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

Dinh, Dũng

ISIKNOWLEDGE; http://repository.vnu.edu.vn/handle/VNU_123/26474

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  • Nhan đề:
    Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation
  • Tác giả: Dinh, Dũng
  • Chủ đề: Linear sampling algorithms; Optimal sampling recovery; Cubature formulas; Optimal cubature; Besov-type spaces of anisotropic smoothness; B-spline quasi-interpolation representations
  • Mô tả: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS Volume: 16 Issue: 5 Pages: 1193-1240 ; TNS06354
    LetXn ={x j } n j=1 be a set ofnpoints in thed-cube I d := [0,1] d , and n ={ϕj} n j=1 a family ofnfunctions onI d . We consider the approximate recovery of functions f onI d from the sampled values f(x 1 ),..., f(x n ), by the linear sam-pling algorithmLn(Xn, n, f) := n j=1 f(x j )ϕj . The error of sampling recovery is measured in the norm of the space Lq(I d )-norm or the energy quasi-norm of the isotropic Sobolev space W γ q(I d )for 1 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces B α,β p,θ of a “hybrid” of mixed smoothnessα>0 and isotropic smoothnessβ∈R, and spaces B a p,θ of a nonuniform mixed smoothnessa∈R d +. We constructed asymptotically optimal linear sampling algorithmsLn(X ∗ n, ∗ n ,·)on special sparse grids X ∗ n and a family ∗ n of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymp-totic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions inB α,β p,θ andB a p,θ . As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integra-tion of functions from the unit ball of these Besov-type spaces.
  • Nơi xuất bản: H. : ĐHQGHN
  • Năm xuất bản: 2016
  • Ngôn ngữ: English
  • Số nhận dạng: ISIKNOWLEDGE; http://repository.vnu.edu.vn/handle/VNU_123/26474

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