Hyperbolic cross approximation in infinite dimensions

• المؤلف الرئيسي: Dinh Dũng , Michael Griebel
• التنسيق: كتاب Book chapter Dataset
• منشور في: Journal of Complexity 2016
• الموضوعات: Infinite-dimensional hyperbolic cross approximation; Mixed Sobolev–Korobov-type smoothness; Mixed Sobolev-analytic-type smoothness; ε-dimension; Parametric and stochastic elliptic PDEs; Linear information
• الوصول للمادة أونلاين: http://repository.vnu.edu.vn/handle/VNU_123/11116
• المؤسسة: Vietnam National University, Hanoi

We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev–Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where specific summability properties of the smoothness indices are fulfilled. These estimates are then applied to the linear approximation of functions from the associated spaces in terms of the ε-dimension of their unit balls. Here, the approximation is based on linear information. Such function spaces appear for example for the solution of parametric and stochastic PDEs. The obtained upper and lower bounds of the approximation error as well as of the associated ε-complexities are completely independent of any parametric or stochastic dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinite-variate parametric or stochastic part of the solution. These parameters are only contained in the order constants. This way, linear approximation theory becomes possible in the infinite-dimensional case and corresponding infinite-dimensional problems get tractable.