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Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

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Abstract

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Haibing Wang & Jijun Liu

  2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, China

    Haibing Wang

Authors
  1. Haibing Wang
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  2. Jijun Liu
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Correspondence to Jijun Liu.

Additional information

Communicated by Yuesheng Xu and Hongqi Yang.

This work is supported by Natural Science Foundation of China (No.11071039).

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Wang, H., Liu, J. Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods. Adv Comput Math 36, 279–297 (2012). https://doi.org/10.1007/s10444-011-9191-6

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  • DOI: https://doi.org/10.1007/s10444-011-9191-6

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