Inverse Problems in Milan: Abstracts

In this work, we investigate the problem of vertical infiltration in a homogeneous, bounded soil profile. The governing evolution equation is formulated in a one-dimensional setting with boundary conditions that model either flooding or rainfall. To solve the direct problem—computing soil moisture given specific soil properties—we employ the unified transform method (also known as the Fokas method). An exact analytical solution is derived and shown to align perfectly with well-known approximate solutions. Furthermore, we address an inverse control problem: determining a boundary control function at the bottom of the profile that drives the solution to zero at a prescribed time. Numerical examples are presented to demonstrate the effectiveness and applicability of the proposed approach.

In this paper, we study an anisotropic Calderón's problem of recovering potentials by the boundary measurements. We consider the Laplace-Beltrami operator with any Riemannian metric which is a small perturbation of the Euclidean one. We prove that the Dirichlet-to-Neumann map associated with the Schrödinger operator determines the potential with a Lipschitz stability estimate in a negative Sobolev norm under a sign condition. In other words, assuming a priori sign condition, we are able to prove two new results in Calderón's problem: small anisotropic media without structure assumption and a Lipschitz stability estimate of reconstructing potentials.

For the reconstruction in linear inverse ill-posed problems, I consider a Tikhonov regularization given by a suitable Hilbert space norm penalization. The regularization coefficient is linked, although in a rather implicit way, to the scale of the reconstructed unknown, that is, to the amount of details recovered.

I consider a multiscale procedure introduced in imaging by Tadmor, Nezzar and Vese and extended to inverse problems by Modin, Nachman and myself. I show that the multiscale procedure provides a more stable reconstruction at a finer scale. Numerical examples confirm the theoretical results.