# American Institute of Mathematical Sciences

November  2012, 11(6): 2327-2349. doi: 10.3934/cpaa.2012.11.2327

## Some inverse problems around the tokamak Tore Supra

 1 INRIA Sophia Antipolis Mediterranee, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis, France 2 ENSIMAG, 681, rue de la passerelle, Domaine universitaire, BP 72, 38402 Saint Martin D'Hères, France 3 ENS Cachan Bretagne, CNRS, Univ. Rennes 1, IRMAR, av. Robert Schuman, F-35170 Bruz

Received  March 2011 Revised  May 2011 Published  April 2012

We consider two inverse problems related to the tokamak Tore Supra through the study of the magnetostatic equation for the poloidal flux. The first one deals with the Cauchy issue of recovering in a two dimensional annular domain boundary magnetic values on the inner boundary, namely the limiter, from available overdetermined data on the outer boundary. Using tools from complex analysis and properties of genereralized Hardy spaces, we establish stability and existence properties. Secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization. Again results about existence and optimality are provided. They give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from Tore Supra.
Citation: Yannick Fischer, Benjamin Marteau, Yannick Privat. Some inverse problems around the tokamak Tore Supra. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2327-2349. doi: 10.3934/cpaa.2012.11.2327
##### References:
 [1] D. Alpay, L. Baratchart and J. Leblond, Some extremal problems linked with identification from a partial frequency data,, in, 185 (1993), 563. doi: 10.1007/BFb0115054. [2] G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002). [3] F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions,, Nuclear Fusion, 26 (1986), 1143. [4] M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas,", Advances in Industrial Control Series, (2008). [5] K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem,, Proceedings of El Escorial, (2006), 127. [6] H. Ben Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity,, Inverse Problems, 20 (2004), 673. doi: 10.1088/0266-5611/20/3/003. [7] L. Baratchart and J. Leblond, Hardy approximation to $L^p$ functions on subsets of the circle with $1 \leq p < \infty$,, Constructive approximation, 14 (1998), 41. doi: 10.1007/s003659900062. [8] L. Baratchart, J. Leblond and J. R. Partington, Hardy approximation to $L^{\infty}$ functions on subsets of the circle with $1 \leq p < \infty$,, Constructive approximation, 12 (1996), 423. doi: 10.1007/s003659900022. [9] L. Baratchart, J. Leblond, S. Rigat and E. Russ, Hardy spaces of the conjugate Beltrami equation,, Journal of Functional Analysis, 259 (2010), 384. doi: 10.1016/j.jfa.2010.04.004. [10] B. Beauzamy, "Introduction to Banach Spaces and their Geometry,", Mathematics studies, (1985). [11] L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications,, Convegno Internazionale Sulle Equazioni Derivate e Parziali, (1954), 111. [12] J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks,", Series in Modern Applied Mathematics, (1989). [13] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse problem obstacle problem,, Inverse Problems and Imaging, 4/3 (2010), 351. doi: 10.3934/ipi.2010.4.351. [14] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction,, Interfaces and Free Boundaries, 5 (2003), 301. doi: 10.4171/IFB/81. [15] M. Burger, Levenberg-Marquardt levet set methods for inverse obstacle problem,, Inverse Problems, 14 (1998), 685. doi: 10.1088/0266-5611/20/1/016. [16] S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity,", Quaderni, (1980). [17] S. Chaabane, M. Jaoua and J. Leblond, "Parameter Identification for Laplace Equation and Approximation in Hardy Classes,", J. Inverse Ill-Posed Probl, 11 (2003), 33. doi: 10.1163/156939403322004928. [18] D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. [19] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization,, SIAM J. Control Optim., 45 (2006), 343. doi: 10.1137/050624108. [20] M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Advances in Design and Control SIAM, (2001). [21] G. Dogǧan, P. Morin, R.H. Nochetto