Publica
Hier finden Sie wissenschaftliche Publikationen aus den FraunhoferInstituten. Hyperbolic cross approximation in infinite dimensions
 Journal of complexity 33 (2016), pp.5588 ISSN: 0885064X 

 English 
 Journal Article 
 Fraunhofer SCAI () 
Abstract
We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev–Korobovtype smoothness and mixed Sobolevanalytictype smoothness in the infinitedimensional 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 εεdimension 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 εεcomplexities are completely independent of any parametric or stochastic dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinitevariate 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 infinitedimensional case and corresponding infinitedimensional problems get tractable.