Identification of a noisy stochastic heat equation with the EM algorithm

Despite the considerable literature on ill-conditioned inverse problem of estimation of spatially varying parameters in partial differential equations an important case has gone untreated. This is when there is noise in both the partial differential equation and the observations. Previous spatially varying parameter estimation work allows one or the other but not both. We show how to deal with this case by developing a penalised EM (expectation-maximization) algorithm for noisy observations of the stochastic heat equation.