  In this paper we extend results taken from compressed sensing to recover
Hilbert-space valued vectors. This is an important problem in parametric
function approximation in particular when the number of parameters is high. By
expanding our target functions in a polynomial chaos and assuming some
compressibility of such an expansion, we can exploit structured sparsity
(typically a group sparsity structure) to recover the sequence of coefficients
with high accuracy.
  While traditional compressed sensing would typically expect a number of
snapshots scaling exponentially with the number of parameters, we can beat this
dependence by adding weights. This anisotropic handling of the parameter space
permits to compute approximations with a number of samples scaling only
linearly (up to log-factors) with the intrinsic complexity of the polynomial
expansion.
  Our results are applied to problems in high-dimensional parametric elliptic
PDEs. We show that under some weighted uniform ellipticity assumptions of a
parametric operator, we are capable of numerically approximating the full
solution (in contrast to the usual quantity of interest) to within a specified
accuracy.
