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Some inverse problems around the tokamak Tore Supra
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 |
References:
[1] |
D. Alpay, L. Baratchart and J. Leblond, Some extremal problems linked with identification from a partial frequency data, in "Lectures Notes in Control and Information Science," Springer Verlag, 185 (1993), 563-573.
doi: 10.1007/BFb0115054. |
[2] |
G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. |
[3] |
F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions, Nuclear Fusion, 26 (1986), 1143-1164. |
[4] |
M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas," Advances in Industrial Control Series, Springer, London, 2008. |
[5] |
K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem, Proceedings of El Escorial, (2006), 127-139. |
[6] |
H. Ben Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems, 20 (2004), 673-696.
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), 41-56.
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), 423-436.
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), 384-427.
doi: 10.1016/j.jfa.2010.04.004. |
[10] |
B. Beauzamy, "Introduction to Banach Spaces and their Geometry," Mathematics studies, North-Holland, New York, 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), 111-138. |
[12] |
J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks," Series in Modern Applied Mathematics, Wiley Gauthier-Villars, Paris, 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), 351-377.
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), 301-329.
doi: 10.4171/IFB/81. |
[15] |
M. Burger, Levenberg-Marquardt levet set methods for inverse obstacle problem, Inverse Problems, 14 (1998), 685-706.
doi: 10.1088/0266-5611/20/1/016. |
[16] |
S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity," Quaderni, Scuola Normale Superiore Pisa, 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), 33-57.
doi: 10.1163/156939403322004928. |
[18] |
D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-289. |
[19] |
F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367.
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, PA, Philadelphia, 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), 3898-3914.
doi: 10.1016/j.cma.2006.10.046. |
[22] |
P. L. Duren, "Theory of $H^p$ Spaces," Pure and Applied Mathematics, Academic Press, New York - London, 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), 264-285.
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., 314 (2006) 126-149.
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), 481-506.
doi: 10.1093/imamat/hxn041. |
[26] |
J. Garnett, "Bounded Analytic Functions," Pure and Applied Mathematics, Academic Press, New York - London, 1981. |
[27] |
H. Haddar and R. Kress, Conformal mappings and inverse boundary value problem, Inverse Problems, 21 (2005), 935-953.
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), 317-330.
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), 231-251.
doi: 10.1023/A:1026095405906. |
[30] |
A. Henrot and M. Pierre, "Variation et Optimisation de Formes," Mathématiques et Applications, Springer, 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), 85-97.
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), 311-323.
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), 2399-2416.
doi: 10.1090/S0002-9947-02-02892-1. |
[34] |
L. Hörmander, Remarks on Holmgren's uniqueness theorem, Annales de l'institut Fourier, 43 (1993), 1223-1251. |
[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), 126-149.
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), 1653-1675.
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), 45-52. |
[38] |
R. Kress, "Linear Integral Equations," 2nd edn Berlin, Springer, 1999. |
[39] |
R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simul., 66 (2004), 255-265.
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., 18 (2012), 157-180.
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), 2497-2505.
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 Numérique de l'Université 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), 49-89.
doi: 10.1016/j.jcp.2003.08.031. |
[45] |
W. Rundell, Recovering an obstacle using integral equations, Inverse Problems and Imaging, 3/2 (2009), 319-332.
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), 761-765. |
[47] |
V. D. Shafranov, On magnetohydrodynamical equilibrium configurations, Soviet Physics JETP, 6 (1958), 545-554. |
[48] |
J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis," Oxford Engineering Science Series, Clarendon Press, Oxford, 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, Springer, Berlin, 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 "Lectures Notes in Control and Information Science," Springer Verlag, 185 (1993), 563-573.
doi: 10.1007/BFb0115054. |
[2] |
G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. |
[3] |
F. Alladio and F. Crisanti, Analysis of MHD equilibria by toroidal multipolar expansions, Nuclear Fusion, 26 (1986), 1143-1164. |
[4] |
M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas," Advances in Industrial Control Series, Springer, London, 2008. |
[5] |
K. Astala and L. Päivärinta, A boundary integral equation for Calderóm's inverse conductivity problem, Proceedings of El Escorial, (2006), 127-139. |
[6] |
H. Ben Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems, 20 (2004), 673-696.
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), 41-56.
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), 423-436.
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), 384-427.
doi: 10.1016/j.jfa.2010.04.004. |
[10] |
B. Beauzamy, "Introduction to Banach Spaces and their Geometry," Mathematics studies, North-Holland, New York, 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), 111-138. |
[12] |
J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics with Applications to Tokamaks," Series in Modern Applied Mathematics, Wiley Gauthier-Villars, Paris, 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), 351-377.
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), 301-329.
doi: 10.4171/IFB/81. |
[15] |
M. Burger, Levenberg-Marquardt levet set methods for inverse obstacle problem, Inverse Problems, 14 (1998), 685-706.
doi: 10.1088/0266-5611/20/1/016. |
[16] |
S. Campanato, "Elliptic Systems in Divergence Form. Interior Regularity," Quaderni, Scuola Normale Superiore Pisa, 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), 33-57.
doi: 10.1163/156939403322004928. |
[18] |
D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-289. |
[19] |
F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367.
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, PA, Philadelphia, 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), 3898-3914.
doi: 10.1016/j.cma.2006.10.046. |
[22] |
P. L. Duren, "Theory of $H^p$ Spaces," Pure and Applied Mathematics, Academic Press, New York - London, 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), 264-285.
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., 314 (2006) 126-149.
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), 481-506.
doi: 10.1093/imamat/hxn041. |
[26] |
J. Garnett, "Bounded Analytic Functions," Pure and Applied Mathematics, Academic Press, New York - London, 1981. |
[27] |
H. Haddar and R. Kress, Conformal mappings and inverse boundary value problem, Inverse Problems, 21 (2005), 935-953.
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), 317-330.
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), 231-251.
doi: 10.1023/A:1026095405906. |
[30] |
A. Henrot and M. Pierre, "Variation et Optimisation de Formes," Mathématiques et Applications, Springer, 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), 85-97.
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), 311-323.
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), 2399-2416.
doi: 10.1090/S0002-9947-02-02892-1. |
[34] |
L. Hörmander, Remarks on Holmgren's uniqueness theorem, Annales de l'institut Fourier, 43 (1993), 1223-1251. |
[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), 126-149.
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), 1653-1675.
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), 45-52. |
[38] |
R. Kress, "Linear Integral Equations," 2nd edn Berlin, Springer, 1999. |
[39] |
R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simul., 66 (2004), 255-265.
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., 18 (2012), 157-180.
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), 2497-2505.
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 Numérique de l'Université 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), 49-89.
doi: 10.1016/j.jcp.2003.08.031. |
[45] |
W. Rundell, Recovering an obstacle using integral equations, Inverse Problems and Imaging, 3/2 (2009), 319-332.
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), 761-765. |
[47] |
V. D. Shafranov, On magnetohydrodynamical equilibrium configurations, Soviet Physics JETP, 6 (1958), 545-554. |
[48] |
J. Sokolowski and J. P. Zolesio, "Introduction to Shape Optimization Shape Sensitivity Analysis," Oxford Engineering Science Series, Clarendon Press, Oxford, 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, Springer, Berlin, 1992. |
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