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August  2022, 16(4): 691-702. doi: 10.3934/ipi.2021071

## The domain derivative for semilinear elliptic inverse obstacle problems

 Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany

*Corresponding author: Frank Hettlich

Received  April 2021 Revised  September 2021 Published  August 2022 Early access  December 2021

We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.

Citation: Frank Hettlich. The domain derivative for semilinear elliptic inverse obstacle problems. Inverse Problems and Imaging, 2022, 16 (4) : 691-702. doi: 10.3934/ipi.2021071
##### References:
 [1] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Applied Mathematical Sciences, 93, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [2] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1998. [3] F. Hagemann and F. Hettlich, Application of the second domain derivative in inverse electromagnetic scattering, Inverse Problems, 36 (2020), 34pp. doi: 10.1088/1361-6420/abaa31. [4] H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering problems, J. Integral Equations Appl., 19 (2007), 237-260.  doi: 10.1216/jiea/1190905486. [5] F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation, Inverse Problems, 17 (2001), 1465-1482.  doi: 10.1088/0266-5611/17/5/315. [6] F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266.  doi: 10.1088/0266-5611/12/3/006. [7] R. Hiptmaier and J. Li, Shape derivatives for scattering problems, Inverse Problems, 34 (2018), 25pp. doi: 10.1088/1361-6420/aad34a. [8] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^{rd}$ edition, Applied Mathematical Sciences, 127, Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5. [9] V. Isakov and A. I. Nachmann, Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.  doi: 10.1090/S0002-9947-1995-1311909-1. [10] V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math., 47 (1994), 1403-1410.  doi: 10.1002/cpa.3160471005. [11] E. Jalade, Inverse problem for a nonlinear Helmholtz equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 517-531.  doi: 10.1016/j.anihpc.2003.07.001. [12] B. Kaltenbacher, Regularization based on all-at-once formulations for inverse problems, SIAM J. Numer. Anal., 54 (2016), 2594-2618.  doi: 10.1137/16M1060984. [13] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276. [14] B. Kaltenbacher and W. Rundell, On the identification of a nonlinear term in a reaction-diffusion equation, Inverse Problems, 35 (2019), 38pp. doi: 10.1088/1361-6420/ab2aab. [15] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. [16] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations. Expansion-, Integral-, and Variational Methods, Applied Mathematical Sciences, 190, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8. [17] R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simulation, 66 (2004), 255-265.  doi: 10.1016/j.matcom.2004.02.006. [18] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1233.  doi: 10.1088/0266-5611/21/4/002. [19] M. Lassas, T. Liimatainen, Y.-H. Lin and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Math. Iberoam., 37 (2021), 1553-1580.  doi: 10.4171/rmi/1242. [20] A. Lechleiter, Explicit characterization of the support of non-linear inclusions, Inverse Probl. Imaging, 5 (2011), 675-694.  doi: 10.3934/ipi.2011.5.675. [21] W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, 24 (2008), 12pp. doi: 10.1088/0266-5611/24/5/055015. [22] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 39pp. doi: 10.1088/0266-5611/25/12/123011.

show all references

##### References:
 [1] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, Applied Mathematical Sciences, 93, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [2] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1998. [3] F. Hagemann and F. Hettlich, Application of the second domain derivative in inverse electromagnetic scattering, Inverse Problems, 36 (2020), 34pp. doi: 10.1088/1361-6420/abaa31. [4] H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering problems, J. Integral Equations Appl., 19 (2007), 237-260.  doi: 10.1216/jiea/1190905486. [5] F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation, Inverse Problems, 17 (2001), 1465-1482.  doi: 10.1088/0266-5611/17/5/315. [6] F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266.  doi: 10.1088/0266-5611/12/3/006. [7] R. Hiptmaier and J. Li, Shape derivatives for scattering problems, Inverse Problems, 34 (2018), 25pp. doi: 10.1088/1361-6420/aad34a. [8] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^{rd}$ edition, Applied Mathematical Sciences, 127, Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5. [9] V. Isakov and A. I. Nachmann, Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.  doi: 10.1090/S0002-9947-1995-1311909-1. [10] V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math., 47 (1994), 1403-1410.  doi: 10.1002/cpa.3160471005. [11] E. Jalade, Inverse problem for a nonlinear Helmholtz equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 517-531.  doi: 10.1016/j.anihpc.2003.07.001. [12] B. Kaltenbacher, Regularization based on all-at-once formulations for inverse problems, SIAM J. Numer. Anal., 54 (2016), 2594-2618.  doi: 10.1137/16M1060984. [13] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276. [14] B. Kaltenbacher and W. Rundell, On the identification of a nonlinear term in a reaction-diffusion equation, Inverse Problems, 35 (2019), 38pp. doi: 10.1088/1361-6420/ab2aab. [15] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. [16] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations. Expansion-, Integral-, and Variational Methods, Applied Mathematical Sciences, 190, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8. [17] R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simulation, 66 (2004), 255-265.  doi: 10.1016/j.matcom.2004.02.006. [18] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1233.  doi: 10.1088/0266-5611/21/4/002. [19] M. Lassas, T. Liimatainen, Y.-H. Lin and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Math. Iberoam., 37 (2021), 1553-1580.  doi: 10.4171/rmi/1242. [20] A. Lechleiter, Explicit characterization of the support of non-linear inclusions, Inverse Probl. Imaging, 5 (2011), 675-694.  doi: 10.3934/ipi.2011.5.675. [21] W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, 24 (2008), 12pp. doi: 10.1088/0266-5611/24/5/055015. [22] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 39pp. doi: 10.1088/0266-5611/25/12/123011.
First and 10. iteration in case of noise free data, and the approximation errors versus iteration steps (reconstructions (red), initial guess (dotted, blue), exact object (dashed, black), residual error (blue), reconstruction error (dashed, red))
The worst and the best result from noisy data
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