  We present and analyze a novel sparse polynomial technique for approximating
high-dimensional Hilbert-valued functions, with application to parameterized
partial differential equations (PDEs) with deterministic and stochastic inputs.
Our theoretical framework treats the function approximation problem as a joint
sparse recovery problem, where the set of jointly sparse vectors is possibly
infinite. To achieve the simultaneous reconstruction of Hilbert-valued
functions in both parametric domain and Hilbert space, we propose a novel
mixed-norm based $\ell_1$ regularization method that exploits both energy and
sparsity. Our approach requires extensions of concepts such as the restricted
isometry and null space properties, allowing us to prove recovery guarantees
for sparse Hilbert-valued function reconstruction. We complement the enclosed
theory with an algorithm for Hilbert-valued recovery, based on standard
forward-backward algorithm, meanwhile establishing its strong convergence in
the considered infinite-dimensional setting. Finally, we demonstrate the
minimal sample complexity requirements of our approach, relative to other
popular methods, with numerical experiments approximating the solutions of
high-dimensional parameterized elliptic PDEs.
