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Image segmentation via $ L_1 $ Monge-Kantorovich problem
Nash strategies for the inverse inclusion Cauchy-Stokes problem
1. | Université Cȏte d'Azur, Inria, CNRS, LJAD, UMR 7351, Parc Valrose, Nice 06108, France |
2. | Université de Tunis El Manar, Ecole Nationale d'Ingénieurs de Tunis, LAMSIN, BP 37, 1002 Tunis Belvedere, Tunisia |
We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.
References:
[1] |
R. Aboulaich, A. Ben Abda and M. Kallel,
A control type method for solving the cauchy-stokes problem, Applied Mathematical Modelling, 37 (2013), 4295-4304.
doi: 10.1016/j.apm.2012.09.014. |
[2] |
G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the cauchy problem for elliptic equations, Inverse problems, 25 (2009), 123004. Google Scholar |
[3] |
G. Allaire, F. Jouve and A.-M. Toader,
Structural optimization using sensitivity analysis and a level-set method, Journal of computational physics, 194 (2004), 363-393.
doi: 10.1016/j.jcp.2003.09.032. |
[4] |
C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega,
Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552.
doi: 10.1088/0266-5611/21/5/003. |
[5] |
C. Alves, R. Kress and A. Silvestre,
Integral equations for an inverse boundary value problem for the two-dimensional stokes equations, Journal of Inverse and Ill-posed Problems Jiip, 15 (2007), 461-481.
doi: 10.1515/jiip.2007.026. |
[6] |
S. Andrieux and A. Ben Abda,
The reciprocity gap: A general concept for flaws identification problems, Mechanics research communications, 20 (1993), 415-420.
doi: 10.1016/0093-6413(93)90032-J. |
[7] |
S. Andrieux, T. Baranger and A. Ben Abda,
Solving cauchy problems by minimizing an energy-like functional, Inverse problems, 22 (2006), 115-133.
doi: 10.1088/0266-5611/22/1/007. |
[8] |
H. Attouch, J. Bolte and P. Redont,
Alternating proximal algorithms for weakly coupled convex minimization problems. applications to dynamical games and pde's, J. Convex Anal., 15 (2008), 485-506.
|
[9] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147, Springer Science & Business Media, 2006. |
[10] |
A. Ballerini, Stable determination of an immersed body in a stationary stokes fluid, Inverse Problems, 26 (2010), 125015(25pp).
doi: 10.1088/0266-5611/26/12/125015. |
[11] |
G. Bastay, T. Johansson, V. Kozlov and D. Lesnic,
An alternating method for the stationary stokes system, ZAMM, 86 (2006), 268-280.
doi: 10.1002/zamm.200410238. |
[12] |
L. Bourgeois,
A mixed formulation of quasi-reversibility to solve the cauchy problem for laplace's equation, Inverse Problems, 21 (2005), 1087-1104.
doi: 10.1088/0266-5611/21/3/018. |
[13] |
L. Bourgeois and J. Dardé,
A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4 (2010), 351-377.
doi: 10.3934/ipi.2010.4.351. |
[14] |
L. Bourgeois and J. Dardé,
The exterior approach to solve the inverse obstacle problem for the stokes system, Inverse Problems and Imaging, 8 (2014), 23-51.
doi: 10.3934/ipi.2014.8.23. |
[15] |
F. Caubet, M. Badra 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. |
[16] |
F. Caubet, Détection d'un Objet Immergé dans un Fluide, PhD thesis, Université de Pau, 2012. Google Scholar |
[17] |
F. Caubet, C. Conca and M. Godoy,
On the detection of several obstacles in 2d stokes flow: Topological sensitivity and combination with shape derivatives, Inverse Problems and Imaging, 10 (2016), 327-367.
doi: 10.3934/ipi.2016003. |
[18] |
F. Caubet, J. Dardé and M. Godoy, On the data completion problem and the inverse obstacle problem with partial cauchy data for laplace's equation, ESAIM: Control, Optimisation and Calculus of Variations, 2017.
