anisotropic smoothness, B-spline quasi-interpolation representations
Issue Date: 2012
Appears in Collections: ITI – Papers

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