  We analyze a novel multi-level version of a recently introduced compressed
sensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab:
Compressive Sensing Petrov-Galerkin approximation of high-dimensional
parametric operator equations, Math. Comp. 304(2017) 661-700] for the solution
of many-parametric partial differential equations. We propose to use
multi-level PG discretizations, based on a hierarchy of nested finite
dimensional subspaces, and to reconstruct parametric solutions at each level
from level-dependent random samples of the high-dimensional parameter space via
CS methods such as weighted l1-minimization. For affine parametric, linear
operator equations, we prove that our approach allows to approximate the
parametric solution with (almost) optimal convergence order as specified by
certain summability properties of the coefficient sequence in a general
polynomial chaos expansion of the parametric solution and by the convergence
order of the PG discretization in the physical variables. The computations of
the parameter samples of the PDE solution is "embarrassingly parallel", as in
Monte-Carlo Methods. Contrary to other recent approaches, and as already noted
in [A. Doostan and H. Owhadi: A non-adapted sparse approximation of PDEs with
stochastic inputs. JCP 230(2011) 3015-3034] the optimality of the computed
approximations does not require a-priori assumptions on ordering and structure
of the index sets of the largest gpc coefficients (such as the "downward
closed" property). We prove that under certain assumptions work versus accuracy
of the new algorithms is asymptotically equal to that of one PG solve for the
corresponding nominal problem on the finest discretization level up to a
constant.
