Real and Complex Analysis

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.

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An $H_\mathsf{div}$-Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems

https://hal.inria.fr/hal-01104629/document

Uniqueness results for inverse Robin problems with bounded coefficient

We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)

Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation

Cette these traite de la resolution theorique et constructive de problemes inverses pour des equations de diffusion isotropes dans des domaines plan simplement et doublement connexes. A partir de donnees de Cauchy (potentiel, flux) disponibles sur une partie de la frontiere du domaine, il s’agit de retrouver ces quantites sur la partie du bord ou l’on ne dispose pas d’information, ainsi qu’a l’interieur du domaine. L’approche mise au point consiste a considerer les solutions de l’equation de diffusion comme les parties reelles des solutions complexes d’une equation de Beltrami conjuguee. Ces fonctions analytiques generalisees d’un type particulier permettent de definir des classes de Hardy, dans lesquelles le probleme inverse est regularise en etant reformule comme un probleme de meilleure approximation sous contrainte (ou encore probleme extremal borne, d’adequation aux donnees). Le caractere bien pose de celui-ci est assure par des resultats d’existence et de regularite auxquels s’ajoutent des proprietes de densite a la frontiere. Une application au calcul de la frontiere libre d’un plasma sous confinement magnetique dans le tokamak Tore Supra (CEA-IRFM Cadarache) est proposee. La resolution du probleme extremal a partir d’une base de fonctions adaptees (harmoniques toroidales) fournit un critere permettant de qualifier les estimations de la frontiere plasma. Un algorithme de descente permet de le faire decroitre, en ameliorant l’estimation de la frontiere. Cette methode, qui ne requiert pas d’integration de l’equation dans le domaine, fournit de tres bons resultats et semble appelee a connaitre des extensions pour d’autres tokamaks tels que JET et ITER.

https://tel.archives-ouvertes.fr/tel-00643239/document

Approximation dans des classes de fonctions analytiques généralisées et résolution de problèmes inverses pour les tokamaks

https://hal.inria.fr/inria-00460820/document

Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains

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}.

https://hal.archives-ouvertes.fr/hal-00537648/document

Some inverse problems around the tokamak Tore Supra

We present a constructive method for the robust approximation to solutions of some elliptic equations in a plane domain from incomplete and corrupted boundary data We state this inverse problem in generalized Hardy spaces of functions satisfying the conjugate Beltrami equation, of which we give some properties, in the Hilbertian framework The issue is then reworded as a constrained approximation (bounded extremal) problem which is shown to be well-posed A practical motivation comes from modelling plasma confinement in a tokamak reactor There, the particular form of the conductivity coefficient leads to Bessel-exponential type families of solutions of which we establish density properties

https://www-sop.inria.fr/members/Yannick.Fischer/Publications/APAM.pdf

Solutions to conjugate Beltrami equations and approximation in generalized Hardy spaces

We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $\nabla.(\sigma \nabla u)=0$ where $\sigma=(1-\nu)/(1+\nu)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.

Yannick Fischer

Papers

Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains

Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation

Some inverse problems around the tokamak Tore Supra

Solutions to conjugate Beltrami equations and approximation in generalized Hardy spaces

Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation