Supervision : Fabienne JÉZÉQUEL

Co-supervision : BOITEAU Olivier, MARY Théo

Using Block Low-Rank compression in mixed precision for sparse direct linear solvers

EDF performs numerical simulations in different domains of physics. Several of its softwares use the library MUMPS in order to perform the costly step of solving sparse linear systems in a way that is generic, robust and efficient. The goal of this work is to develop new techniques for improving the performance gains of an existing functionality of MUMPS, the Block Low-Rank (BLR) compression. By combining several formats of floating-point numbers (mixed precision), it is possible to reduce the time and memory complexities, without compromising the accuracy of the result. Based on an error analysis, we design new variants of the LU factorization of dense matrices. We then adapt this work to the case of a sparse matrix factorization with MUMPS. A first implementation uses our mixed-precision BLR compression as a storage format, thus reducing the memory consumption of MUMPS. A second implementation allows combining these memory gains with time gains during the resolution of triangular systems. Finally, we study new techniques for improving the data locality of the BLR triangular solve with multiple right-hand sides, and obtain time reductions within MUMPS.

2021-2024 Publications

• 2024
  • P. Amestoy, O. Boiteau, A. Buttari, L. Combe, F. Faucher, M. Gerest, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary, S. Operto, Ch. Puglisi : “Recent advances on sparse direct solver using Block Low-Rank and mixed precision for large scale applications”, SIAM Conference on Applied Linear Algebra (LA24), Paris, France (2024)
  • P. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.‑Y. L’excellent, Th. Mary : “Communication Avoiding Block Low-Rank Parallel Multifrontal Triangular Solve with Many Right-Hand Sides”, SIAM Journal on Matrix Analysis and Applications, vol. 45 (1), pp. 148-166, (Society for Industrial and Applied Mathematics) (2024)
• 2023
  • M. Gerest : “Using Block Low-Rank compression in mixed precision for sparse direct linear solvers”, thesis, phd defence 11/08/2023, supervision Jézéquel, Fabienne, co-supervision : Boiteau, Olivier, Mary, Théo (2023)
  • S. Operto, P. Amestoy, S. Beller, A. Buttari, L. Combe, V. Dolean, M. Gerest, G. Guo, P. Jolivet, J.‑Y. L'Excellent, F. Mamfoumbi, Th. Mary, Ch. Puglisi, A. Ribodetti, P.‑H. Tournier : “Is 3D frequency-domain FWI of full-azimuth/long-offset OBN data feasible? The Gorgon case study”, Leading Edge, vol. 42 (3), pp. 173-183, (Society of Exploration Geophysicists) (2023)
  • P. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary : “Reducing communications and memory costs of parallel Block Low-Rank solvers”, SIAM Conference on Computational Science and Engineering (CSE 2023), Amsterdam, Netherlands (2023)
• 2022
  • P. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary : “Mixed Precision Low Rank Approximations and their Application to Block Low Rank LU Factorization”, IMA Journal of Numerical Analysis, (Oxford University Press (OUP)) (2022)
  • M. Gerest, P. Amestoy, O. Boiteau, A. Buttari, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary : “Block low-rank compression in mixed precision for the solution of sparse linear systems”, Sparse Days conference 2022, Saint-Girons, France (2022)
  • P. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary : “Solving linear systems efficiently using block low-rank compression in mixed precision”, 45^e Congrès National d'Analyse Numérique (CANUM 2022), Évian-les-Bains, France (2022)
  • P. Amestoy, A. Buttari, F. Faucher, J.‑Y. L'Excellent, M. Gerest, Th. Mary : “Mixed precision sparse direct solver applied to 3D wave propagation”, 8^th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2022), Oslo, Norway (2022)
• 2021
  • P. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.‑Y. L'Excellent, Th. Mary : “Solving linear systems efficiently using mixed-precision Block Low-Rank compression”, Rencontres Arithmétique de l'informatique Mathématique (RAIM 2021), Paris, France (2021)