Topological derivatives for shape reconstruction.

*(English)* Zbl 1144.65307
Bonilla, Luis L. (ed.), Inverse problems and imaging. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, September 15–21, 2002. Berlin: Springer; Florenz: Fondazione CIME Roberto Conti (ISBN 978-3-540-78545-3/pbk). Lecture Notes in Mathematics 1943, 85-133 (2008).

Summary: Topological derivative methods to solve constrained optimization reformulations of inverse scattering problems are analyzed. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.

##### MSC:

65N21 | Inverse problems (BVP of PDE, numerical methods) |

35J05 | Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation |

35P25 | Scattering theory (PDE) |

35R30 | Inverse problems for PDE |

92C50 | Medical applications of mathematical biology |