January  2022, 15(1): 23-40. doi: 10.3934/dcdss.2021069

A new coupled complex boundary method (CCBM) for an inverse obstacle problem

LMA, Faculty of Sciences and Technology, University of Sultan Moulay Slimane, Béni Mellal, Morroco

* Corresponding author: Lekbir Afraites

Received  October 2020 Revised  April 2021 Published  January 2022 Early access  June 2021

In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion $ \omega $ contained in a larger bounded domain $ \Omega $ via boundary measurements on $ \partial \Omega $. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [16]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion $ \omega $. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain $ \omega $. We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.

Citation: Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 23-40. doi: 10.3934/dcdss.2021069
References:
[1]

L. AfraitesM. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement, Numerical Functional Analysis and Optimization, 28 (2007), 519-551.  doi: 10.1080/01630560701381005.

[2]

L. AfraitesM. DambrineK. Eppler and D. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete and Continuous Dynamical Systems-Series B, 8 (2007), 389-416.  doi: 10.3934/dcdsb.2007.8.389.

[3]

L. AfraitesM. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography, SIAM J. CONTROL OPTIM., 47 (2008), 1556-1590.  doi: 10.1137/070687438.

[4]

L. Afraites, C. Masnaoui and M. Nachaoui, Shape optimization method for an inverse geometric source problem and stability at critical shape, Discrete and Continuous Dynamical Systems-Series S. doi: 10.3934/dcdss.2021006.

[5]

G. AlessandriniV. Isakov and J. Powell, Local uniqueness in the inverse problem with one measurement, Trans. Am. Math. Soc., 347 (1995), 3031-3041.  doi: 10.1090/S0002-9947-1995-1303113-8.

[6]

G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim., 34 (1996), 913-921.  doi: 10.1137/S0363012994262853.

[7]

H. Azegami and Z. Takeuchi, A smoothing method for shape optimization : Traction method using the robin condition, Int. J. Comput. Methods, 3 (2006), 21-33.  doi: 10.1142/S0219876206000709.

[8]

M. BadraF. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101.  doi: 10.1142/S0218202511005660.

[9]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Probl. Imaging, 4 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351.

[10]

F. Caubet, Instability of an inverse problem for the stationary Navier Stokes equations, SIAM J. Control Optim., 51 (2013), 2949-2975.  doi: 10.1137/110836857.

[11]

F. CaubetM. DambrineD. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157.  doi: 10.3934/ipi.2013.7.123.

[12]

F. Caubet, M. Dambrine and D. Kateb, Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions, Inverse Problems, 29 (2013), 115011. doi: 10.1088/0266-5611/29/11/115011.

[13]

A. ChakibA. EllabibA. Nachaoui and M. Nachaoui, A shape optimization formulation of weld pool determination, Appl. Math. Lett., 25 (2012), 374-379.  doi: 10.1016/j.aml.2011.09.017.

[14]

A. ChakibA. Nachaoui and M. Nachaoui, Approximation and numerical realization of an optimal design welding problem, Numer. Methods Partial Differential Eq., 29 (2013), 1563-1586.  doi: 10.1002/num.21767.

[15]

A. ChakibA. Nachaoui and M. Nachaoui, Existence analysis of an optimal shape design problem with non coercive state equation, Nonlinear Anal. Real World Appl., 28 (2016), 171-183.  doi: 10.1016/j.nonrwa.2015.09.009.

[16]

X. L. Cheng, R. F. Gong, W. Han and X. Zheng, A novel coupled complex boundary method for solving inverse source problems, Inverse Problems, 30 (2014), 055002. doi: 10.1088/0266-5611/30/5/055002.

[17]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 2, Springer, Berlin, 1998.

[18]

M. Delfour and J.-P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, USA, 2001.

[19]

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using Hessian information, Control and Cybernetics, 34 (2005), 203-225. 

[20]

M. GiacominiO. Pantz and K. Trabelsi, Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators, ESAIM Control Optimisation and Calculus of Variations, 23 (2017), 977-1001.  doi: 10.1051/cocv/2016021.

[21]

R. Gong, X. Cheng and W. Han, A coupled complex boundary method for an inverse conductivity problem with one measurement, Applicable Analysis An International Journal, 96 (2017). doi: 10.1080/00036811.2016.1165215.

[22]

F. Hecht, Finite Element Library FREEFEM++., Available from: http://www.freefem.org/ff++/.

[23]

A. Henrot and M. Pierre, Variation et optimisation de formes, Springer Mathḿatiques et Applications, 48, (2005). doi: 10.1007/3-540-37689-5.

[24]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67-82.  doi: 10.1088/0266-5611/14/1/008.

[25]

V. Isakov, Inverse Problems for Partial Differential Equations, 127, Springer Science & Business Media, 2006.

[26]

V. Maz'ya and T. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Monographs and Studies in Mathematics, 23, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[27]

F. Murat and J. Simon, Sur le Contôle par Domaine Géométrique, Rapport du L.A. 189, Université de Paris VI, 1976.

[28]

J. J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687.  doi: 10.1080/01630563.1980.10120631.

[29]

J. Simon, Second variation for domain optimization problems, International Series of Numerical Mathematics, 91 (1989), 361-378. 

[30]

J. Sokolowski and J-P Zolesio, Introduction to shape optimization shape sensitivity analysis, Springer-Verlag Springer Series in Computational Mathematics, 16 (1991). doi: 10.1007/978-3-642-58106-9.

[31]

X. Zheng, X. Cheng and R. Gong, A coupled complex boundary method for parameter identification in elliptic problems, International Journal of Computer Mathematics, 97 (2020). doi: 10.1080/00207160.2019.1601181.

show all references

References:
[1]

L. AfraitesM. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement, Numerical Functional Analysis and Optimization, 28 (2007), 519-551.  doi: 10.1080/01630560701381005.

[2]

L. AfraitesM. DambrineK. Eppler and D. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete and Continuous Dynamical Systems-Series B, 8 (2007), 389-416.  doi: 10.3934/dcdsb.2007.8.389.

[3]

L. AfraitesM. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography, SIAM J. CONTROL OPTIM., 47 (2008), 1556-1590.  doi: 10.1137/070687438.

[4]

L. Afraites, C. Masnaoui and M. Nachaoui, Shape optimization method for an inverse geometric source problem and stability at critical shape, Discrete and Continuous Dynamical Systems-Series S. doi: 10.3934/dcdss.2021006.

[5]

G. AlessandriniV. Isakov and J. Powell, Local uniqueness in the inverse problem with one measurement, Trans. Am. Math. Soc., 347 (1995), 3031-3041.  doi: 10.1090/S0002-9947-1995-1303113-8.

[6]

G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim., 34 (1996), 913-921.  doi: 10.1137/S0363012994262853.

[7]

H. Azegami and Z. Takeuchi, A smoothing method for shape optimization : Traction method using the robin condition, Int. J. Comput. Methods, 3 (2006), 21-33.  doi: 10.1142/S0219876206000709.

[8]

M. BadraF. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101.  doi: 10.1142/S0218202511005660.

[9]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Probl. Imaging, 4 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351.

[10]

F. Caubet, Instability of an inverse problem for the stationary Navier Stokes equations, SIAM J. Control Optim., 51 (2013), 2949-2975.  doi: 10.1137/110836857.

[11]

F. CaubetM. DambrineD. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157.  doi: 10.3934/ipi.2013.7.123.

[12]

F. Caubet, M. Dambrine and D. Kateb, Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions, Inverse Problems, 29 (2013), 115011. doi: 10.1088/0266-5611/29/11/115011.

[13]

A. ChakibA. EllabibA. Nachaoui and M. Nachaoui, A shape optimization formulation of weld pool determination, Appl. Math. Lett., 25 (2012), 374-379.  doi: 10.1016/j.aml.2011.09.017.

[14]

A. ChakibA. Nachaoui and M. Nachaoui, Approximation and numerical realization of an optimal design welding problem, Numer. Methods Partial Differential Eq., 29 (2013), 1563-1586.  doi: 10.1002/num.21767.

[15]

A. ChakibA. Nachaoui and M. Nachaoui, Existence analysis of an optimal shape design problem with non coercive state equation, Nonlinear Anal. Real World Appl., 28 (2016), 171-183.  doi: 10.1016/j.nonrwa.2015.09.009.

[16]

X. L. Cheng, R. F. Gong, W. Han and X. Zheng, A novel coupled complex boundary method for solving inverse source problems, Inverse Problems, 30 (2014), 055002. doi: 10.1088/0266-5611/30/5/055002.

[17]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 2, Springer, Berlin, 1998.

[18]

M. Delfour and J.-P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, USA, 2001.

[19]

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using Hessian information, Control and Cybernetics, 34 (2005), 203-225. 

[20]

M. GiacominiO. Pantz and K. Trabelsi, Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators, ESAIM Control Optimisation and Calculus of Variations, 23 (2017), 977-1001.  doi: 10.1051/cocv/2016021.

[21]

R. Gong, X. Cheng and W. Han, A coupled complex boundary method for an inverse conductivity problem with one measurement, Applicable Analysis An International Journal, 96 (2017). doi: 10.1080/00036811.2016.1165215.

[22]

F. Hecht, Finite Element Library FREEFEM++., Available from: http://www.freefem.org/ff++/.

[23]

A. Henrot and M. Pierre, Variation et optimisation de formes, Springer Mathḿatiques et Applications, 48, (2005). doi: 10.1007/3-540-37689-5.

[24]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67-82.  doi: 10.1088/0266-5611/14/1/008.

[25]

V. Isakov, Inverse Problems for Partial Differential Equations, 127, Springer Science & Business Media, 2006.

[26]

V. Maz'ya and T. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Monographs and Studies in Mathematics, 23, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[27]

F. Murat and J. Simon, Sur le Contôle par Domaine Géométrique, Rapport du L.A. 189, Université de Paris VI, 1976.

[28]

J. J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687.  doi: 10.1080/01630563.1980.10120631.

[29]

J. Simon, Second variation for domain optimization problems, International Series of Numerical Mathematics, 91 (1989), 361-378. 

[30]

J. Sokolowski and J-P Zolesio, Introduction to shape optimization shape sensitivity analysis, Springer-Verlag Springer Series in Computational Mathematics, 16 (1991). doi: 10.1007/978-3-642-58106-9.

[31]

X. Zheng, X. Cheng and R. Gong, A coupled complex boundary method for parameter identification in elliptic problems, International Journal of Computer Mathematics, 97 (2020). doi: 10.1080/00207160.2019.1601181.

Figure 1.  Reconstruction of circular shape and evolution of cost function and shape gradient with respect to iterations
Figure 2.  Reconstruction of ellipse shape and evolution of cost function and shape gradient with respect to iterations
Figure 3.  Reconstruction of different shapes with medium configuration
Figure 4.  Reconstruction of more complex configurations
Figure 5.  Reconstruction of more complex shapes
Figure 6.  Reconstruction of simple shapes with noise 3$ \% $
Figure 7.  Reconstruction of different configurations with noise 3$ \% $
Figure 8.  Reconstruction of more complex shapes with noise 3$ \% $
Figure 9.  Reconstruction for more complex shapes with noise 5$ \% $
Figure 10.  Reconstruction of more complex shapes with noise 10$ \% $
Figure 11.  The comparison between the evolution of the cost function and the gradient with respect the iteration number
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