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Linear Algebra

Numerical linear algebra lies at the core of virtually all scientific computing. It provides the mathematical basis for operating on all geometric and image data, it is the foundation for the numerical solution of ordinary and partial differential equations, and it has become the central toolbox for the current hype-theme in computing: big data analytics.

The deep relationship between numerical linear algebra and differential as well as integral equations remains important to engineering and science. Unless explicit methods are being used, the discretization of PDEs leads to systems of linear or nonlinear equations. The nonlinear case requires the repeated solution of linear equations. For most equations and discretization methods, the linear matrix systems are sparse. CERFACS has thirty years of expertise in solving such systems and is internationally acknowledged as one of the leading research centres in this field. During the last decade, work has centred on developing direct methods with a high degree of parallelism. More recently, the focus at Cerfacs lies on highly-parallel iterative solvers and on using these in hybrid methods to extend the capability of direct methods. Furthermore, the Algo team concentrates on multigrid and multilevel methods which are among the primary candidates to achieve the scalability and performance needed for future extreme scale computing. To show the applicability to real-world engineering problems we have been testing a range of all these solution methods for large scale problems in collaboration with our academic and industrial partners.


Some examples of our recent research work:

Sparse direct methods, hybrid direct-iterative and iterative methods

We focus on core problems in numerical linear algebra that are relevant for both CERFACS, its shareholders and are of academic interest. For example, we worked on solution methods for linear systems arising in the structural analysis of a reactor containment building with our industrial partner EDF. The developed iterative solver showed excellent convergence in only a few iterations for this problem [2]. It has been made available to EDF's finite-element code code_aster through the implementation in the open source linear algebra library PETSc. An example of our academic research work is the recently finished PhD thesis of Philippe Leleux on Hybrid direct and iterative solvers for sparse indefinite and overdetermined systems on future exascale architectures, in which attention is given to iterative and hybrid methods.



Multilevel and multigrid methods

Multigrid methods belong to the few solvers that have scalable numerical complexity and are thus among the primary candidates to exploit future extreme scale computing to its full potential. A current research topic is to develop such hierarchical algorithms and the software techniques to implement them on massively parallel and heterogeneous computer systems. The algo team takes part in the H2020 Energy oriented Centre of Excellence EoCoE 2. Here, a tailored multigrid solver for the gyrokinetic Poisson equation on disk-like geometries described by curvilinear coordinates has been developed [1]. This algorithm has the potential for highly parallel scalability and, since it is a matrix free algorithm, it has a low memory footprint.


[1] Kuehn, M., Kruse, C., Ruede, U., Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation, 2020, https://hal.archives-ouvertes.fr/hal-02941899/document

[2] Kruse, C., Darrigrand, V., Tardieu, N. et al. Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints. Adv. Model. and Simul. in Eng. Sci. 7, 45 (2020). https://doi.org/10.1186/s40323-020-00181-2


The AVBP code from CERFACS at the heart of three PRACE projects from the 23rd call

CERFACS |  30 September 2021

Cerfacs is involved in three PRACE projects of the 23rd call for which hour allocation runs from 01/10/2021 to 30/09/2022. Researchers from ECL/LMFA UMR5509 (Ecole Centrale de Lyon) and IMFT (UMR 5502) laboratories have earned projects entirely based on the use of the LES solver developed by Cerfacs AVBP and involve the support of experts from the CFD and COOP teams underling the importance and effectiveness of collaborations between French labs and Cerfacs. Alexis Giauque from ECL/LMFA UMR5509 (Ecole Centrale de Lyon) has obtained not only one but two PRACE projects! The first project LESFAN (RA0101, 30 000 000 CPU hours on Irene/Rome TGCC) is based on the use of AVBP in the turbomachinery version to study the generation of noise by a fan of a real airplane engine. The second, PRACE-EDGES (RA0101, 40 000 000 CPU hours on Irene/Rome TGCC) focuses on LES modeling of dens gas in complex geometries. To do so, the LMFA Team has developed advanced thermodynamic closures in AVBP allowing the direct simulation of such flows. Laurent Selle from IMFT (UMR 5502) has received CPU hours for the GASTON project (RA0061, 30 000 000 CPU hours on Marenostrum BSC) which aims to study the structure of hydrogen flames in porous materials. For this, IMFT and Cerfacs will perform coupled simulations considering the reactive flow with AVBP as well as the conduction in the porous medium with AVTP which is known to play an central role in the flame stabilization process.Read more

watch the online contest ”my thesis in 180 sec” organized for Marie Curie actions

CERFACS |  24 September 2021

Next Thursday, Sept 30th the #FallingWallsLab #MSCA research presentation competition will take place, . This is a competition in which 15 researchers compete to deliver the best presentation of their research topic in the format "my thesis in 180 seconds" . Javier Crespo-Anadon, 3rd year PhD student in the CFD team is one of the 15 finalists and he will present this thesis topic: ignition in spinning combustion engines for a reduction in CO2 emissions! You can watch it online on the above website on Sept 30th at 3pm. I encourage you to go to the website and register!  You'll be able to listen to the other participants' pitches and vote for your favorite. Come and see for yourself what kind of research the EU funds!Read more