The current research in the field is significantly advancing the methodologies for solving linear systems and inverse problems. There is a notable shift towards integrating dynamic and adaptive algorithms that reduce computational complexity while maintaining high accuracy. These methods often combine traditional iterative techniques with modern optimization strategies, such as reinforcement learning and particle swarm optimization, to enhance performance and versatility. Notably, the field is witnessing innovative approaches to parameter estimation in differential equations, which are overcoming the limitations of traditional methods by leveraging advanced optimization techniques. Additionally, there is a growing focus on deriving convergence rates for iterative methods in Banach spaces, with new strategies being developed to handle variable step sizes and stochastic systems. These developments collectively indicate a move towards more efficient, adaptive, and robust solutions in the field of linear systems and inverse problems.