Bayesian approach to a nonlinear inverse problem for a time-space fractional diffusion equation

Inverse problems for fractional differential equations have become a promising research area because of their wide applications in many scientific and engineering fields. In particular, the correct orders of fractional derivatives are hard to know as they are usually determined by experimental data and contain non-negligible uncertainty. Therefore, research on inverse problems involving the orders is necessary. Furthermore, problems involving the inversion of fractional orders are essentially nonlinear. Since classical methods may find it hard to provide satisfactory approximations and fail to capture the relevant uncertainty, a natural way to solve such inverse problems is through a Bayesian approach. In this paper, we consider an inverse problem of simultaneously recovering the source function and the orders of both time and space fractional derivatives for a time-space fractional diffusion equation. The problem will be formulated in the Bayesian framework, where the solution is the posterior distribution incorporating the prior information about the unknown and the noisy data. Under the considered infinite-dimensional function space setting, we prove that the corresponding Bayesian inverse problem is well-defined based on a proof of the continuity of the forward mapping. In addition, we also prove that the posterior distribution depends continuously on the data with respect to the Hellinger distance. Moreover, we adopt the iterative regularizing ensemble Kalman method to provide a numerical implementation of the considered inverse problem for the one-dimensional case. The numerical results shed light on the viability and efficiency of the method.