January  2022, 15(1): 1-21. doi: 10.3934/dcdss.2021006

Shape optimization method for an inverse geometric source problem and stability at critical shape

Laboratory of Mathematics and Applications (LMA), Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco

* Corresponding author: lekbir.afraites@gmail.com

Received  August 2020 Revised  November 2020 Published  January 2022 Early access  January 2021

This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

Citation: Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006
References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.

[2]

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

[3]

L. AfraitesM. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement, Numer. Funct. Anal. Optim., 28 (2007), 519-551.  doi: 10.1080/01630560701381005.

[4]

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.

[5]

C. J. S. AlvesM. J. ColacoV. M. A. LeitãoN. F. M. MartinsH. R. B. Orlande and N. C. Roberty, Recovering the source term in a linear diffusion problem by the method of fundamental solutions, Inverse Probl. Sci. Eng., 16 (2008), 1005-1021.  doi: 10.1080/17415970802083243.

[6]

C. J. S. Alves, R. Mamud, N. F. M. Martins and N. C. Roberty, On inverse problems for characteristic sources in Helmholtz equations, Math. Probl. Eng., 2017 (2017), Art. ID 2472060, 16 pp. doi: 10.1155/2017/2472060.

[7]

H. Azegami and K. 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]

F. Ben Belgacem and F. Jelassi, Identifiability of surface sources from Cauchy data, Inverse Probl., 25 (2009), 075007, 14 pp. doi: 10.1088/0266-5611/25/7/075007.

[10]

K. A. BerdawoodA. NachaouiR. SaeedM. Nachaoui and F. Aboud, An aternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation, Advanced Mathematical Models & Applications, 5 (2020), 131-139. 

[11]

B. Bin-Mohsin and D. Lesnic, Reconstruction of a source domain from boundary measurements, Applied Mathematical Modelling, 45 (2017), 925-939.  doi: 10.1016/j.apm.2017.01.021.

[12]

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.

[13]

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 (26pp). doi: 10.1088/0266-5611/29/11/115011.

[14]

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.

[15]

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.

[16]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[17]

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

[18]

A. El Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Probl., 14 (1998), 883-891.  doi: 10.1088/0266-5611/14/4/008.

[19]

A. El Badia and T. Nara, An inverse source problem for Helmholtz's equation from the Cauchy data with a single wave number, Inverse Problems, 27 (2011), 105001. doi: 10.1088/0266-5611/27/10/105001.

[20]

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

[21]

R. Fiorenza, H$\ddot{o}$lder and Locally H$\ddot{o}$lder Continuous Functions, and Open Sets of Class ${{C}^{\hat{\ }}}k,{{C}^{\hat{\ }}}\{k,lambda\} $, Birkh$\ddot{a}$user, 2017. doi: 10.1007/978-3-319-47940-8.

[22]

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.

[23] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923. 
[24]

M. H$\ddot{a}$m$\ddot{a}$l$\ddot{a}$inenR. HariR. J. IlmoniemiJ. Knuutila and O. V. Lounasmaa, Magnetoencephalography theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, 65 (1993), 413-497. 

[25]

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

[26]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, volume 48 of Mathématiques & Applications (Berlin)[Mathematics & Applications], Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[27]

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.

[28]

F. Hettlich and W. Rundell, Identification of a discontinuity source in the heat equation, Inverse Problems, 17 (2001), 1465-1482.  doi: 10.1088/0266-5611/17/5/315.

[29]

Y. C. HonM. Li and Y. A. Melnikov, Inverse source identification by Green's function, Eng. Anal. Bound. Elem., 34 (2010), 352-358.  doi: 10.1016/j.enganabound.2009.09.009.

[30]

M. Hrizi and M. Hassine, One–iteration reconstruction algorithm for geometric inverse source problem, Journal of Elliptic and Parabolic Equations, 4 (2018), 177-205.  doi: 10.1007/s41808-018-0015-4.

