Regularization Methods for Solving Inverse Problems: A Comprehensive Review

Inverse problems arise in various scientific and engineering disciplines, presenting challenges of ill-posedness and sensitivity to noise. This work conducts a comprehensive review of regularization methods aimed at stabilizing solutions to inverse problems. Focusing on techniques such as Tikhonov regularization, machine learning-based regularization, and Bayesian regularization, we explore their mathematical foundations, numerical implementations, and applications in diverse fields. Numerical implementation aspects, addressing discretization, stability, and computational efficiency, are presented to guide researchers in the practical application of regularization methods.