Increasing stability in the linearized inverse Schr\"{o}dinger potential problem with power type nonlinearities

TL;DR

This paper investigates methods to enhance stability in the inverse Schrödinger potential problem with nonlinearities by using linearization techniques at large wavenumbers, leading to improved reconstruction algorithms.

Contribution

It introduces two linearization approaches for nonlinear inverse Schrödinger problems and demonstrates their effectiveness in increasing stability and developing practical reconstruction algorithms.

Findings

Higher order linearization improves stability for large wavenumbers.

Linearization with respect to the potential benefits quadratic nonlinearity.

Numerical algorithms successfully reconstruct potentials using boundary measurements.

Abstract

We consider increasing stability in the inverse Schr\"{o}dinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schr\"{o}dinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearity in solving the inverse Schr\"{o}dinger potential problem. Noticing that both linearization approaches can be numerically approximated, we provide several reconstruction algorithms for the quadratic and general power type nonlinearity terms, where one of these algorithms is designed based on boundary measurements of multiple wavenumbers. Several numerical examples shed light on the efficiency of our proposed algorithms.