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Shape and parameter reconstruction for the Robin transmission inverse problem

Antoine Laurain and Houcine Meftahi

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

G:[0,ε]×X×Y

for some ε>0 and the Banach spaces X and Y. For each t[0,ε], define

(A.1)g(t)=infxXsupyYG(t,x,y),h(t)=supyYinfxXG(t,x,y),

and the associated sets

X(t)={xtX:supyYG(t,xt,y)=g(t)},
Y(t)={ytY:infxXG(t,x,yt)=h(t)}.

Note that inequality h(t)g(t) holds. If h(t)=g(t), the set of saddle points is given by

S(t):=X(t)×Y(t).

We state now a simplified version of a result from [18] derived from [16] which gives realistic conditions that allows to differentiate g(t) at t=0. The main difficulty is to obtain conditions which allow to exchange the derivative with respect to t and the inf-sup in (A.1).

Theorem 1

Theorem 1 (Correa and Seeger [16, 19])

Let X, Y, G and ε be given as above. Assume that the following conditions hold:

  1. ($H$1)

    S(t) for 0tε.

  2. ($H$2)

    The partial derivative tG(t,x,y) exists for all (t,x,y)[0,ε]×X×Y.

  3. ($H$3)

    For any sequence {tn}n, with tn0, there exist a subsequence {tnk}k and x0X(0), xnkX(tnk) such that for all yY(0),

    limt0,ktG(t,xnk,y)=tG(0,x0,y).
  4. ($H$4)

    For any sequence {tn}n, with tn0, there exist a subsequence {tnk}k and y0Y(0), ynkY(tnk) such that for all xX(0),

    limt0,ktG(t,x,ynk)=tG(0,x,y0).

Then there exists (x0,y0)X(0)×Y(0) such that

dgdt(0)=tG(0,x0,y0).

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Received: 2015-1-21
Accepted: 2016-2-24
Published Online: 2016-5-1
Published in Print: 2016-12-1

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