and M. Verani, Discrete Gradient Flows for Shape Optimization and Applications,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3898. doi: 10.1016/j.cma.2006.10.046. [22] P. L. Duren, "Theory of $H^p$ Spaces,", Pure and Applied Mathematics, (1970). [23] Y. Fischer, J. Leblond, J. R. Partington and E. Sincich, Bounded extremal problems in Hardy spaces for the conjugate Beltrami equations in simply connected domains,, Applied and Computational Harmonic Analysis, 31 (2011), 264. doi: 10.1016/j.acha.2011.01.003. [24] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems,, J. Math. Anal. Appl., (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [25] M. Jaoua, J. Leblond, M. Mahjoub and J. R. Partington, Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains,, IMA Journal of Applied Mathematics, 74 (2009), 481. doi: 10.1093/imamat/hxn041. [26] J. Garnett, "Bounded Analytic Functions,", Pure and Applied Mathematics, (1981). [27] H. Haddar and R. Kress, Conformal mappings and inverse boundary value problem,, Inverse Problems, 21 (2005), 935. doi: 10.1088/0266-5611/21/3/009. [28] J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type,, Interfaces Free Bound., 11 (2009), 317. doi: 10.4171/IFB/213. [29] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type,, Comput. Optim. Appl., 26 (2003), 231. doi: 10.1023/A:1026095405906. [30] A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Math\'ematiques et Applications, (2005). [31] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. I. The exterior convex case,, J. Reine Angew. Math., 521 (2000), 85. doi: 10.1515/crll.2000.031. [32] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. II. The interior convex case,, Indiana Univ. Math. J., 49 (2000), 311. doi: 10.1512/iumj.2000.49.1711. [33] A. Henrot and H. Shahgholian, The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition,, Trans. Amer. Math. Soc., 354 (2002), 2399. doi: 10.1090/S0002-9947-02-02892-1. [34] L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Annales de l'institut Fourier, 43 (1993), 1223. [35] K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems,, J. Math. Anal. Appl., 314 (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [36] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equations,, SIAM J. Appl. Math., 51 (1991), 1653. doi: 10.1137/0151085. [37] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the cauchy problem for elliptic equation,, Comput. Math. Phys., 31 (1991), 45. [38] R. Kress, "Linear Integral Equations,", 2nd edn Berlin, (1999). [39] R. Kress, Inverse Dirichlet problem and conformal mapping,, Math. Comput. Simul., 66 (2004), 255. doi: 10.1016/j.matcom.2004.02.006. [40] R. Lattès and J. L. Lions, "Méthode de Quasi-réversibilité et Applications,", Dunod, (1967). [41] A. Laurain and Y. Privat, On a Bernoulli problem with geometric constraints,, ESAIM Control Optim. Calc. Var., 1 (2012), 157. doi: 10.1016/j.matcom.2004.02.006. [42] E. Lindgren and Y. Privat, A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition,, Nonlinear Anal., 67 (2007), 2497. doi: 10.1016/j.na.2006.08.045. [43] F. Murat and J. Simon, "Sur le contrôle par un domaine géométrique,", Publication du Laboratoire d'Analyse Num\'erique de l'Universit\'e Paris 6, 189 (1976). [44] B. Protas, T-R Bewley and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale pde systems,, J. Comput. Phys., 195 (2004), 49. doi: 10.1016/j.jcp.2003.08.031. [45] W. Rundell, Recovering an obstacle using integral equations,, Inverse Problems and Imaging, 3/2 (2009), 319. doi: 10.3934/ipi.2009.3.319. [46] F. Saint-Laurent and G. Martin, Real time determination and control of the plasma localisation and internal inductance in Tore Supra,, Fusion Engineering and Design, 56-57 (2001), 56. [47] V. D. Shafranov, On magnetohydrodynamical equilibrium configurations,, Soviet Physics JETP, 6 (1958), 545. [48] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis,", Oxford Engineering Science Series, (1987). [49] A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems,", Winstons and Sons, (1977). [50] J. Wesson, "Tokamaks,", Series in Computational Mathematics, (1992).

