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Solving inhomogeneous inverse problems by topological derivative methods

Carpio Rodríguez, Ana María and Rapún , M.L. (2008) Solving inhomogeneous inverse problems by topological derivative methods. Inverse problems, 24 (4). ISSN 0266-5611

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Abstract

We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

Item Type:Article
Uncontrolled Keywords:Level set methods; Transmission problems; Obstacle scattering; Helmotz-equation; Shape optimization; Integral-equations; Tomography; Uniqueness; Reconstruction; Approximation
Subjects:Sciences > Mathematics > Differential equations
ID Code:14934
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