doi: 10.1051/cocv/2017056. |
[19] |
R. Chamekh, A. Habbal, M. Kallel and N. Zemzemi, A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology, Mathematical Modelling of Natural Phenomena, 14 (2019), Art. 201, 15 pp.
doi: 10.1051/mmnp/2018059. |
[20] |
D. Chenais,
Optimal design of midsurface of shells: Differentiability proof and sensitivity computation, Applied Mathematics and Optimization, 16 (1987), 93-133.
doi: 10.1007/BF01442187. |
[21] |
A. Cimetiere, F. Delvare, M. Jaoua and F. Pons,
Solution of the cauchy problem using iterated tikhonov regularization, Inverse Problems, 17 (2001), 553-570.
doi: 10.1088/0266-5611/17/3/313. |
[22] |
P. Constantin and C. Foias, Navier-stokes Equations, University of Chicago Press, 1988.
![]() |
[23] |
X.-B. Duan, Y.-C. Ma and R. Zhang,
Shape-topology optimization of stokes flow via variational level set method, Applied Mathematics and Computation, 202 (2008), 200-209.
doi: 10.1016/j.amc.2008.02.014. |
[24] |
C. Fabre and G. Lebeau,
Unique continuation property of solutions of the stokes equation, Communications in Partial Differential Equations, 21 (1996), 573-596.
doi: 10.1080/03605309608821198. |
[25] |
R. Falk and P. Monk,
Logarithmic convexity for discrete harmonic functions and the approximation of the cauchy problem for poisson's equation, Mathematics of Computation, 47 (1986), 135-149.
doi: 10.2307/2008085. |
[26] |
P. C. Franzone and E. Magenes,
On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538.
doi: 10.1007/BF02576643. |
[27] |
G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Science & Business Media, 2011.
doi: 10.1007/978-0-387-09620-9. |
[28] |
A. Habbal and M. Kallel,
Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems, SIAM Journal on Control and Optimization, 51 (2013), 4066-4083.
doi: 10.1137/120869808. |
[29] |
J. Hadamard, The Cauchy Problem and the Linear Hyperbolic Partial Differential Equations, Dover, New York, 1953. Google Scholar |
[30] |
F. Hecht,
New development in freefem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
[31] |
T. Johansson and D. Lesnic,
Reconstruction of a stationary flow from incomplete boundary data using iterative methods, European Journal of Applied Mathematics, 17 (2006), 651-663.
doi: 10.1017/S0956792507006791. |
[32] |
M. Kallel, M. Moakher and A. Theljani,
The cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting, Inverse Problems and Imaging, 9 (2015), 853-874.
doi: 10.3934/ipi.2015.9.853. |
[33] |
S. Katz and C. F. Landefeld (eds.), On the detection, Behaviour and Control of Inclusions in Liquid Metals, Springer US, Boston, MA, 1988,447–466. Google Scholar |
[34] |
G. Kawchuk, J. Fryer, J. L. Jaremko, H. Zeng, L. Rowe and R. Thompson,
Real-time visualization of joint cavitation, PloS one, 10 (2015), e0119470.
doi: 10.1371/journal.pone.0119470. |
[35] |
R. V. Kohn and M. Vogelius,
Relaxation of a variational method for impedance computed tomography, Communications on Pure and Applied Mathematics, 40 (1987), 745-777.
doi: 10.1002/cpa.3160400605. |
[36] |
V. Kozlov, V. Maz'ya and A. Fomin,
An iterative method for solving the cauchy problems for elliptic equations, Comput. Math. Phys., 31 (1991), 45-52.
|
[37] |
S. Li and T. Bașar,
Distributed algorithms for the computation of noncooperative equilibria, Automatica, 23 (1987), 523-533.
