Determining the random source and initial value simultaneously in stochastic fractional diffusion equations

In this paper, we investigate the inverse problem of recovering unknown random source and initial value simultaneously from statistical measurement data in a time-fractional stochastic diffusion equation. Based on the eigenfunction expansions, we first establish the statistical moments estimate for the solution of direct problem. Then the conditional stability for inverse problem is also proved. Furthermore, to address the issue of ill-posedness of inverse problem, the Tikhonov regularization method is adopted, and an a priori and a posteriori convergence rate estimates are obtained. Finally, several numerical results are presented to illustrate the effectiveness of the proposed method.