Addressing large-scale linear systems derived from partial differential equations presents numerous challenges in both computational complexity and memory storage. However, these challenges create a fertile ground for the exploration of innovative approaches, particularly in the realm of extensive data processing and the utilization of massively parallel supercomputers [1]. Over the last decade, a paradigm shift has emerged, introducing a novel methodology that incorporates randomization to expedite linear algebra operations [2]. This groundbreaking approach exhibits remarkable efficiency, achieving a linear complexity of o(N) with respect to problem size. This efficacy has been validated across various dense linear systems, encompassing integral equations, statistics, and machine learning domains. In contrast to our prior works [1,3], which primarily dealt with dense and low-rank systems, this presentation delves into the opportunities and challenges inherent in extending the randomization approach to intricate sparse systems arising from fluid and solid mechanics problems. These systems, characterized by sparsity and ill-conditioning, demand a nuanced exploration of algorithmic adaptations. The central focus will be on the integration of appropriate sketching techniques into the construction of the krylov method, thereby optimizing the performance of the underlying iterative solver. Throughout the discussion, particular attention will be given to the embedding sketching methodologies and their impact on enhancing the efficiency of the krylov method. The talk will also showcase the applicability of this innovative paradigm through the presentation of 2d and 3d applications. These real-world examples aim to provide a comprehensive assessment of the newfound efficiency and adaptability of the proposed approach within the context of solving complex linear systems derived from fluid and solid mechanics problems. REFERENCES [1] R. Alomairy, W. Bader, H. Ltaief, Y. Mesri, D. Keyes, High-performance 3D Unstructured Mesh Deformation Using Rank Structured Matrix Computations, ACM Trans. Parallel Comput. 9(1), 2022, pp. 1-23 [2] N. Halko, P.G. Martinsson, J. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions., SIAM Review, 53(2), 2011, pp. 217-288. [3] W. Daldoul, E. Hachem, Y. Mesri, A ‘R-to-H’Mesh Adaptation Approach for Moving Immersed Complex Geometries Using Parallel Computers., International