doi: 10.1016/0005-1098(87)90081-1. |
[38] |
C.-W. Lo, S.-F. Chen, C.-P. Li and P.-C. Lu,
Cavitation phenomena in mechanical heart valves: Studied by using a physical impinging rod system, Annals of biomedical engineering, 38 (2010), 3162-3172.
doi: 10.1007/s10439-010-0070-y. |
[39] |
R. Malladi, J. A. Sethian and B. C. Vemuri,
Shape modeling with front propagation: A level set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17 (1995), 158-175.
doi: 10.1109/34.368173. |
[40] |
B. Rousselet,
Note on the design differentiability of the static response of elastic structures, Journal of Structural Mechanics, 10 (1982), 353-358.
doi: 10.1080/03601218208907417. |
[41] |
F. Santosa,
A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1996), 17-33.
doi: 10.1051/cocv:1996101. |
[42] |
Y. Son and K. B. Migler,
Cavitation of polyethylene during extrusion processing instabilities, Journal of Polymer Science Part B: Polymer Physics, 40 (2002), 2791-2799.
doi: 10.1002/polb.10314. |
[43] |
T. Stieger, H. Agha, M. Schoen, M. G. Mazza and A. Sengupta,
Hydrodynamic cavitation in stokes flow of anisotropic fluids, Nature communications, 8 (2017), 15550.
doi: 10.1038/ncomms15550. |
[44] |
M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational physics, 114 (1994), 146-159. Google Scholar |
[45] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001.
doi: 10.1090/chel/343. |
show all references
References:
[1] |
R. Aboulaich, A. Ben Abda and M. Kallel,
A control type method for solving the cauchy-stokes problem, Applied Mathematical Modelling, 37 (2013), 4295-4304.
doi: 10.1016/j.apm.2012.09.014. |
[2] |
G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the cauchy problem for elliptic equations, Inverse problems, 25 (2009), 123004. Google Scholar |
[3] |
G. Allaire, F. Jouve and A.-M. Toader,
Structural optimization using sensitivity analysis and a level-set method, Journal of computational physics, 194 (2004), 363-393.
doi: 10.1016/j.jcp.2003.09.032. |
[4] |
C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega,
Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552.
doi: 10.1088/0266-5611/21/5/003. |
[5] |
C. Alves, R. Kress and A. Silvestre,
Integral equations for an inverse boundary value problem for the two-dimensional stokes equations, Journal of Inverse and Ill-posed Problems Jiip, 15 (2007), 461-481.
doi: 10.1515/jiip.2007.026. |
[6] |
S. Andrieux and A. Ben Abda,
The reciprocity gap: A general concept for flaws identification problems, Mechanics research communications, 20 (1993), 415-420.
doi: 10.1016/0093-6413(93)90032-J. |
[7] |
S. Andrieux, T. Baranger and A. Ben Abda,
Solving cauchy problems by minimizing an energy-like functional, Inverse problems, 22 (2006), 115-133.
doi: 10.1088/0266-5611/22/1/007. |
[8] |
H. Attouch, J. Bolte and P. Redont,
Alternating proximal algorithms for weakly coupled convex minimization problems. applications to dynamical games and pde's, J. Convex Anal., 15 (2008), 485-506.
|
[9] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147, Springer Science & Business Media, 2006. |
[10] |
A. Ballerini, Stable determination of an immersed body in a stationary stokes fluid, Inverse Problems, 26 (2010), 125015(25pp).
doi: 10.1088/0266-5611/26/12/125015. |
[11] |
G. Bastay, T. Johansson, V. Kozlov and D. Lesnic,
An alternating method for the stationary stokes system, ZAMM, 86 (2006), 268-280.
doi: 10.1002/zamm.200410238. |
[12] |
L. Bourgeois,
A mixed formulation of quasi-reversibility to solve the cauchy problem for laplace's equation, Inverse Problems, 21 (2005), 1087-1104.
doi: 10.1088/0266-5611/21/3/018. |
[13] |
L. Bourgeois and J. Dardé,
A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4 (2010), 351-377.
