
-
Previous Article
Extended Krylov subspace methods for solving Sylvester and Stein tensor equations
- DCDS-S Home
- This Issue
-
Next Article
Shape optimization method for an inverse geometric source problem and stability at critical shape
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 |
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 [
References:
[1] |
L. Afraites, M. 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. Afraites, M. Dambrine, K. 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. Afraites, M. 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. Alessandrini, V. 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. Badra, F. 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. Caubet, M. Dambrine, D. 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. Chakib, A. Ellabib, A. 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. Chakib, A. 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. Chakib, A. 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. Giacomini, O. 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. Afraites, M. 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. Afraites, M. Dambrine, K. 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. Afraites, M. 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. Alessandrini, V. 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. Badra, F. 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. Caubet, M. Dambrine, D. 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. Chakib, A. Ellabib, A. 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. Chakib, A. 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. Chakib, A. 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. Giacomini, O. 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. |











[1] |
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 |
[2] |
Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations and Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011 |
[3] |
Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069 |
[4] |
Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825 |
[5] |
Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733 |
[6] |
Youness El Yazidi, Abdellatif ELLABIB. A new hybrid method for shape optimization with application to semiconductor equations. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021034 |
[7] |
Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks and Heterogeneous Media, 2021, 16 (1) : 1-29. doi: 10.3934/nhm.2020031 |
[8] |
John Sebastian Simon, Hirofumi Notsu. A shape optimization problem constrained with the Stokes equations to address maximization of vortices. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022003 |
[9] |
Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623 |
[10] |
Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control and Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017 |
[11] |
Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations and Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010 |
[12] |
Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055 |
[13] |
Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625 |
[14] |
Julius Fergy T. Rabago, Jerico B. Bacani. Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2683-2702. doi: 10.3934/cpaa.2018127 |
[15] |
Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1 |
[16] |
Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems and Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217 |
[17] |
Pierre-Étienne Druet. Some mathematical problems related to the second order optimal shape of a crystallisation interface. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2443-2463. doi: 10.3934/dcds.2015.35.2443 |
[18] |
Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems and Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37 |
[19] |
Lekbir Afraites, Marc Dambrine, Karsten Eppler, Djalil Kateb. Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 389-416. doi: 10.3934/dcdsb.2007.8.389 |
[20] |
Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]