  For finite-dimensional problems, stochastic approximation methods have long
been used to solve stochastic optimization problems. Their application to
infinite-dimensional problems is less understood, particularly for nonconvex
objectives. This paper presents convergence results for the stochastic proximal
gradient method applied to Hilbert spaces, motivated by optimization problems
with partial differential equation (PDE) constraints with random inputs and
coefficients. We study stochastic algorithms for nonconvex and nonsmooth
problems, where the nonsmooth part is convex and the nonconvex part is the
expectation, which is assumed to have a Lipschitz continuous gradient. The
optimization variable is an element of a Hilbert space. We show almost sure
convergence of strong limit points of the random sequence generated by the
algorithm to stationary points. We demonstrate the stochastic proximal gradient
algorithm on a tracking-type functional with a $L^1$-penalty term constrained
by a semilinear PDE and box constraints, where input terms and coefficients are
subject to uncertainty. We verify conditions for ensuring convergence of the
algorithm and show a simulation.
