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Journal ArticleDOI

Some inverse problems around the tokamak Tore Supra

01 Apr 2012-Communications on Pure and Applied Mathematics (Communications on Pure & Applied Analysis)-Vol. 11, Iss: 6, pp 2327-2349
TL;DR: In this paper, two inverse problems related to the tokamak \textsl{Tore Supra} through the study of the magnetostatic equation for the poloidal flux are considered.
Abstract: We consider two inverse problems related to the tokamak \textsl{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 \textsl{Tore Supra}.
Citations
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Journal ArticleDOI
TL;DR: A new quasi-reversibility approach is introduced for approximating the solution of the ill-posed Cauchy problem in a regularized manner based on a well-posed mixed variational problem on H^1\times H_\mathsf{div}$ with the corresponding solution pair converging monotonically to the Solution of the Cauche problem and the associated flux, if they exist.
Abstract: This work considers the Cauchy problem for a second order elliptic operator in a bounded domain. A new quasi-reversibility approach is introduced for approximating the solution of the ill-posed Cauchy problem in a regularized manner. The method is based on a well-posed mixed variational problem on $H^1\times H_\mathsf{div}$ with the corresponding solution pair converging monotonically to the solution of the Cauchy problem and the associated flux, if they exist. It is demonstrated that the regularized problem can be discretized using Lagrange and Raviart--Thomas finite elements. The functionality of the resulting numerical algorithm is tested via three-dimensional numerical experiments based on simulated data. Both the Cauchy problem and a related inverse obstacle problem for the Laplacian are considered.

32 citations

Journal ArticleDOI
TL;DR: In this article, Hardy spaces of the conjugate Beltrami equation over Dini-smooth finitely connected domains were studied for real contractive contracts with real contractivity in the range of r/(r-1)
Abstract: We study Hardy spaces $H^p_ u$ of the conjugate Beltrami equation $\bar{\partial} f= u\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $ u\in W^{1,r}$ with $r>2$, in the range $r/(r-1)

16 citations

Posted Content
TL;DR: In this article, an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle were proved for the Dirichlet problem with weighted boundary data for 2-D isotropic conductivity equations.
Abstract: We study Hardy classes on the disk associated to the equation $\bar\d w=\alpha\bar w$ for $\alpha\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.

13 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied Hardy classes on the disk associated to the equa- tion of Dirichlet problems for 2-D isotropic conductivity equations whose coefficients have logarithm in W 1,2.
Abstract: We study Hardy classes on the disk associated to the equa- tion ¯ @w = � ¯ w for � 2 L r with 2 � r < 1. The paper seems to be the first to deal with the case r = 2. We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity prin- ciple. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth do- mains of the Dirichlet problem with weighted L p boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in W 1,2 . In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for W 1,2 0 -functions.

11 citations

Journal ArticleDOI
14 Jul 2015
TL;DR: In this article, the authors considered the generalized axisymmetric potentials (GASP) and proved a new decomposition theorem for the GASP in annular domains.
Abstract: We consider the Weinstein equation, also known as the equation governing generalized axisymmetric potentials (GASP), with complex coefficients $$L_mu={\Delta } u+(m/x)\partial _x u =0$$ , $$m\in {\mathbb {C}}$$ . We generalize results known for $$m\in {\mathbb {R}}$$ to the case $$m\in {\mathbb {C}}$$ . In particular, explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities are presented, a Green’s formula for GASP in the right half-plane $${\mathbb {H}}^+$$ for $$\mathrm{Re}\,m<1$$ is established. We prove a new decomposition theorem for the GASP in annular domains for $$m\in {\mathbb {C}}$$ , which is in fact a generalization of the Bocher’s decomposition theorem. In particular, using bipolar coordinates, it is proved for annuli that a family of solutions for the GASP equation in terms of associated Legendre functions of first and second kind is complete. This family is shown to be a Riesz basis in some non-concentric circular annuli.

6 citations

References
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Book
19 Nov 2010
TL;DR: In this article, the Corona construction was used to construct Douglas algebra and interpolating sequences and Maximal Ideals were used to solve a set of problems in the Corona Construction.
Abstract: Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals.

3,585 citations

Book
01 Jan 1989
TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Abstract: Normed Spaces.- Bounded and Compact Operators.- Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature Methods.- Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.

2,323 citations

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