The remarkable progress of Artificial Intelligence (AI) in recent years is starting to greatly influence research across a wide range of disciplines. As Numerical Linear Algebra plays a crucial role in Deep Learning models, this trend presents unprecedented opportunities for experts in numerical analysis and linear algebra to contribute to ongoing AI research. This proposal represents a step toward capitalizing on this opportunity. The focus of the proposed work is not on applying AI to solve a specific problem, but rather on enhancing AI methods themselves by exploiting insights from numerical methods to optimize the Deep Learning process. This process is time-consuming, energy-intensive, resource-demanding, and overall very costly. Therefore, any improvements that can speed up the process are likely to have a significant impact. The investigators will leverage their experience in numerical methods to develop a number of techniques for accelerating the training of large AI models. The project aims to develop techniques that exploit both accelerators and preconditioners to speed up iterative procedures used in training deep learning models. The same combination of preconditioning and acceleration techniques is central to the effectiveness of iterative solution methods for linear systems. Acceleration methods such as Anderson/Pulay mixing or the Reduced Rank Extrapolation method, among others, have had immense success across various fields of science and engineering. Howev