Reconstruction algorithms for the fractional Laplacian and applications to inverse problems

Abstract

We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space $\mathbb{R}^n$ from local data $(u|_W, [(-\Delta)^s u]|_W)$, where $W$ is an arbitrarily small nonempty open subset of $\mathbb R^n$ and $(-\Delta)^s$ denotes the fractional Laplace operator of order $s\in (0,1)$. These procedures rely crucially on the weak Unique Continuation Property (UCP) for the fractional Laplacian. We apply these schemes to two distinct inverse problems. Following the seminal work from Ghosh et al., the first one concerns the recovery of a potential (Calder\'on-type problem) from the fractional Schr\"odinger equation under nonlocal Robin-type exterior conditions. The second one involves recovering the solution of the space-fractional heat equation in $\mathbb{R}^n$ from localized time-dependent measurements within a ball. To tackle these problems, we introduce new analytical tools such as a generalized weak Kelvin transform and a fractional Robin-to-Robin map. Finally, we provide numerical simulations for one of the reconstruction methods, illustrating the stability issues and the severe ill-posedness inherent to such inverse problems.