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**Level set methods for inverse scattering.**
*(English)*
Zbl 1191.35272

Summary: We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwell’s equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of F. L. Santosa [ESAIM, Control Optim. Calc. Var. 1, 17–33 (1996; Zbl 0870.49016)] for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

### MSC:

35R30 | Inverse problems for PDEs |

35P25 | Scattering theory for PDEs |

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

49N45 | Inverse problems in optimal control |

65J22 | Numerical solution to inverse problems in abstract spaces |

65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |