# 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).

### References

[1] Afraites L., Dambrine M. and Kateb D., Shape methods for the transmission problem with a single measurement, Numer. Funct. Anal. Optim. 28 (2007), no. 5–6, 519–551. Search in Google Scholar

[2] Alessandrini G. and Di Cristo M., Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal. 37 (2005), no. 1, 200–217. Search in Google Scholar

[3] Ammari H. and Kang H., Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Math. 1846, Springer, Berlin, 2004. Search in Google Scholar

[4] Arridge S. R., Dorn O., Kaipio J. P., Kolehmainen V., Schweiger M., Tarvainen T., Vauhkonen M. and Zacharopoulos A., Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data, Inverse Problems 22 (2006), no. 6, 2175–2196. Search in Google Scholar

[5] Barceló B., Fabes E. and Seo J. K., The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc. 122 (1994), no. 1, 183–189. Search in Google Scholar

[6] Belhachmi Z. and Meftahi H., Shape sensitivity analysis for an interface problem via minimax differentiability, Appl. Math. Comput. 219 (2013), no. 12, 6828–6842. Search in Google Scholar

[7] Belhachmi Z. and Meftahi H., Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement, Math. Methods Appl. Sci. 38 (2015), no. 2, 221–240. Search in Google Scholar

[8] Bellout H., Friedman A. and Isakov V., Stability for an inverse problem in potential theory, Trans. Amer. Math. Soc. 332 (1992), no. 1, 271–296. Search in Google Scholar

[9] Canelas A., Laurain A. and Novotny A. A., A new reconstruction method for the inverse potential problem, J. Comput. Phys. 268 (2014), 417–431. Search in Google Scholar

[10] Canelas A., Laurain A. and Novotny A. A., A new reconstruction method for the inverse source problem from partial boundary measurements, Inverse Problems 31 (2015), no. 7, Article ID 075009. Search in Google Scholar

[11] Céa J., Conception optimale ou identification de formes, calcul rapide de la derivee dircetionelle de la fonction cout, Math. Modell. Numer. Anal. 20 (1986), 371–402. Search in Google Scholar

[12] Chaabane S., Elhechmi C. and Jaoua M., A stable recovery method for the Robin inverse problem, Math. Comput. Simulation 66 (2004), no. 4–5, 367–383. Search in Google Scholar

[13] Chaabane S. and Jaoua M., Identification of robin coefficients by the means of boundary measurements, Inverse Problems 15 (1999), no. 6, 1425–1438. Search in Google Scholar

[14] Cheng J., Choulli M. and Yang X., An iterative BEM for the inverse problem of detecting corrosion in a pipe, Frontiers and Prospects of Contemporary Applied Mathematics, Ser. Contemp. Appl. Math. CAM 6, Higher Education Press, Beijing (2005), 1–17. Search in Google Scholar

[15] Choulli M., An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl. 12 (2004), no. 4, 349–367. Search in Google Scholar

[16] Correa R. and Seeger A., Directional derivative of a minimax function, Nonlinear Anal. 9 (1985), no. 1, 13–22. Search in Google Scholar

[17] Delfour M., Introduction to Optimization and Semidifferential Calculus, MOS-SIAM Ser. Optim., SIAM, Philadelphia, 2012. Search in Google Scholar

[18] Delfour M. C. and Zolésio J.-P., Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim. 26 (1988), no. 4, 834–862. Search in Google Scholar

[19] Delfour M. C. and Zolésio J.-P., Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed., Adv. Des. Control 22, SIAM, Philadelphia, 2011. Search in Google Scholar

[20] Ekeland I. and Temam R., Analyse convexe et problèmes variationnels, Études Math., Dunod, Paris, 1974. Search in Google Scholar

[21] Fan Q., Jiao Y., Lu X. and Sun Z., Lq-regularization for the inverse Robin problem, J. Inverse Ill-Posed Probl. 24 (2014), no. 1, 3–12. Search in Google Scholar

[22] Freiberger M., Laurain A., Hintermüller M., Köstinger A. and Scharfetter H., Using the topological derivative for initializing a Markov-chain Monte Carlo reconstruction in fluorescence tomography, Diffuse Optical Imaging III (Munich 2011), Proc. SPIE 8088, SPIE Press, Bellingham (2011), DOI 10.1117/12.889609. Search in Google Scholar

[23] Friedman A. and Isakov V., On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J. 38 (1989), no. 3, 563–579. Search in Google Scholar

[24] Fulmański P., Laurain A., Scheid J.-F. and Sokołowski J., A level set method in shape and topology optimization for variational inequalities, Int. J. Appl. Math. Comput. Sci. 17 (2007), no. 3, 413–430. Search in Google Scholar

[25] Fulmański P., Laurain A., Scheid J.-F. and Sokołowski J., Level set method with topological derivatives in shape optimization, Int. J. Comput. Math. 85 (2008), no. 10, 1491–1514. Search in Google Scholar

