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Hier finden Sie wissenschaftliche Publikationen aus den FraunhoferInstituten. εdimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs
 Journal of complexity 46 (2018), pp.6689 ISSN: 0885064X 

 English 
 Journal Article, Electronic Publication 
 Fraunhofer SCAI () 
Abstract
In this article, we present a cost–benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolevanalytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the dimension in Hilbert spaces which gives the lowest number of linear information that is needed to approximate an element from the unit ball in a Hilbert space up to an accuracy with respect to the norm of a Hilbert space . From a practical point of view this means that we a priori fix an accuracy and ask for the amount of information to achieve this accuracy. Such an analysis usually requires sharp estimates on the cardinality of certain index sets which are in our case infinitedimensional hyperbolic crosses. As main result, we obtain sharp bounds of the dimension of the Sobolevanalytictype function classes which depend only on the smoothness differences in the Sobolev spaces and the dimension of the finite dimensional domain where these spaces are defined. This implies in particular that, up to constants, the costs of the infinite dimensional (analytical) approximation problem is dominated by the finitevariate Sobolev approximation problem. We demonstrate this procedure with examples of functions spaces stemming from the regularity theory of parametric partial differential equations.