show all references

##### References:
 [1] D. Alpay, L. Baratchart and J. Leblond, Some extremal problems linked with identification from a partial frequency data,, in, 185 (1993), 563. doi: 10.1007/BFb0115054. [2] G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002). [3] F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions,, Nuclear Fusion, 26 (1986), 1143. [4] M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas,", Advances in Industrial Control Series, (2008). [5] K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem,, Proceedings of El Escorial, (2006), 127. [6] H. Ben Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity,, Inverse Problems, 20 (2004), 673. doi: 10.1088/0266-5611/20/3/003. [7] L. Baratchart and J. Leblond, Hardy approximation to $L^p$ functions on subsets of the circle with $1 \leq p < \infty$,, Constructive approximation, 14 (1998), 41. doi: 10.1007/s003659900062. [8] L. Baratchart, J. Leblond and J. R. Partington, Hardy approximation to $L^{\infty}$ functions on subsets of the circle with $1 \leq p < \infty$,, Constructive approximation, 12 (1996), 423. doi: 10.1007/s003659900022. [9] L. Baratchart, J. Leblond, S. Rigat and E. Russ, Hardy spaces of the conjugate Beltrami equation,, Journal of Functional Analysis, 259 (2010), 384. doi: 10.1016/j.jfa.2010.04.004. [10] B. Beauzamy, "Introduction to Banach Spaces and their Geometry,", Mathematics studies, (1985). [11] L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications,, Convegno Internazionale Sulle Equazioni Derivate e Parziali, (1954), 111. [12] J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks,", Series in Modern Applied Mathematics, (1989). [13] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse problem obstacle problem,, Inverse Problems and Imaging, 4/3 (2010), 351. doi: 10.3934/ipi.2010.4.351. [14] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction,, Interfaces and Free Boundaries, 5 (2003), 301. doi: 10.4171/IFB/81. [15] M. Burger, Levenberg-Marquardt levet set methods for inverse obstacle problem,, Inverse Problems, 14 (1998), 685. doi: 10.1088/0266-5611/20/1/016. [16] S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity,", Quaderni, (1980). [17] S. Chaabane, M. Jaoua and J. Leblond, "Parameter Identification for Laplace Equation and Approximation in Hardy Classes,", J. Inverse Ill-Posed Probl, 11 (2003), 33. doi: 10.1163/156939403322004928. [18] D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. [19] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization,, SIAM J. Control Optim., 45 (2006), 343. doi: 10.1137/050624108. [20] M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Advances in Design and Control SIAM, (2001). [21] G. Dogǧan, P. Morin, R.H. Nochetto and M. Verani, Discrete Gradient Flows for Shape Optimization and Applications,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3898. doi: 10.1016/j.cma.2006.10.046. [22] P. L. Duren, "Theory of $H^p$ Spaces,", Pure and Applied Mathematics, (1970). [23] Y. Fischer, J. Leblond, J. R. Partington and E. Sincich, Bounded extremal problems in Hardy spaces for the conjugate Beltrami equations in simply connected domains,, Applied and Computational Harmonic Analysis, 31 (2011), 264. doi: 10.1016/j.acha.2011.01.003. [24] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems,, J. Math. Anal. Appl., (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [25] M. Jaoua, J. Leblond, M. Mahjoub and J. R. Partington, Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains,, IMA Journal of Applied Mathematics, 74 (2009), 481. doi: 10.1093/imamat/hxn041. [26] J. Garnett, "Bounded Analytic Functions,", Pure and Applied Mathematics, (1981). [27] H. Haddar and R. Kress, Conformal mappings and inverse boundary value problem,, Inverse Problems, 21 (2005), 935. doi: 10.1088/0266-5611/21/3/009. [28] J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type,, Interfaces Free Bound., 11 (2009), 317. doi: 10.4171/IFB/213. [29] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type,, Comput. Optim. Appl., 26 (2003), 231. doi: 10.1023/A:1026095405906. [30] A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Math\'ematiques et Applications, (2005). [31] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. I. The exterior convex case,, J. Reine Angew. Math., 521 (2000), 85. doi: 10.1515/crll.2000.031. [32] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. II. The interior convex case,, Indiana Univ. Math. J., 49 (2000), 311. doi: 10.1512/iumj.2000.49.1711. [33] A. Henrot and H. Shahgholian, The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition,, Trans. Amer. Math. Soc., 354 (2002), 2399. doi: 10.1090/S0002-9947-02-02892-1. [34] L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Annales de l'institut Fourier, 43 (1993), 1223. [35] K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems,, J. Math. Anal. Appl., 314 (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [36] M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equations,, SIAM J. Appl. Math., 51 (1991), 1653. doi: 10.1137/0151085. [37] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the cauchy problem for elliptic equation,, Comput. Math. Phys., 31 (1991), 45. [38] R. Kress, "Linear Integral Equations,", 2nd edn Berlin, (1999). [39] R. Kress, Inverse Dirichlet problem and conformal mapping,, Math. Comput. Simul., 66 (2004), 255. doi: 10.1016/j.matcom.2004.02.006. [40] R. Lattès and J. L. Lions, "Méthode de Quasi-réversibilité et Applications,", Dunod, (1967). [41] A. Laurain and Y. Privat, On a Bernoulli problem with geometric constraints,, ESAIM Control Optim. Calc. Var., 1 (2012), 157. doi: 10.1016/j.matcom.2004.02.006. [42] E. Lindgren and Y. Privat, A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition,, Nonlinear Anal., 67 (2007), 2497. doi: 10.1016/j.na.2006.08.045. [43] F. Murat and J. Simon, "Sur le contrôle par un domaine géométrique,", Publication du Laboratoire d'Analyse Num\'erique de l'Universit\'e Paris 6, 189 (1976). [44] B. Protas, T-R Bewley and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale pde systems,, J. Comput. Phys., 195 (2004), 49. doi: 10.1016/j.jcp.2003.08.031. [45] W. Rundell, Recovering an obstacle using integral equations,, Inverse Problems and Imaging, 3/2 (2009), 319. doi: 10.3934/ipi.2009.3.319. [46] F. Saint-Laurent and G. Martin, Real time determination and control of the plasma localisation and internal inductance in Tore Supra,, Fusion Engineering and Design, 56-57 (2001), 56. [47] V. D. Shafranov, On magnetohydrodynamical equilibrium configurations,, Soviet Physics JETP, 6 (1958), 545. [48] J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis,", Oxford Engineering Science Series, (1987). [49] A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems,", Winstons and Sons, (1977). [50] J. Wesson, "Tokamaks,", Series in Computational Mathematics, (1992).
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