Abstract
In this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization approach and compare the results obtained using two different types of shape functionals. The reformulation of the shape optimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup combined with a function space parameterization technique. The reconstruction is then performed by means of an iterative algorithm based on a conjugate shape gradient method combined with a level set approach. To conclude we give and discuss several numerical examples.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: MATHEON-Project C37
Funding statement: The authors acknowledge financial support from the DFG Research Center MATHEON “Mathematics for key technologies” through the MATHEON-Project C37 “Shape/Topology optimization methods for inverse problems”.
A Appendix
A.1 An abstract differentiability result
In this section we give an abstract result for differentiating Lagrangian functionals with respect to a parameter. This result is used to prove Proposition 1. We first introduce some notations. Consider the functional
for some
and the associated sets
Note that inequality
We state now a simplified version of a result from [18] derived from [16] which gives realistic conditions that allows to differentiate
Theorem 1 (Correa and Seeger [16, 19])
Let X, Y, G and ε be given as above. Assume that the following conditions hold:
The partial derivative
For any sequence
For any sequence
Then there exists
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