doi: 10.3934/ipi.2010.4.351. |
[14] |
L. Bourgeois and J. Dardé,
The exterior approach to solve the inverse obstacle problem for the stokes system, Inverse Problems and Imaging, 8 (2014), 23-51.
doi: 10.3934/ipi.2014.8.23. |
[15] |
F. Caubet, M. Badra 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. |
[16] |
F. Caubet, Détection d'un Objet Immergé dans un Fluide, PhD thesis, Université de Pau, 2012. Google Scholar |
[17] |
F. Caubet, C. Conca and M. Godoy,
On the detection of several obstacles in 2d stokes flow: Topological sensitivity and combination with shape derivatives, Inverse Problems and Imaging, 10 (2016), 327-367.
doi: 10.3934/ipi.2016003. |
[18] |
F. Caubet, J. Dardé and M. Godoy, On the data completion problem and the inverse obstacle problem with partial cauchy data for laplace's equation, ESAIM: Control, Optimisation and Calculus of Variations, 2017.
doi: 10.1051/cocv/2017056. |
[19] |
R. Chamekh, A. Habbal, M. Kallel and N. Zemzemi, A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology, Mathematical Modelling of Natural Phenomena, 14 (2019), Art. 201, 15 pp.
doi: 10.1051/mmnp/2018059. |
[20] |
D. Chenais,
Optimal design of midsurface of shells: Differentiability proof and sensitivity computation, Applied Mathematics and Optimization, 16 (1987), 93-133.
doi: 10.1007/BF01442187. |
[21] |
A. Cimetiere, F. Delvare, M. Jaoua and F. Pons,
Solution of the cauchy problem using iterated tikhonov regularization, Inverse Problems, 17 (2001), 553-570.
doi: 10.1088/0266-5611/17/3/313. |
[22] |
P. Constantin and C. Foias, Navier-stokes Equations, University of Chicago Press, 1988.
![]() |
[23] |
X.-B. Duan, Y.-C. Ma and R. Zhang,
Shape-topology optimization of stokes flow via variational level set method, Applied Mathematics and Computation, 202 (2008), 200-209.
doi: 10.1016/j.amc.2008.02.014. |
[24] |
C. Fabre and G. Lebeau,
Unique continuation property of solutions of the stokes equation, Communications in Partial Differential Equations, 21 (1996), 573-596.
doi: 10.1080/03605309608821198. |
[25] |
R. Falk and P. Monk,
Logarithmic convexity for discrete harmonic functions and the approximation of the cauchy problem for poisson's equation, Mathematics of Computation, 47 (1986), 135-149.
doi: 10.2307/2008085. |
[26] |
P. C. Franzone and E. Magenes,
On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538.
doi: 10.1007/BF02576643. |
[27] |
G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Science & Business Media, 2011.
doi: 10.1007/978-0-387-09620-9. |
[28] |
A. Habbal and M. Kallel,
Neumann-dirichlet nash strategies for the solution of elliptic cauchy problems, SIAM Journal on Control and Optimization, 51 (2013), 4066-4083.
doi: 10.1137/120869808. |
[29] |
J. Hadamard, The Cauchy Problem and the Linear Hyperbolic Partial Differential Equations, Dover, New York, 1953. Google Scholar |
[30] |
F. Hecht,
New development in freefem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
[31] |
T. Johansson and D. Lesnic,
Reconstruction of a stationary flow from incomplete boundary data using iterative methods, European Journal of Applied Mathematics, 17 (2006), 651-663.
doi: 10.1017/S0956792507006791. |
[32] |
M. Kallel, M. Moakher and A. Theljani,
The cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting, Inverse Problems and Imaging, 9 (2015), 853-874.
