A tutorial on inverse problems for anomalous diffusion processes

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications.At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space.The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level.The new physics can change dramatically the behavior of the forward problems.For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity.Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness.The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest.In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of illposedness of several 'classical' inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations.We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville Inverse Problems