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n-Widths and epsilon-dimensions for high-dimensional sparse approximations / Dinh Dũng, Tino Ullrich

Tác giả : Dinh Dũng, Tino Ullrich

Nhà xuất bản : Hausdorff Research Institute for Mathematics

Năm xuất bản : 2012

Chủ đề : 1. High-dimensional approximation, Trigonometric hyperbolic cross space, Kolmogorov n-widths, e-dimensions, Sobolev space, Function classes with anisotropic smoothness.. 2. Dataset.

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In this paper, we study linear trigonometric hyperbolic cross approximations, Kol-mogorov n-widths dn(W; H ), and "-dimensions n"(W; H ) of periodic d-variate func-tion classes W with anisotropic smoothness, where d may be large. We are interested in nding the accurate dependence of dn(W; H ) and n"(W; H ) as a function of two variables n, d and ", d, respectively. Recall that n, the dimension of the approximat-ing subspace, is the main parameter in the study of convergence rates with respect to n going to in nity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonomet-ric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H . Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths dn(W; H ) and "-dimensions n"(W; H ). Some of the re-ceived results imply that the curse of dimensionality can be broken in some relevant situations.
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https://repository.vnu.edu.vn/handle/VNU_123/10996

 

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Sách, tuyển tập

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square / Dinh Dũng, Tino Ullrich

Tác giả : Dinh Dũng, Tino Ullrich

Nhà xuất bản : Mathematische Nachrichten

Năm xuất bản : 2015