2010, 4(3): 351-377. doi: 10.3934/ipi.2010.4.351

A quasi-reversibility approach to solve the inverse obstacle problem

1. 

Laboratoire POEMS, ENSTA, 32, Boulevard Victor, 75739 Paris Cedex 15, France

Received  May 2009 Revised  November 2009 Published  July 2010

We introduce a new approach based on the coupling of the method of quasi-reversibility and a simple level set method in order to solve the inverse obstacle problem with Dirichlet boundary condition. We provide a theoretical justification of our approach and illustrate its feasibility with the help of numerical experiments in $2D$.
Citation: Laurent Bourgeois, Jérémi Dardé. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Problems & Imaging, 2010, 4 (3) : 351-377. doi: 10.3934/ipi.2010.4.351
[1]

Jérémi Dardé. Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Problems & Imaging, 2016, 10 (2) : 379-407. doi: 10.3934/ipi.2016005

[2]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479

[3]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[4]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[5]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[6]

Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225

[7]

Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems & Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971

[8]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[9]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[10]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

[11]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[12]

Kim Knudsen, Matti Lassas, Jennifer L. Mueller, Samuli Siltanen. Regularized D-bar method for the inverse conductivity problem. Inverse Problems & Imaging, 2009, 3 (4) : 599-624. doi: 10.3934/ipi.2009.3.599

[13]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[14]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[15]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211

[16]

Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems & Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23

[17]

Cheng-Dar Liou. Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method". Journal of Industrial & Management Optimization, 2012, 8 (3) : 727-732. doi: 10.3934/jimo.2012.8.727

[18]

Kuo-Hsiung Wang, Chuen-Wen Liao, Tseng-Chang Yen. Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method. Journal of Industrial & Management Optimization, 2010, 6 (1) : 197-207. doi: 10.3934/jimo.2010.6.197

[19]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006

[20]

Masaru Ikehata, Mishio Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method. Inverse Problems & Imaging, 2018, 12 (5) : 1173-1198. doi: 10.3934/ipi.2018049

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]