Convergence Rates of Levenberg–Marquardt Regularization Under General Source Conditions

Abstract

We investigate the convergence of the Levenberg–Marquardt (LM) iterative regularization method for linear ill-posed inverse problems in Hilbert spaces under general source conditions characterized by admissible index functions. We introduce admissibility conditions tailored to the spectral filter structure of the LM iteration, extending the classical convergence theory beyond Hölder-type source conditions to a broader class of index functions, including logarithmic source conditions relevant to severely ill-posed inverse problems. Using the filter representation of the LM iteration, we derive convergence rate estimates under both a priori and an a posteriori parameter choice rules. The analysis provides a unified framework for convergence analysis that covers a broad class of admissible index functions. Numerical experiments involving a time-fractional backward heat problem and a Fredholm integral equation of the first kind are consistent with the theoretical convergence results and demonstrate close agreement between the predicted error estimates and the observed reconstruction errors under both Hölder-type and logarithmic source conditions.