BAYESIAN INVERSION TECHNIQUES FOR STOCHASTIC

. We consider the problem of recovering the initial condition for a class of stochastic partial differential equations under Gaussian additive noise. We assume that the covariance operator of the noise is unknown. We develop an adapted Bayesian regularisation strategy, which incorporates the estimation of the unknown parameters into the computation of the initial condition posterior distribution. The proposed method allows estimation of the initial condition curve as well as construction of forecasts of the entire state curve, although the observed data may include only partial observations of the system state. We prove that, under certain conditions, the posterior distribution converges to that under known parameter values when the sample size is large. We also compare the performance of the proposed method to that of Tikhonov regularization on simulated data.