Restarted inverse Born series for the Schrödinger problem with discrete internal measurements

Abstract

Convergence and stability results for the inverse Born series (Moskow and Schotland 2008 Inverse Problems 24 065005) are generalized to mappings between Banach spaces. We show that by restarting the inverse Born series one obtains a class of iterative methods containing the Gauss–Newton and Chebyshev–Halley methods. We use the generalized inverse Born series results to show convergence of the inverse Born series for the Schrödinger problem with discrete internal measurements. In this problem, the Schrödinger potential is to be recovered from a few measurements of solutions to the Schrödinger equation resulting from a few different source terms. An application of this method to a problem related to transient hydraulic tomography is given, where the source terms model injection and measurement wells.