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**Shape reconstruction of buried obstacles by controlled evolution of a level set: From a min-max formulation to numerical experimentation.**
*(English)*
Zbl 0994.35121

Summary: The nonlinearized reconstruction of the cross-sectional contour of a homogeneous, possibly multiply connected obstacle buried in a half-space from time-harmonic wave field data collected above this half-space in both transverse magnetic (TM) and transverse electric (TE) polarization cases is investigated. The reconstruction is performed via controlled evolution of a level set that was pioneered in [A. Litman, D. Lesselier and F. Santosa, Inverse Probl. 14, No. 3, 685-706 (1998; Zbl 0912.35158)] but at this time restricted to free space for TM data collected all around the sought obstacle.

The main novelty of the investigation lies in the following points: from the rigorous contrast-source domain integral formulation (TM) and integral-differential formulation (TE) of the direct scattering problems in the buried obstacle configuration, and from appropriately cast adjoint scattering problems, we demonstrate, by processing min-max formulations of an objective functional \(J\) made of the data error to be minimized, that its derivatives with respect to the evolution time \(t\) are given in closed form. They are contour integrals involving the normal component of the velocity field of evolution times the product of direct and adjoint fields (TM), or of the scalar product of gradients of such fields (TE) at \(t\).

This approach only calls for the analysis of the well-posed direct and adjoint scattering problems formulated from the TM and TE Green systems of the unperturbed layered environment and, unusually, it avoids the differentiation of state fields.

The main novelty of the investigation lies in the following points: from the rigorous contrast-source domain integral formulation (TM) and integral-differential formulation (TE) of the direct scattering problems in the buried obstacle configuration, and from appropriately cast adjoint scattering problems, we demonstrate, by processing min-max formulations of an objective functional \(J\) made of the data error to be minimized, that its derivatives with respect to the evolution time \(t\) are given in closed form. They are contour integrals involving the normal component of the velocity field of evolution times the product of direct and adjoint fields (TM), or of the scalar product of gradients of such fields (TE) at \(t\).

This approach only calls for the analysis of the well-posed direct and adjoint scattering problems formulated from the TM and TE Green systems of the unperturbed layered environment and, unusually, it avoids the differentiation of state fields.

### MSC:

35R30 | Inverse problems for PDEs |

49K35 | Optimality conditions for minimax problems |

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |

78A45 | Diffraction, scattering |

78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |