Range-relaxed strategy applied to the Levenberg–Marquardt method with uniformly convex penalization term in Banach spaces

In this paper we propose the employment of the so-called range-relaxed criteria Boiger et al (2020 IMA J. Numer. Anal. 40 606–627) for choosing the regularization parameters (or equivalently, the Lagrange multipliers) of the Levenberg–Marquardt method for solving nonlinear ill-posed problems in Banach spaces. The proposed algorithm employs the Bregman distance induced by a uniformly convex functional and allows the use of a penalization generated from the total variation semi-norm. We present a geometrical interpretation of the method and deliver a complete convergence analysis, including stability and regularization properties. Further, we show that our new method is competitive by testing it with real data in the complete electrode model of 2D electrical impedance tomography.