doi: 10.3934/ipi.2021071
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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 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 & Imaging, 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.  Google Scholar

[2]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Hiptmaier and J. Li, Shape derivatives for scattering problems, Inverse Problems, 34 (2018), 25pp. doi: 10.1088/1361-6420/aad34a.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simulation, 66 (2004), 255-265.  doi: 10.1016/j.matcom.2004.02.006.  Google Scholar

[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.  Google Scholar

[19]

M. LassasT. LiimatainenY.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 39pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

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.  Google Scholar

[2]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Hiptmaier and J. Li, Shape derivatives for scattering problems, Inverse Problems, 34 (2018), 25pp. doi: 10.1088/1361-6420/aad34a.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simulation, 66 (2004), 255-265.  doi: 10.1016/j.matcom.2004.02.006.  Google Scholar

[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.  Google Scholar

[19]

M. LassasT. LiimatainenY.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 39pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

Figure 1.  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))
Figure 2.  The worst and the best result from noisy data
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