American Institute of Mathematical Sciences

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August  2019, 13(4): 827-862. doi: 10.3934/ipi.2019038

Nash strategies for the inverse inclusion Cauchy-Stokes problem

Abderrahmane Habbal 1,, , Moez Kallel 2, and  Marwa Ouni 2,

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

* Corresponding author: A. Habbal

Received  October 2018 Revised  March 2019 Published  May 2019

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.

Keywords: Data completion, Cauchy-Stokes problem, shape identification, Nash games.
Mathematics Subject Classification: Primary: 49J20, 65K10; Secondary: 65N06, 90C30.
Citation: Abderrahmane Habbal, Moez Kallel, Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem. Inverse Problems and Imaging, 2019, 13 (4) : 827-862. doi: 10.3934/ipi.2019038
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. 

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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. 

[45]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.

Figure 1.  An example of the geometric configuration of the problem : the whole domain including cavities is denoted by $ \Omega $. It contains an inclusion $ \omega^* $. The boundary of $ \Omega $ is composed of $ \Gamma_{\!\! c} $, an accessible part where over-specified data are available, and an inaccessible part $ \Gamma_{\!\!i } $ where the data are missing
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Figure 2.  Different situations
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Figure 5.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_1$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$
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Figure 6.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$
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Figure 7.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with noise levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$
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Figure 8.  Test case B. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$
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Figure 9.  Test case B. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$
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Figure 10.  Test case C. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$
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Figure 11.  Test case C. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$
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Figure 3.  Test case A. (Left) sensitivity of the reconstruction w.r.t. the mesh size (on abscissae : the number of F.E. nodes on the boundary $\partial \Omega$). (Right) sensitivity of the reconstruction w.r.t. the distance to the inaccessible boundary $ \Gamma_{i }$ (on abscissae : the distance of the center of the circular inclusion from $ \Gamma_{i }$)
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Figure 4.  Test case C. (Left) Mesh used for solving the direct problem with the P1bubble-P1 finite element, in order to construct the synthetic data. (Right) Mesh used for solving the coupled inverse problem with P2-P1 finite element, using the P1 bubble-P1 synthetic data
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Figure 12.  Assessing Inverse-Crime-Free reconstruction. Test case C. Top: initial and optimal contour. Middle: the two components of the velocity on $\Gamma_i$. Bottom: the two components of the normal stress on $\Gamma_i$ ($err_D = 0.0615048,$ $err_N = 0.124296,$ and $err_O = 0.113156)$
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Table 1.  Test-case A. $L^2$ relative errors on missing data on $\Gamma_i$ (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape of the inclusion for various noise levels
Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
$err_D$ 0.010 0.015 0.039 0.063
$err_N$ 0.031 0.033 0.051 0.07
$err_O$ 0.032 0.043 0.066 0.117
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Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
$err_D$ 0.010 0.015 0.039 0.063
$err_N$ 0.031 0.033 0.051 0.07
$err_O$ 0.032 0.043 0.066 0.117
Table 2.  Test-case C. $L^2$-errors on missing data over $ \Gamma_{i }$ (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape for various noise levels
Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
$err_D$ 0.042 0.044 0.046 0.08
$err_N$ 0.095 0.1 0.13 0.16
$err_O$ 0.099 0.11 0.13 0.15
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Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
$err_D$ 0.042 0.044 0.046 0.08
$err_N$ 0.095 0.1 0.13 0.16
$err_O$ 0.099 0.11 0.13 0.15
Table 3.  Relative errors on the reconstructed missing data and inclusion shape for the Stokes problem (with noise free measurements), compared for a classical Nash algorithm and Algorithm 2: (left) test-case A (right) test-case C
Case A Classical algorithm Algorithm 2
$err_D$ 0.058 0.033
$err_N$ 0.106 0.032
$err_O$ 0.358 0.140
Case C Classical algorithm Algorithm 2
$err_D$ 0.067 0.058
$err_N$ 0.208 0.122
$err_O$ 0.566 0.167
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Case A Classical algorithm Algorithm 2
$err_D$ 0.058 0.033
$err_N$ 0.106 0.032
$err_O$ 0.358 0.140
Case C Classical algorithm Algorithm 2
$err_D$ 0.067 0.058
$err_N$ 0.208 0.122
$err_O$ 0.566 0.167
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