Applying the Lagrange multiplier technique to reconstruct a force-free magnetic field

Abstract

Context. The magnetic field in the solar corona can be reconstructed by extrapolating the data obtained by measuring the magnetic field in the photosphere. It is widely thought that the dynamics of the solar corona is governed by the magnetic field. The magnetic field in the corona can therefore be reconstructed using the physics of force-free fields. Several models have been developed so far to reconstruct the magnetic field, but shortcomings in this regard remain. Therefore, alternative models in this respect can still be proposed. Here we apply the Lagrange multiplier technique to render an optimization model. Aims. The main aim of the present paper is to propose a method of constrained optimization using the vector Lagrange multiplier and compare the results with those of preexisting models. Methods. Our main focus is on the conceptual modification of the optimization models. Since these models are computationally efficient and easy to implement, any possible progressive step would be welcome. The Lagrange multiplier technique is a powerful mathematical tool that has been successfully applied to many areas in physics. It may serve this purpose. In the absence of nonmagnetic forces such as pressure, gravity, and dissipative forces, the coronal medium is dominated by magnetic force. Thus, the function that is considered to be minimized may include a divergence term subject to the constraint force-free term, which yields a solenoidal and force-free magnetic field by an iterative process. Results. The numerical analysis of the proposed model was conducted through an artificial magnetogram and an observational vector magnetogram obtained by SDO/HMI images. The results obtained confirm that (i) the Lagrangian to be minimized in the present model converges slightly faster toward zero, at least for initial iteration steps, (ii) the energy variation during the optimization is compatible with the variations in previous studies, (iii) the numerical results seem to be compatible with a semi-analytical solution as the test case, and (iv) the model is applied to a real magnetogram, and relevant quantities such as magnetic energy content, the current-weighted angle between the current density vector and magnetic field, and the fractional flux errors are computed. Conclusions. The methods and techniques that convert a constrained problem into an unconstrained one are the penalty method, the barrier method, the augmented Lagrangian method, and the Lagrange multiplier technique, for instance. We have employed the Lagrange multiplier technique, by which any constrained condition could be added to the Lagrangian by an appropriate Lagrange multiplier. In our case, the constraint is the force-freeness of the magnetic field and is therefore a special case. The method has the following advantages: (i) The convergence rate is slightly higher for the initial iteration steps, which may help us for time-series events, while several magnetograms must be considered and limited steps of iteration may be needed. (ii) The Lagrangian is introduced based on the Lagrange multiplier technique, which facilitates first fixing a physical compromise such as a divergence-free condition, and subsequently adding any given constraint term. (iii) The quantities obtained by the constrained optimization with the vector Lagrange multiplier model, that is, the relative magnetic energy, the ratio of the magnetic energy to its initial value, the angle between the electric current and the magnetic field, fractional flux errors, the normalized vector error, the vector correlation, and the Cauchy-Schwarz indicator, are comparable with those of the comparison models considered in this article.