Some inverse problems around the tokamak Tore Supra

International audienceWe 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}

Multidomain Spectral Method for the Helically Reduced Wave Equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the "eigenspectral method." Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes revisions based on referee reports and has two extra figure

Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell's Equations

This work is on the numerical approximation of incoming solutions to Maxwell's equations with dissipative boundary conditions whose energy decays exponentially with time. Such solutions are called asymptotically disappearing (ADS) and they play an importarnt role in inverse back-scatering problems. The existence of ADS is a difficult mathematical problem. For the exterior of a sphere, such solutions have been constructed analytically by Colombini, Petkov and Rauch [7] by specifying appropriate initial conditions. However, for general domains of practical interest (such as Lipschitz polyhedra), the existence of such solutions is not evident. This paper considers a finite-element approximation of Maxwell's equations in the exterior of a polyhedron, whose boundary approximates the sphere. Standard Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to approximate the electric and magnetic fields. Discrete initial conditions interpolating the ones chosen in [7] are modified so that they are (weakly) divergence-free. We prove that with such initial conditions, the approximation to the electric field is weakly divergence-free for all time. Finally, we show numerically that the finite-element approximations of the ADS also decay exponentially with time when the mesh size and the time step become small.Comment: 15 pages, 3 figure

Two nonlinear systems from mathematical physics

The dissertation is divided into two chapters. In the first one, we consider the 2-Vortex problem for two point vortices in a complex domain. The Hamiltonian of the system contains the regular part of a hydrodynamic Green’s function, the Robin function h and two coefficinets which are the strengths of the point vortices. We prove the existence of infinitely many periodic solutions with minimal period T which are a superposition of a slow motion of the center of vorticity along a level line of h and of a fast rotation of the two vortices around their center of vorticity. These vortices move in a prescribed subset of the domain that has to satisfy a geometric condition. The minimal period can be any T in a certain interval. Subsets to which our results apply can be found in any generic bounded domain. The proofs are based on a recent higher dimensional version of the Poincaré-Birkhoff theorem due to Fonda and Ureña. In the second part, we study bifurcations of a multi-component Schrödinger system. We construct a solution branch synchronized to a positive solution of a simpler system. From this branch, we find a sequence of local bifurcation values in the one dimensional case and also in the general case provided that the positive solution is nondegenerate

Two nonlinear systems from mathematical physics

The dissertation is divided into two chapters. In the first one, we consider the 2-Vortex problem for two point vortices in a complex domain. The Hamiltonian of the system contains the regular part of a hydrodynamic Green’s function, the Robin function h and two coefficinets which are the strengths of the point vortices. We prove the existence of infinitely many periodic solutions with minimal period T which are a superposition of a slow motion of the center of vorticity along a level line of h and of a fast rotation of the two vortices around their center of vorticity. These vortices move in a prescribed subset of the domain that has to satisfy a geometric condition. The minimal period can be any T in a certain interval. Subsets to which our results apply can be found in any generic bounded domain. The proofs are based on a recent higher dimensional version of the Poincaré-Birkhoff theorem due to Fonda and Ureña. In the second part, we study bifurcations of a multi-component Schrödinger system. We construct a solution branch synchronized to a positive solution of a simpler system. From this branch, we find a sequence of local bifurcation values in the one dimensional case and also in the general case provided that the positive solution is nondegenerate

Harmonic fields on the extended projective disc and a problem in optics

The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1-fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the non-characteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptic-hyperbolic boundary-value problems is formulated for those equations as well. For both classes of boundary-value problems, an arbitrarily small lower-order perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.Comment: 30 pages; Section 3.3 has been revise