[31]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, Vol. 127, Springer–Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[32]

R. Kress and W. Rundell, A nonlinear integral equation and an iterative algorithm for an inverse source problem, Journal of Integral Equations and Applications, 27 (2015), 179-197.  doi: 10.1216/JIE-2015-27-2-179.

[33]

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

[34]

V. Michel and A. S. Fokas, A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods, Inverse Probl., 24 (2008), 1-25.  doi: 10.1088/0266-5611/24/4/045019.

[35]

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

[36]

M. NachaouiA. Chakib and A. Nachaoui, An efficient evolutionary algorithm for a shape optimization problem, Appl. Comput. Math., 19 (2020), 220-244. 

[37]

A. Nachaoui, M. Nachaoui, A. Chakib and M. A. Hilal, Some novel numerical techniques for an inverse Cauchy problem, Journal of Computational and Applied Mathematics, (2020), 113030. doi: 10.1016/j.cam.2020.113030.

[38]

P. Novikov, Sur le probleme inverse du potentiel, Dokl. Akad. Nauk., 18 (1938), 165-168. 

[39]

N. C. Roberty and C. J. Alves, On the identification of star–shape sources from boundary measurements using a reciprocity functional, Inverse Problems in Science and Engineering, 17 (2009), 187-202.  doi: 10.1080/17415970802082799.

[40]

J. R. Roche and J. Sokolowski, Numerical methods for shape identification problems, Control and Cybernetics, 25 (1996), 867-894. 

[41]

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.

[42]

J. Simon, Second variations for domain optimization problems, Control Theory of Distributed Parameter Systems and Applications, 91 (1989), 361-378. 

show all references

References:
[1]

B. AbdelazizA. El Badia and A. El Hajj, Direct algorithm for multipolar sources reconstruction, J. Math. Anal. Appl., 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013.

[2]

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

[3]

L. AfraitesM. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement, Numer. Funct. Anal. Optim., 28 (2007), 519-551.  doi: 10.1080/01630560701381005.

[4]

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.

[5]

C. J. S. AlvesM. J. ColacoV. M. A. LeitãoN. F. M. MartinsH. R. B. Orlande and N. C. Roberty, Recovering the source term in a linear diffusion problem by the method of fundamental solutions, Inverse Probl. Sci. Eng., 16 (2008), 1005-1021.  doi: 10.1080/17415970802083243.

[6]

C. J. S. Alves, R. Mamud, N. F. M. Martins and N. C. Roberty, On inverse problems for characteristic sources in Helmholtz equations, Math. Probl. Eng., 2017 (2017), Art. ID 2472060, 16 pp. doi: 10.1155/2017/2472060.

[7]

H. Azegami and K. 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]

F. Ben Belgacem and F. Jelassi, Identifiability of surface sources from Cauchy data, Inverse Probl., 25 (2009), 075007, 14 pp. doi: 10.1088/0266-5611/25/7/075007.

[10]

K. A. BerdawoodA. NachaouiR. SaeedM. Nachaoui and F. Aboud, An aternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation, Advanced Mathematical Models & Applications, 5 (2020), 131-139. 

[11]

B. Bin-Mohsin and D. Lesnic, Reconstruction of a source domain from boundary measurements, Applied Mathematical Modelling, 45 (2017), 925-939.  doi: 10.1016/j.apm.2017.01.021.

[12]

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.

[13]

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 (26pp). doi: 10.1088/0266-5611/29/11/115011.

[14]

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.

[15]

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.

[16]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[17]

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

[18]

A. El Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Probl., 14 (1998), 883-891.  doi: 10.1088/0266-5611/14/4/008.

[19]

A. El Badia and T. Nara, An inverse source problem for Helmholtz's equation from the Cauchy data with a single wave number, Inverse Problems, 27 (2011), 105001. doi: 10.1088/0266-5611/27/10/105001.

[20]

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

[21]

R. Fiorenza, H$\ddot{o}$lder and Locally H$\ddot{o}$lder Continuous Functions, and Open Sets of Class ${{C}^{\hat{\ }}}k,{{C}^{\hat{\ }}}\{k,lambda\} $, Birkh$\ddot{a}$user, 2017. doi: 10.1007/978-3-319-47940-8.

[22]

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.

[23] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923. 
[24]

M. H$\ddot{a}$m$\ddot{a}$l$\ddot{a}$inenR. HariR. J. IlmoniemiJ. Knuutila and O. V. Lounasmaa, Magnetoencephalography theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, 65 (1993), 413-497. 

[25]

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

[26]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, volume 48 of Mathématiques & Applications (Berlin)[Mathematics & Applications], Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[27]

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.

[28]

F. Hettlich and W. Rundell, Identification of a discontinuity source in the heat equation, Inverse Problems, 17 (2001), 1465-1482.  doi: 10.1088/0266-5611/17/5/315.

[29]

Y. C. HonM. Li and Y. A. Melnikov, Inverse source identification by Green's function, Eng. Anal. Bound. Elem., 34 (2010), 352-358.  doi: 10.1016/j.enganabound.2009.09.009.

[30]

M. Hrizi and M. Hassine, One–iteration reconstruction algorithm for geometric inverse source problem, Journal of Elliptic and Parabolic Equations, 4 (2018), 177-205.  doi: 10.1007/s41808-018-0015-4.

[31]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, Vol. 127, Springer–Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[32]

R. Kress and W. Rundell, A nonlinear integral equation and an iterative algorithm for an inverse source problem, Journal of Integral Equations and Applications, 27 (2015), 179-197.  doi: 10.1216/JIE-2015-27-2-179.

[33]

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

[34]

V. Michel and A. S. Fokas, A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods, Inverse Probl., 24 (2008), 1-25.  doi: 10.1088/0266-5611/24/4/045019.

[35]

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

[36]

M. NachaouiA. Chakib and A. Nachaoui, An efficient evolutionary algorithm for a shape optimization problem, Appl. Comput. Math., 19 (2020), 220-244. 

[37]

A. Nachaoui, M. Nachaoui, A. Chakib and M. A. Hilal, Some novel numerical techniques for an inverse Cauchy problem, Journal of Computational and Applied Mathematics, (2020), 113030. doi: 10.1016/j.cam.2020.113030.

[38]

P. Novikov, Sur le probleme inverse du potentiel, Dokl. Akad. Nauk., 18 (1938), 165-168. 

[39]

N. C. Roberty and C. J. Alves, On the identification of star–shape sources from boundary measurements using a reciprocity functional, Inverse Problems in Science and Engineering, 17 (2009), 187-202.  doi: 10.1080/17415970802082799.

[40]

J. R. Roche and J. Sokolowski, Numerical methods for shape identification problems, Control and Cybernetics, 25 (1996), 867-894. 

[41]

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.

[42]

J. Simon, Second variations for domain optimization problems, Control Theory of Distributed Parameter Systems and Applications, 91 (1989), 361-378. 

Figure 1.  Reconstruction of the first domain by Kohn-Vogelius and least squares methods
Figure 2.  Evolution of the Kohn-Vogelius and least squares cost functions and their gradients according to number of iterations
Figure 3.  Reconstruction of the second domain by Kohn-Vogelius and least squares methods
Figure 4.  Evolution of the Kohn-Vogelius and least squares cost functions, and their associated gradients according to the number of iterations
Figure 5.  Reconstruction of the third domain by Kohn-Vogelius and least squares methods
Figure 6.  Reconstruction of the fourth domain by Kohn-Vogelius and least squares methods
Figure 7.  Reconstruction of the fifth domain by Kohn-Vogelius and least squares methods
Figure 8.  Reconstruction of the first domain by Kohn-Vogelius and least squares methods with noise
Figure 9.  Reconstruction of the second domain by Kohn-Vogelius and least squares methods with noise
Figure 10.  Reconstruction of the third domain by Kohn-Vogelius and least squares methods with noise
Figure 11.  Reconstruction of the fourth domain by Kohn-Vogelius and least squares methods with noise
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