A frozen Levenberg-Marquardt-Kaczmarz method with convex penalty terms and two-point gradient strategy for ill-posed problems

In this paper, we present a frozen iteratively regularized approach for solving ill-posed problems and conduct a thorough analysis of its performance. This method involves incorporating Nesterov's acceleration strategy into the Levenberg-Marquardt-Kaczmarz method and maintaining a constant Fréchet derivative of $F_i$ at an initial approximation solution $x_0$ throughout the iterative process, which called the frozen strategy. Moreover, convex functions are employed as penalty terms to capture the distinctive features of solutions. We establish convergence and regularization analysis by leveraging some classical assumptions and properties of convex functions. These theoretical findings are further supported by a number of numerical studies, which demonstrate the efficacy of our approach. Additionally, to verify the impact of initial values on the accuracy of reconstruction, the data-driven strategy is adopted in the third numerical example for comparison.