<figure><img src="image_1.png" /></figure>

HAL
open science

# Innovating Linear System Solutions in Fluid and Solid Mechanics through Randomized Algorithms

Youssef Mesri, Wael Bader, Sébastien da Viga, Augustin Parret-Fréaud

► To cite this version:

Youssef Mesri, Wael Bader, Sébastien da Viga, Augustin Parret-Fréaud. Innovating Linear System Solutions in Fluid and Solid Mechanics through Randomized Algorithms. ECCOMAS -9th European Congress on Computational Methods In Applied Science and Engineering, Jun 2024, Lisbon, Portugal. hal-04888396

HAL Id: hal-04888396

https://hal.science/hal-04888396v1

Submitted on 15 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Innovating Linear System Solutions in Fluid and Solid Mechanics through Randomized Algorithms

Y. Mesri*, W. Bader, S. Da Veiga, A. Parret-Fréaud

* Mines Paris - PSL - CEMEF
Campus Pierre Laffitte, 06904 Sophia Antipolis, France

## ABSTRACT

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 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[4-10]. 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.

## REFERENCES

<footnote>[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</footnote>

<footnote>[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.</footnote>

<footnote>[3] W. Daldoul, E. Hachem, Y. Mesri, A ‘R-to-H’Mesh Adaptation Approach for Moving Immersed Complex Geometries Using Parallel Computers., International Journal of Computational Fluid Dynamics 34 (7-8), 2020, pp. 597-609.</footnote>

<footnote>[4] Y. Mesri, H. Guillard, Coupez T. Automatic coarsening of three dimensional anisotropic unstructured meshes for multigrid applications. Appl Math Comput., 218(21), 2012, pp. 10500-10519.</footnote>

<footnote>[5] Y. Mesri, W. Zerguine, H. Digonnet, L. Silva, T. Coupez, Dynamic parallel adaption for three dimensional unstructured meshes: Application to interface tracking, in: Proceedings of the 17th International Meshing Roundtable, Springer, 2008, pp. 195-212.</footnote>

<footnote>[6] A Bazile, E Hachem, JC Larroya-Huguet, Y Mesri, Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: application to convection-diffusion problems., CMAME 331, 2018, pp. 94-115.</footnote>

<footnote>[7] Y. Yang, Y. Mesri, Learning by neural networks under physical constraints for simulation in fluid mechanics., Comp. & Fluids 248, 2022, pp. 105632.</footnote>

<footnote>[8] J Sari, F Cremonesi, M Khalloufi, F Cauneau, P Meliga, Y. Mesri, E. Hachem. Anisotropic adaptive stabilized finite element solver for RANS models., IJNMF, 86(11), 2018, pp. 717-736</footnote>

<footnote>[9] Y. Mesri, J.-M. Gratien, O. Ricois, R. Gayno. Parallel Adaptive Mesh Refinement for Capturing Front Displacements: Application to Thermal EOR Processes. SPE, SPE-166058, 2013</footnote>

<footnote>[10] Mesri, Y., Digonnet, H., Coupez, T.: Hierarchical adaptive multi-mesh partitioning algorithm on heterogeneous systems. Lect. Notes Comput. Sci. Eng. n74, 299-306 (2010)</footnote>