Étude de la propagation des erreurs d’arrondi induit par résolution haute performance de systèmes linéaires creux

Numerical verification of industrial codes, such as those developed at EDF R&D, is required to estimate the precision and the quality of computed results, even more for code running in HPC environments where millions of instructions are performed each second. These programs usually use external libraries (MPI, BLACS, BLAS, LAPACK}. In this context, it is required to have a tool as nonintrusive as possible to avoid rewriting the original code. In this regard, the CADNA library, which implement the Discrete Stochastic Arithmetic, appears to be one of a promising approach for industrial applications. In the first part of this work, we are interested in an efficient implementation of the BLAS routine DGEMM (General Matrix Multiply) implementing Discrete Stochastic Arithmetic. The implementation of a basic algorithm for matrix product using stochastic types leads to an overhead greater than 1000 for a matrix of 1024*1024 compared to the standard version and commercial versions of xGEMM. Here, we detail different solutions to reduce this overhead and the results we have obtained. A new routine DgemmCADNA have been designed. This routine has allowed to reduce the overhead from 1100 to 35 compare to optimized BLAS implementations (GotoBLAS).

Then, we focus on the numerical verification of Telemac-2D computed results. Performing a numerical validation with the CADNA library shows that more than 30% of the numerical instabilities occurring during an execution come from the dot product function. A more accurate implementation of the dot product with compensated algorithms is presented in this work. We show that implementing these kind of algorithms, in other to improve the accuracy of computed results does not alter the code performance.

2012-2013 Publications

• 2013
  • S. Montan : “Étude de la propagation des erreurs d’arrondi induit par résolution haute performance de systèmes linéaires creux”, thesis, phd defence 10/25/2013, supervision Chesneaux, Jean-Marie (2013)
  • S. Montan, J.‑M. Chesneaux, Ch. Denis, J.‑L. Lamotte : “Étude de la propagation des erreurs d’arrondi dans un code d’hydrodynamique parallèle”, Conférence d'informatique en Parallélisme, Architecture et Systeme, COMPAS 2013, Grenoble, France (2013)
• 2012
  • S. Montan, J.‑M. Chesneaux, Ch. Denis, J.‑L. Lamotte : “Towards an efficient implementation of CADNA in the BLAS : Example of DgemmCADNA routine.”, 15^th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN), Novosibirsk, Russian Federation (2012)
  • Ch. Denis, S. Montan : “Numerical Verification of Industrial Numerical Codes”, Congrès National de Mathématiques Appliquées et Industrielles, vol. 35, ESAIM: Proceedings, Guidel, France, pp. 107-113 (2012)