On an inverse Cauchy problem arising in tokamaks Yannick Fischer - PowerPoint PPT Presentation

On an inverse Cauchy problem arising in tokamaks Yannick Fischer INRIA Sophia-Antipolis projet APICS joint work with L. Baratchart*, J. Leblond* *INRIA (Sophia-Antipolis) SMAI - 23 Mai 2011

Cauchy problem Ω ⊂ R 2 : annular domain with smooth boundary ∂ Ω = Γ i ∪ Γ e σ : smooth function (Lipschitz) with 0 < c ≤ σ ≤ C ∇ · ( σ ∇ u ) = 0 a.e in Ω with u and ∂ n u prescribed on I ⊆ ∂ Ω Can we recover u and ∂ n u on J = ∂ Ω \ I ?

Application to tokamak Physical motivation : application to Tokamak (Tore Supra) z z Axisymmetric Γ e configuration Ω l Γ l (3D-problem) Ω p y r ⇒ 0 = ρ e Γ p ρ l x r Study of the ϕ equilibrium in poloidal section (2D-problem) R Maxwell equation in the vacuum Ω l : ∇ . ( 1 r ∇ u ) = 0 where u ( r , z ) is the magnetic poloidal flux and σ = 1 r is regular in Ω l . How to recover u and ∂ n u on Γ l from (finite) measurements on Γ e ?

The conjugate Beltrami equation Idea : Astala and P¨ aiv¨ arinta (2006) From real equation σ ∈ W 1 , ∞ ∇ · ( σ ∇ u ) = 0 a.e in Ω ( CD ) (Ω) R to complex equation (but R -linear) ν ∈ W 1 , ∞ ∂ f = ν∂ f a.e in Ω ( CB ) (Ω) R Proposition f = u + iv ∈ W 1 , 2 (Ω) ν = 1 − σ satisfies (CB) with 1 + σ ⇒ ∇ . ( σ ∇ u ) = ∇ . ( σ − 1 ∇ v ) = 0 = a.e in Ω and � ∂ x v = − σ∂ y u a.e in Ω ( or ∂ t v = σ∂ n u a.e on ∂ Ω) ∂ y v = σ∂ x u

The conjugate Beltrami equation Advantages : • Symmetric roles played by u and v Dirichlet + Neumann conditions for u ⇓ � Dirichlet condition for u + Dirichlet condition for v = ∂ Ω σ∂ n u ⇓ ONLY Dirichlet conditions for f • Allow regularization of the Cauchy problem for data in L 2 ( ∂ Ω)

Generalized Hardy classes Definition H 2 ν (Ω) = Lebesgue measurable fonctions f on Ω such that � 1 / p � 2 π � 1 | f ( re i θ ) | 2 d θ � f � H 2 ν (Ω) := ess sup < + ∞ (1) 2 π 0 ̺< r < 1 and solving (CB) in the sense of distributions in Ω H 2 ν (Ω) is a Hilbert space. When ν = 0 and Ω = D , recover the classical H 2 ( D ) space of holomorphic functions in D satisfying (1) � . � H 2 ν (Ω ) ∼ � . � L 2 ( ∂ Ω) tr H 2 ν (Ω) is closed subspace of L 2 ( ∂ Ω)

Density result Theorem Let I ⊂ ∂ Ω be a mesurable subset such that | I | , | J | > 0 tr H 2 ν (Ω) | I is dense in L 2 ( I ) As tr H 2 ν (Ω) is a closed subset of L 2 ( ∂ Ω) , if ( f k ) k ≥ 1 ∈ H 2 ν (Ω) is such that � tr f k − f � L 2 ( I ) − → 0 , k there are only two possibilities : f = ( tr F ) | I with F ∈ H 2 ν (Ω) or � tr f k � L 2 ( J ) → + ∞ This leads to a bounded extremal problem.

Bounded extremal problems... If ( u , v ) are compatible data on I − → unique solution by extrapolation If ( u , v ) are NOT compatible data on I Idea : constrain solutions on J . Definition For M > 0 and ϕ ∈ L 2 R ( J ) � � f ∈ tr H 2 | I ⊂ L 2 ( I ) B = ν (Ω); � f − ϕ � L 2 ( J ) ≤ M → well-posed L 2 approximation problem extrapolation problem ←

...Bounded extremal problems Then the approximation problem admits a unique solution Theorem Fix M > 0 ∀ f ∈ L 2 ( I ) , ∃ ! g 0 ∈ B / � f − g 0 � L 2 ( I ) = min g ∈B � f − g � L 2 ( I ) Moreover, if f / ∈ B , then � g 0 − ϕ � L 2 ( J ) = M The solution g 0 is given by g 0 = g 0 ( λ ) = ( I + λ P ν χ J ) − 1 P ν ( χ I f ∨ ( 1 + λ ) ϕ ) with λ ∈ ( − 1 , ∞ ) .

Algorithm plasma boundary = outermost closed magnetic surface in the limiter 1) u = � N i = 1 α i b i from u and ∂ n u on Γ ext 0.8 2) u 0 = max u on the limiter and 0.6 Γ 1 int = { ( x , y ); u ( x , y ) = u 0 } 0.4 ∂ Ω 1 = Γ ext ∪ Γ 1 0.2 int 0 3) BEP in Ω 1 −0.2 ⇒ g 0 = min � f − g � with −0.4 � Re g 0 − u 0 � = M −0.6 4) u 1 = max Re g 0 on the limiter −0.8 and Γ 2 int = { ( x , y ); Re g 0 = u 1 } 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 etc...

Simulations 0.9 BEP0_6 EFIT_REF LIM 0.85 0.8 ρ (m) 0.75 0.7 0.65 0 pi/2 pi 3pi/2 2pi Θ Fig. : Graphe de θ �→ ρ ( θ ) pour les fronti` eres plasma et le limiteur.

Simulations 0.9 BEP0_6 BEP1_6_6 EFIT_REF LIM 0.85 0.8 ρ (m) 0.75 0.7 0.65 0 pi/2 pi 3pi/2 2pi Θ Fig. : Graphe de θ �→ ρ ( θ ) pour les fronti` eres plasma et le limiteur.

Simulations 0.9 BEP0_6 BEP1_6_6 BEP1_6_10 EFIT_REF 0.85 LIM 0.8 ρ (m) 0.75 0.7 0.65 0 pi/2 pi 3pi/2 2pi Θ Fig. : Graphe de θ �→ ρ ( θ ) pour les fronti` eres plasma et le limiteur.

Simulations 0.9 BEP0_6 BEP1_6_6 BEP1_6_10 BEP1_6_14 0.85 EFIT_REF LIM 0.8 ρ (m) 0.75 0.7 0.65 0 pi/2 pi 3pi/2 2pi Θ Fig. : Graphe de θ �→ ρ ( θ ) pour les fronti` eres plasma et le limiteur.

Simulations 0.9 BEP0_6 BEP1_6_6 BEP1_6_10 BEP1_6_14 0.85 BEP1_6_18 EFIT_REF LIM 0.8 ρ (m) 0.75 0.7 0.65 0 pi/2 pi 3pi/2 2pi Θ Fig. : Graphe de θ �→ ρ ( θ ) pour les fronti` eres plasma et le limiteur.

Conclusion Fast method (no mesh) compact representation with solutions of the equation evaluation of the poloidal flux between the plasma and the outside boundary Work in progress : Optimization of the Lagrange parameter λ pathological cases

Thank you for your attention