This work deals with a geometric inverse source problem. It consists in recovering the characteristic function of an unknown inclusion based on boundary measurements. We propose a new reconstruction method based on the CCBM and the shape gradient method, the inverse problem is formulated as a shape optimization one, corresponding to a coupled complex boundary state problem. Well posedness and existence results are presented. A computed expression for the shape gradient is used to implement a gradient algorithm. The efficiency and accuracy of the reconstruction algorithm are illustrated by some numerical results, and a comparison between CCBM, Least-squares and Kohn-Vogeluis methods is presented.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/ro/2022168
Keywords: Coupled complex boundary method, inverse geometric source problem, shape optimization, shape derivative, adjoint method
@article{RO_2022__56_5_3689_0,
author = {Afraites, Lekbir and Masnaoui, Chorouk and Nachaoui, Mourad},
title = {Coupled complex boundary method for a geometric inverse source problem},
journal = {RAIRO. Operations Research},
pages = {3689--3709},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/ro/2022168},
mrnumber = {4502918},
zbl = {1518.65058},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022168/}
}
TY - JOUR AU - Afraites, Lekbir AU - Masnaoui, Chorouk AU - Nachaoui, Mourad TI - Coupled complex boundary method for a geometric inverse source problem JO - RAIRO. Operations Research PY - 2022 SP - 3689 EP - 3709 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022168/ DO - 10.1051/ro/2022168 LA - en ID - RO_2022__56_5_3689_0 ER -
%0 Journal Article %A Afraites, Lekbir %A Masnaoui, Chorouk %A Nachaoui, Mourad %T Coupled complex boundary method for a geometric inverse source problem %J RAIRO. Operations Research %D 2022 %P 3689-3709 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022168/ %R 10.1051/ro/2022168 %G en %F RO_2022__56_5_3689_0
Afraites, Lekbir; Masnaoui, Chorouk; Nachaoui, Mourad. Coupled complex boundary method for a geometric inverse source problem. RAIRO. Operations Research, Tome 56 (2022) no. 5, pp. 3689-3709. doi: 10.1051/ro/2022168
[1] , A new coupled complex boundary method (ccbm) for an inverse obstacle problem. Discrete Contin. Dyn. Syst.-S 15 (2022) 23. | MR | Zbl | DOI
[2] and , Parameters identification in the mathematical model of immune competition cells. J. Inverse Ill-Posed Probl. 23 (2015) 323–337. | MR | Zbl | DOI
[3] , , and , Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discrete Contin. Dyn. Syst.-B 8 (2007) 389. | MR | Zbl
[4] , and , Shape methods for the transmission problem with a single measurement. Numer. Funct. Anal. Optim. 28 (2007) 519–551. | MR | Zbl | DOI
[5] , and , Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete Contin. Dyn. Syst.-S 15 (2022) 1. | MR | Zbl | DOI
[6] , , and , On inverse problems for characteristic sources in helmholtz equations. Math. Probl. Eng. 2017 (2017). | MR | Zbl
[7] and , A smoothing method for shape optimization: traction method using the robin condition. Int. J. Comput. Methods 3 (2006) 21–33. | MR | Zbl | DOI
[8] , , and , A kohn-vogelius formulation to detect an obstacle immersed in a fluid. Inverse Probl. Imaging 7 (2013) 123. | MR | Zbl | DOI
[9] , , and , A novel coupled complex boundary method for solving inverse source problems. Inverse Probl. 30 (2014) 055002. | MR | Zbl | DOI
[10] , Lectures on cauchy’s problem in linear partial differential equations. Math. Gaz. 12 (1924) 173–174.
[11] and , Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications, Vol. 3. Springer Science & Business Media (1999). | MR
[12] and , Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. SIAM (2011). | MR | Zbl | DOI
[13] and , Some remarks on the problem of source identification from boundary measurements. Inverse Probl. 14 (1998) 883. | MR | Zbl | DOI
[14] and , An inverse source problem for helmholtz’s equation from the cauchy data with a single wave number. Inverse Probl. 27 (2011) 105001. | MR | Zbl | DOI
[15] and , A regularized newton method in electrical impedance tomography using shape hessian information. Control Cybern. 34 (2005) 203. | MR | Zbl
[16] , and , Certified descent algorithm for shape optimization driven by fully-computable a posteriori error estimators. ESAIM: Control Optim. Calc. Var. 23 (2017) 977–1001. | MR | Zbl | Numdam
[17] , and , A coupled complex boundary method for an inverse conductivity problem with one measurement. Appl. Anal. 96 (2017) 869–885. | MR | Zbl | DOI
[18] and , Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, 2003. | MR | Zbl | DOI
[19] , Finite Element Library Freefem++.
[20] and , One-iteration reconstruction algorithm for geometric inverse source problem. J. Elliptic Parabol. Equ. 4 (2018) 177–205. | MR | Zbl | DOI
[21] and , A nonlinear integral equation and an iterative algorithm for an inverse source problem. J. Integral Equ. Appl. 27 (2015) 179–198. | MR | Zbl | DOI
[22] and , A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods. Inverse Probl. 24 (2008) 045019. | MR | Zbl | DOI
[23] and , Sur le contrôle par un domaine géométrique. Rapport du LA 189 (1976) 76015.
[24] , , and , An optimal control approach for determining the source term in fractional diffusion equation by different cost functionals. Appl. Numer. Math. 181 (2022) 647–664. | MR | Zbl | DOI
[25] and , Shape Variation and Optimization: A Geometrical Analysis (2018). | MR | Zbl
[26] , On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems. Preprint (2022). | arXiv | MR | Zbl
[27] and , On the identification of star-shape sources from boundary measurements using a reciprocity functional. Inverse Probl. Sci. Eng. 17 (2009) 187–202. | MR | Zbl | DOI
[28] and , Shapes and Geometries: Analysis, Differential Calculus, and Optimization. SIAM (2001). | MR | Zbl
Cité par Sources :
