Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities
Lu, S., Salo, M., & Xu, B. (2022). Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities. Inverse Problems, 38(6), Article 065009. https://doi.org/10.1088/1361-6420/ac637a
Published inInverse Problems
DisciplineInversio-ongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematics
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© 2022 IOP Publishing Ltd.
We consider increasing stability in the inverse Schrödinger 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 Schrödinger 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 Schrödinger 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. ...
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Related funder(s)European Commission; Academy of Finland
Funding program(s)Centre of Excellence, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingM Salo is supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, Grant 284715) and by the European Research Council under Horizon 2020 (ERC CoG 770924). B Xu is supported by NSFC (No. 12171301 and No. 11801351).
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