AN ITERATIVELY REGULARIZED STOCHASTIC GRADIENT METHOD FOR ESTIMATING A RANDOM PARAMETER IN A STOCHASTIC PDE. A VARIATIONAL INEQUALITY APPROACH

We develop a variational inequality approach for the inverse problem of identifying a stochastic parameter in a stochastic partial differential equation. An iteratively regularized projected stochastic gradient scheme for variational inequalities posed in a Hilbert space is proposed. By employing the martingale theory, we give a complete convergence analysis for the iterative scheme under weaker conditions on the random noise than those commonly imposed in the available literature. Preliminary numerical results on the inverse problem demonstrate the efficacy of the developed framework.