Linearized inverse Schrödinger potential problem with partial data and its deep neural network inversion

Abstract

<p style='text-indent:20px;'>We study the linearized inverse Schrödinger potential problem with (many) partial boundary data. By fixing specific partial boundary these measurements are realized by the linearized local Dirichlet-to-Neumann map. When the wavenumber is assumed to be large, we verify a Hölder type increasing stability by constructing the complex exponential solutions in a reflection form. Meanwhile, the linearized inverse Schrödinger potential problem admits an integral equation where the unknown potential function is indirectly contained there. Such a formulation allows us to adopt a deep neural network inversion algorithm. Numerical examples show that one can reconstruct the unknown potential function stably within the partial boundary data setting.</p>