Some inverse problems around the tokamak <em>Tore Supra</em>

Yannick Fischer; Benjamin Marteau; Yannick Privat

doi:10.3934/cpaa.2012.11.2327

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

Some inverse problems around the tokamak Tore Supra

Yannick Fischer ^1, , Benjamin Marteau ^2, and Yannick Privat ^3,

1.	INRIA Sophia Antipolis Mediterranee, 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'Hères, 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

Abstract
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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.

Keywords: least square problems, conjugate harmonic function, Hardy spaces, inverse problems, shape optimization, bounded extremal problem.

Mathematics Subject Classification: Primary: 30H10, 49J20, 65N21; Secondary: 30H05, 42A50, 35N05.

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. Google Scholar
[2]	G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002). Google Scholar
[3]	F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions,, Nuclear Fusion, 26 (1986), 1143. Google Scholar
[4]	M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas,", Advances in Industrial Control Series, (2008). Google Scholar
[5]	K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem,, Proceedings of El Escorial, (2006), 127. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	B. Beauzamy, "Introduction to Banach Spaces and their Geometry,", Mathematics studies, (1985). Google Scholar
[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. Google Scholar
[12]	J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks,", Series in Modern Applied Mathematics, (1989). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity,", Quaderni, (1980). Google Scholar
[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. Google Scholar
[18]	D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. Google Scholar
[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. Google Scholar
[20]	M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Advances in Design and Control SIAM, (2001). Google Scholar
[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. Google Scholar
[22]	P. L. Duren, "Theory of $H^p$ Spaces,", Pure and Applied Mathematics, (1970). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. Garnett, "Bounded Analytic Functions,", Pure and Applied Mathematics, (1981). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Math\'ematiques et Applications, (2005). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Annales de l'institut Fourier, 43 (1993), 1223. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	R. Kress, "Linear Integral Equations,", 2nd edn Berlin, (1999). Google Scholar
[39]	R. Kress, Inverse Dirichlet problem and conformal mapping,, Math. Comput. Simul., 66 (2004), 255. doi: 10.1016/j.matcom.2004.02.006. Google Scholar
[40]	R. Lattès and J. L. Lions, "Méthode de Quasi-réversibilité et Applications,", Dunod, (1967). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[45]	W. Rundell, Recovering an obstacle using integral equations,, Inverse Problems and Imaging, 3/2 (2009), 319. doi: 10.3934/ipi.2009.3.319. Google Scholar
[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. Google Scholar
[47]	V. D. Shafranov, On magnetohydrodynamical equilibrium configurations,, Soviet Physics JETP, 6 (1958), 545. Google Scholar
[48]	J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis,", Oxford Engineering Science Series, (1987). Google Scholar
[49]	A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems,", Winstons and Sons, (1977). Google Scholar
[50]	J. Wesson, "Tokamaks,", Series in Computational Mathematics, (1992). Google Scholar

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. Google Scholar
[2]	G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002). Google Scholar
[3]	F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions,, Nuclear Fusion, 26 (1986), 1143. Google Scholar
[4]	M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas,", Advances in Industrial Control Series, (2008). Google Scholar
[5]	K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem,, Proceedings of El Escorial, (2006), 127. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	B. Beauzamy, "Introduction to Banach Spaces and their Geometry,", Mathematics studies, (1985). Google Scholar
[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. Google Scholar
[12]	J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks,", Series in Modern Applied Mathematics, (1989). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity,", Quaderni, (1980). Google Scholar
[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. Google Scholar
[18]	D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. Google Scholar
[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. Google Scholar
[20]	M. Delfour and J. P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization,", Advances in Design and Control SIAM, (2001). Google Scholar
[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. Google Scholar
[22]	P. L. Duren, "Theory of $H^p$ Spaces,", Pure and Applied Mathematics, (1970). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. Garnett, "Bounded Analytic Functions,", Pure and Applied Mathematics, (1981). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Math\'ematiques et Applications, (2005). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Annales de l'institut Fourier, 43 (1993), 1223. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	R. Kress, "Linear Integral Equations,", 2nd edn Berlin, (1999). Google Scholar
[39]	R. Kress, Inverse Dirichlet problem and conformal mapping,, Math. Comput. Simul., 66 (2004), 255. doi: 10.1016/j.matcom.2004.02.006. Google Scholar
[40]	R. Lattès and J. L. Lions, "Méthode de Quasi-réversibilité et Applications,", Dunod, (1967). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[45]	W. Rundell, Recovering an obstacle using integral equations,, Inverse Problems and Imaging, 3/2 (2009), 319. doi: 10.3934/ipi.2009.3.319. Google Scholar
[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. Google Scholar
[47]	V. D. Shafranov, On magnetohydrodynamical equilibrium configurations,, Soviet Physics JETP, 6 (1958), 545. Google Scholar
[48]	J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis,", Oxford Engineering Science Series, (1987). Google Scholar
[49]	A. N. Tikhonov and V. Y. Arsenin, "Solutions of Ill-Posed Problems,", Winstons and Sons, (1977). Google Scholar
[50]	J. Wesson, "Tokamaks,", Series in Computational Mathematics, (1992). Google Scholar

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