# American Institute of Mathematical Sciences

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

## 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

* 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.

Citation: Abderrahmane Habbal, Moez Kallel, Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem. Inverse Problems & 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.
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
Different situations
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$
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 $\Gamma_{i }$)
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
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 $err_O = 0.113156)$
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
 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
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
 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
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
 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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