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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


CMIP6 debriefing workshop – May 2021: a stocktaking and foresight exercise that highlighted the major and long-term investment of the CNRM-CERFACS group in the latest climate model intercomparison international exercise

CERFACS |  9 June 2021

The participation of the CNRM-CERFACS group in the international coupled global climate model intercomparison exercise CMIP6 was debriefed on 10 and 11 May. Since the mid-1990s, the various phases of the Coupled Model Intercomparison Project (CMIP), supported by the World Climate Research Programme (WCRP), have enabled climate science to make significant progress, feeding into the IPCC reports, supporting regional climate modelling activities and downstream climate services. On the balance sheet, the participation of CNRM-CERFACS in the extremely demanding CMIP6 exercise has resulted in a large-scale, high-quality contribution in terms of climate simulations, thanks in particular to the strong commitment of researchers and engineers, the quality of support services and national and international collaborations. The activities mainly took place over the period 2013-2020. Initially, the production environment ("workflow") was set up and the coupled climate models were assembled by capitalising on the development of components (atmosphere, ocean and continental surfaces) carried out over the long term. The actual production of simulations was spread over 2018-2020. During the debriefing, perspectives (scientific objectives, model development, organisation) were drawn up with a view to a new participation of CNRM and CERFACS in CMIPRead more

CERFACS Combustion paper on rocket engines selected as Distinguished Paper at the last Int. Symp. on Comb. in Adelaide

CERFACS |  14 April 2021

The paper of C. Laurent 'Heat-release dynamics in a doubly- transcritical LO2/LCH4 cryogenic coaxial jet flame subjected to fuel inflow acoustic modulation’  has been selected at the Distinguished Paper in the Gas Turbine and Rocket Engine Combustion colloquium for the 38th International Symposium on Combustion. This paper authored by Laurent, Staffelbach, Nicoud  and  T. Poinsot  is available here:  describes the first LES of a forced doubly transcritical flame.Read more