[26] Gangl P., Langer U., Laurain A., Meftahi H. and Sturm K., Shape optimization of an electric motor subject to nonlinear magnetostatics, SIAM J. Sci. Comput. 37 (2015), no. 6, B1002–B1025. Search in Google Scholar

[27] Hintermüller M. and Laurain A., Electrical impedance tomography: From topology to shape, Control Cybernet. 37 (2008), no. 4, 913–933. Search in Google Scholar

[28] Hintermüller M. and Laurain A., Multiphase image segmentation and modulation recovery based on shape and topological sensitivity, J. Math. Imaging Vision 35 (2009), no. 1, 1–22. Search in Google Scholar

[29] Hintermüller M., Laurain A. and Novotny A. A., Second-order topological expansion for electrical impedance tomography, Adv. Comput. Math. 36 (2012), no. 2, 235–265. Search in Google Scholar

[30] Hintermüller M., Laurain A. and Yousept I., Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model, Inverse Problems 31 (2015), no. 6, Article ID 065006. Search in Google Scholar

[31] Hiptmair R., Paganini A. and Sargheini S., Comparison of approximate shape gradients, technical report 2013-30, ETH Zürich, Zürich, 2013. Search in Google Scholar

[32] Hu N., Fukunaga H., Kameyama M., Aramaki Y. and Chang F., Vibration analysis of delaminated composite beams and plates using a higher-order finite element, Comm. Pure Appl. Math. 44 (2002), no. 7, 1479–1503. Search in Google Scholar

[33] Isakov V., On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math. 41 (1988), no. 7, 865–877. Search in Google Scholar

[34] Isakov V., Inverse Problems for Partial Differential Equations, 2nd ed., Appl. Math. Sci. 127, Springer, New York, 2006. Search in Google Scholar

[35] Isakov V. and Powell J., Corrigendum: “On the inverse conductivity problem with one measurement”, Inverse Problems 6 (1990), no. 3, 479–479. Search in Google Scholar

[36] Ito K., Kunisch K. and Peichl G. H., Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var. 14 (2008), no. 3, 517–539. Search in Google Scholar

[37] Jaoua M., Chaabane S., Elhechmi C., Leblond J., Mahjoub M. and Partington J. R., On some robust algorithms for the Robin inverse problem, ARIMA Rev. Afr. Rech. Inform. Math. Appl. 9 (2008), 287–307. Search in Google Scholar

[38] Kang H. and Seo J. K., The layer potential technique for the inverse conductivity problem, Inverse Problems 12 (1996), no. 3, 267–278. Search in Google Scholar

[39] Kang H. and Seo J. K., A note on uniqueness and stability for the inverse conductivity problem with one measurement, J. Korean Math. Soc. 38 (2001), no. 4, 781–791. Search in Google Scholar

[40] Kasumba H. and Kunisch K., On shape sensitivity analysis of the cost functional without shape sensitivity of the state variable, Control Cybernet. 14 (2011), no. 4, 989–1017. Search in Google Scholar

[41] Kasumba H. and Kunisch K., On computation of the shape Hessian of the cost functional without shape sensitivity of the state variable, J. Optim. Theory Appl. 162 (2014), no. 3, 779–804. Search in Google Scholar

[42] Laurain A., Hintermüller M., Freiberger M. and Scharfetter H., Topological sensitivity analysis in fluorescence optical tomography, Inverse Problems 29 (2013), no. 2, Article ID 025003. Search in Google Scholar

[43] Laurain A. and Sturm K., Distributed shape derivative via averaged adjoint method and applications, ESAIM Math. Model. Numer. Anal. (2015), 10.1051/m2an/2015075. 10.1051/m2an/2015075Search in Google Scholar

[44] Lionheart W. R. B., Boundary shape and electrical impedance tomography, Inverse Problems 14 (1998), no. 1, 139–147. Search in Google Scholar

[45] Logg A., Mardal K.-A. and Wells G. N., Automated Solution of Differential Equations by the Finite Element Method, Lect. Notes Comput. Sci. Eng. 84, Springer, Berlin, 2012. Search in Google Scholar

[46] Mueller J. L. and Siltanen S., Linear and Nonlinear Inverse Problems with Practical Applications, Comput. Sci. Eng. 10, SIAM, Philadelphia, 2012. Search in Google Scholar

[47] Osher S. and Sethian J. A., Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. Search in Google Scholar

[48] Osher S. and Shu C.-W., High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 907–922. Search in Google Scholar

[49] Pantz O., Sensibilité de l’équation de la chaleur aux sauts de conductivité, C. R. Math. Acad. Sci. Paris 341 (2005), no. 5, 333–337. Search in Google Scholar

[50] Sokołowski J. and Zolésio J.-P., Introduction to Shape Optimization, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992. Search in Google Scholar

[51] Sturm K., Minimax lagrangian approach to the differentiability of nonlinear PDE constrained shape functions without saddle point assumption, SIAM J. Control Optim. 53 (2015), no. 4, 2017–2039. Search in Google Scholar