D. Dũng, M. Griebel, V. N. Huy and C. Rieger $\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs Journal of Complexity, 46: 66--89 2018 10.1016/j.jco.2017.12.001
2015
Dinh Dũng and Michael Griebel Hyperbolic cross approximation in infinite dimensions Journal of Complexity 2015 http://arxiv.org/pdf/1501.01119v1
Abstract: 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-analytictype
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.