University of Oulu

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15(5): 1015-1033. doi: 10.3934/ipi.2021026

Two-dimensional inverse scattering for quasi-linear biharmonic operator

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Author: Harju, Markus1; Kultima, Jaakko2; Serov, Valery2;
Organizations: 1Biomimetics and Intelligent Systems Group, P.O. BOX 8000, FIN-90014 University of Oulu, Finland
2Research Unit of Mathematical Sciences, P.O. BOX 3000, FIN-90014 University of Oulu, Finland
3Department of Mathematics and Statistics, P.O. BOX 68, FI-00014 University of Helsinki
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 2.6 MB)
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Language: English
Published: American Institute of Mathematical Sciences, 2021
Publish Date: 2023-06-01


The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space \(W_{∞}^{1}(\mathbb{R}^{2})\). Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.

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Series: Inverse problems and imaging
ISSN: 1930-8337
ISSN-E: 1930-8345
ISSN-L: 1930-8337
Volume: 15
Issue: 5
Pages: 1015 - 1033
DOI: 10.3934/ipi.2021026
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: This work was supported by the Academy of Finland (grant number 312123, the Centre of Excellence of Inverse Modelling and Imaging 2018-2025). T. T. was supported by the Academy of Finland (grant number 312119, Centre of Excellence in Inverse Modelling and Imaging, 2018-2025).
Academy of Finland Grant Number: 312123
Detailed Information: 312123 (Academy of Finland Funding decision)
Copyright information: Inverse Problems & Imaging © 2021 Published by AIMS. All rights reserved.