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We analyze a novel multi-level version of a recently introduced compressedsensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab:Compressive Sensing Petrov-Galerkin approximation of high-dimensionalparametric operator equations, Math. Comp. 304(2017) 661-700] for the solutionof many-parametric partial differential equations. We propose to usemulti-level PG discretizations, based on a hierarchy of nested finitedimensional subspaces, and to reconstruct parametric solutions at each levelfrom level-dependent random samples of the high-dimensional parameter space viaCS methods such as weighted l1-minimization. For affine parametric, linearoperator equations, we prove that our approach allows to approximate theparametric solution with (almost) optimal convergence order as specified bycertain summability properties of the coefficient sequence in a generalpolynomial chaos expansion of the parametric solution and by the convergenceorder of the PG discretization in the physical variables. The computations ofthe parameter samples of the PDE solution is "embarrassingly parallel", as inMonte-Carlo Methods. Contrary to other recent approaches, and as already notedin [A. Doostan and H. Owhadi: A non-adapted sparse approximation of PDEs withstochastic inputs. JCP 230(2011) 3015-3034] the optimality of the computedapproximations does not require a-priori assumptions on ordering and structureof the index sets of the largest gpc coefficients (such as the "downwardclosed" property). We prove that under certain assumptions work versus accuracyof the new algorithms is asymptotically equal to that of one PG solve for thecorresponding nominal problem on the finest discretization level up to aconstant.

Multi-level Compressed Sensing Petrov-Galerkin discretization of high-dimensional parametric PDEs