Abstract
We consider an inverse problem in scattering of plane waves by a two dimensional radially-symmetric potential. The mathematical model of our problem is described by the Helmholtz equation \(\Delta u+k^2p(x)u=0\), in \(\textrm{R}^2\) where \(u=u^i+u^s\) and \(u^i\) is the incident wave. The scattered wave \(u^s\) satisfies the Sommerfeld radiation condition at infinity. Our purpose is to determine the scattering potential \(p(r)=1-q(r)\) from the far-field pattern \(u^\infty \). In the forward problem we use the Born approximation. The inverse problem is then reduced to a nonlinear integral equation \({\mathcal {F}}(q)=u^ \infty \). For the resolution we use Gauss-Newton method with a regularization. Some numerical examples show that the reconstruction is effective.
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References
Bao, G., Li, P.: Inverse medium scattering for the Helmholtz equation at fixed frequency. Inverse Prob. 21, 1621–1641 (2005)
Bao, G., Li, P., Lin, J., Triki, F.: Inverse scattering problems with multi-frequencies. Inverse Prob. 31, 093001 (2015)
Borges, C., Gillman, A., Greengard, L.: High resolution inverse scattering in two dimensions using recursive linearization. SIAM J. Imaging Sci. 10(2), 641–664 (2017)
Bukhgeim, A.: Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Illposed Probl. 16, 19–33 (2008)
Cakoni, F., Colton, D.: A Qualitative Approach to Inverse Scattering Theory. Springer, New York (2014)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, New York (2013)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrales, Series and Products. Academic Press (2007)
Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6, 1–35 (1994)
Kirsh, A.: An Introduction to The Mathematical Theory of Inverse Problems, vol. 120. Springer Science and Business Media, Berlin (2011)
Nanfuka, M., Berntsson, F., Mango, J.: Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines. J. Appl. Math. Comput. (2021)
Shin, J., Arhin, E.: Determining radially symmetric potential from near-field scattering data. J. Appl. Math. Comput. 62, 511–524 (2020). https://doi.org/10.1007/s12190-019-01294-7
Shurup, A.S.: Numerical comparison of iterative and functional-analytical algorithms for inverse acoustic scattering. Eur. J. Math. Comput. Appl. 10(1), 790–99 (2022)
Acknowledgements
The work is supported by the national organism DGRST (mesrs.dz), project PRFU under Grant C00L03UN230120180007.
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Djerrar, I., Khélifa, I. & Chorfi, L. Determining radially symmetric potential from far-field scattering data. Afr. Mat. 34, 13 (2023). https://doi.org/10.1007/s13370-023-01050-y
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DOI: https://doi.org/10.1007/s13370-023-01050-y