Posterior consistency for subdiffusion inverse problems

Abstract: We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle--Matérn process is assigned to the unknown initial condition. We prove the (H^{2+κ})-regularity of the solution when the nonlinearity satisfies a Lipschitz condition in the (H^κ)-norm. We then establish posterior contraction rates for the prediction error in the (L^2)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback--Leibler divergences.