doi: 10.3934/ipi.2015.9.853. |
[33] |
S. Katz and C. F. Landefeld (eds.), On the detection, Behaviour and Control of Inclusions in Liquid Metals, Springer US, Boston, MA, 1988,447–466. Google Scholar |
[34] |
G. Kawchuk, J. Fryer, J. L. Jaremko, H. Zeng, L. Rowe and R. Thompson,
Real-time visualization of joint cavitation, PloS one, 10 (2015), e0119470.
doi: 10.1371/journal.pone.0119470. |
[35] |
R. V. Kohn and M. Vogelius,
Relaxation of a variational method for impedance computed tomography, Communications on Pure and Applied Mathematics, 40 (1987), 745-777.
doi: 10.1002/cpa.3160400605. |
[36] |
V. Kozlov, V. Maz'ya and A. Fomin,
An iterative method for solving the cauchy problems for elliptic equations, Comput. Math. Phys., 31 (1991), 45-52.
|
[37] |
S. Li and T. Bașar,
Distributed algorithms for the computation of noncooperative equilibria, Automatica, 23 (1987), 523-533.
doi: 10.1016/0005-1098(87)90081-1. |
[38] |
C.-W. Lo, S.-F. Chen, C.-P. Li and P.-C. Lu,
Cavitation phenomena in mechanical heart valves: Studied by using a physical impinging rod system, Annals of biomedical engineering, 38 (2010), 3162-3172.
doi: 10.1007/s10439-010-0070-y. |
[39] |
R. Malladi, J. A. Sethian and B. C. Vemuri,
Shape modeling with front propagation: A level set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, 17 (1995), 158-175.
doi: 10.1109/34.368173. |
[40] |
B. Rousselet,
Note on the design differentiability of the static response of elastic structures, Journal of Structural Mechanics, 10 (1982), 353-358.
doi: 10.1080/03601218208907417. |
[41] |
F. Santosa,
A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1996), 17-33.
doi: 10.1051/cocv:1996101. |
[42] |
Y. Son and K. B. Migler,
Cavitation of polyethylene during extrusion processing instabilities, Journal of Polymer Science Part B: Polymer Physics, 40 (2002), 2791-2799.
doi: 10.1002/polb.10314. |
[43] |
T. Stieger, H. Agha, M. Schoen, M. G. Mazza and A. Sengupta,
Hydrodynamic cavitation in stokes flow of anisotropic fluids, Nature communications, 8 (2017), 15550.
doi: 10.1038/ncomms15550. |
[44] |
M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational physics, 114 (1994), 146-159. Google Scholar |
[45] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001.
doi: 10.1090/chel/343. |












Noise level | ||||
0.010 | 0.015 | 0.039 | 0.063 | |
0.031 | 0.033 | 0.051 | 0.07 | |
0.032 | 0.043 | 0.066 | 0.117 |
Noise level | ||||
0.010 | 0.015 | 0.039 | 0.063 | |
0.031 | 0.033 | 0.051 | 0.07 | |
0.032 | 0.043 | 0.066 | 0.117 |
Noise level | ||||
0.042 | 0.044 | 0.046 | 0.08 | |
0.095 | 0.1 | 0.13 | 0.16 | |
0.099 | 0.11 | 0.13 | 0.15 |
Noise level | ||||
0.042 | 0.044 | 0.046 | 0.08 | |
0.095 | 0.1 | 0.13 | 0.16 | |
0.099 | 0.11 | 0.13 | 0.15 |
Case A | Classical algorithm | Algorithm 2 |
0.058 | 0.033 | |
0.106 | 0.032 | |
0.358 | 0.140 | |
Case C | Classical algorithm | Algorithm 2 |
|
0.067 | 0.058 |
|
0.208 | 0.122 |
|
0.566 | 0.167 |
Case A | Classical algorithm | Algorithm 2 |
0.058 | 0.033 | |
0.106 | 0.032 | |
0.358 | 0.140 | |
Case C | Classical algorithm | Algorithm 2 |
|
0.067 | 0.058 |
|
0.208 | 0.122 |
|
0.566 | 0.167 |
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