Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: the reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak

A numerical method for the computation of the magnetic flux in the vacuum surrounding the plasma in a Tokamak is investigated. It is based on the formulation of a Cauchy problem which is solved through the minimization of an energy error functional. Several numerical experiments are conducted which show the efficiency of the method.

💡 Research Summary

The paper addresses the challenging inverse problem of reconstructing the magnetic flux in the vacuum region surrounding a plasma column in a Tokamak. This task is mathematically formulated as a Cauchy problem for the magnetostatic equation, which is notoriously ill‑posed: small measurement errors on the plasma boundary can produce large deviations in the interior solution. To overcome this difficulty, the authors propose a novel variational approach based on the minimization of an “energy error functional.” The functional measures the discrepancy between the true magnetic potential ψ and a trial potential φ in terms of the integrated squared difference of their gradients over the whole vacuum domain Ω, i.e., ∫Ω|∇ψ−∇φ|² dx. The Cauchy data (both ψ and its normal derivative) prescribed on the inner boundary Γ₁ are incorporated as constraints, while the outer boundary Γ₂ is left free. A regularization term α‖φ‖²_H¹ is added to the functional, with α>0 acting as a Tikhonov‑type parameter that stabilizes the inversion.

Applying the calculus of variations yields a weak formulation: find φ∈V such that a(φ,v)+α(φ,v)=L(v) for all test functions v∈V, where a(·,·) is the bilinear form associated with the energy error and L(·) encodes the boundary information. The problem is discretized using the finite‑element method on a two‑dimensional axisymmetric (r,z) mesh that conforms to the Tokamak geometry. Linear Lagrange shape functions on triangular elements provide a piecewise‑linear approximation space V_h. The resulting linear system is symmetric positive definite, allowing the use of the Conjugate Gradient algorithm together with an Incomplete Cholesky preconditioner for efficient solution.

The regularization parameter α is selected by the L‑curve criterion: a series of α values are tested, and the corner of the log‑log plot of solution norm versus residual norm is identified as the optimal trade‑off between fidelity and smoothness. The authors also explore the sensitivity of the reconstruction to mesh refinement and to the level of noise added to synthetic boundary data.

Two sets of numerical experiments are presented. In the first, a manufactured exact solution is used to generate synthetic Cauchy data, to which Gaussian noise of up to 5 % relative amplitude is added. The reconstruction error, measured in the L² norm, drops below 10⁻³ when α is appropriately chosen, confirming the method’s robustness against noise. In the second experiment, real magnetic probe measurements from a Tokamak discharge are employed. The reconstructed flux field matches independent internal diagnostics and exhibits smoother contours than those obtained with a classical Tikhonov regularization applied to the same data. Quantitatively, the proposed method reduces the average error by roughly 30 % and requires about 20 % fewer iterations to converge.

The discussion highlights several practical aspects: (i) the importance of an accurate estimate of α, (ii) the computational cost versus accuracy trade‑off inherent in mesh density, and (iii) the potential extension to fully nonlinear current density profiles and three‑dimensional configurations. The authors conclude that minimizing the energy error functional provides a stable, accurate, and computationally feasible framework for solving the Cauchy problem in Tokamak vacuum magnetic reconstruction. This approach could be integrated into real‑time plasma control systems and aid in the design and optimization of future fusion devices.