A Tutorial on Inverse Problems for Anomalous Diffusion Processes

TL;DR

This paper analyzes the ill-posedness of inverse problems related to anomalous diffusion modeled by fractional differential equations, revealing that anomalous diffusion can either improve or worsen problem conditioning depending on data and interest.

Contribution

It provides a formal analytic and numerical examination of the ill-posedness of classical inverse problems for fractional diffusion equations, highlighting new features of fractional inverse problems.

Findings

Anomalous diffusion's impact on ill-posedness varies with data type and quantity of interest.

Fractional inverse problems exhibit distinct new features compared to classical cases.

The influence of fractional derivatives on problem stability is not uniformly beneficial or detrimental.

Abstract

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. 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 to examine the degree of ill-posedness 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 problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems.