Numerical Verification of Industrial Numerical Codes

Several approximations occur during a numerical simulation: physical effects mapy be discarded, continuous functions replaced by discretized ones and real numbers replaced by finite-precision representations. The use of the floating point arithmetic generates round-off errors at each arithmetical expression and some mathematical properties are lost. The aim of the numerical verification activity at EDF R&D is to study the effect of the round-off error propagation on the results of a numerical simulation. It is indeed crucial to perform a numerical verification of industrial codes such as developped at EDF R&D even more for code running in HPC environments. This paper presents some recent studies around the numerical verification at EDF R&D. Résumé. Le résultat d’un code de simulation numérique subit plusieurs approximations effectuées lors de la modélisation mathématique du problème physique, de la discrétisation du modèle mathématique et de la résolution numérique en arithmétique flottante. L’utilisation de l’arithmétique flottante génère en effet des erreurs d’arrondi lors de chaque opération flottante et des propriétés mathématiques sont perdues. Il existe à EDF R&D une activité transverse de vérification numérique consistant à étudier l’effet de la propagation des erreurs d’arrondi sur les résultats des simulations. Il est en effet important de vérifier numériquement des codes industriels et ce d’autant plus s’ils sont éxécutés dans environnements de calcul haute performance. Ce papier présente des études récentes autour de la vérification numérique à EDF R&D. Introduction : the numerical verification at EDF R&D Several approximations occur during a numerical simulation. For example physical effects may be discarded, continuous functions replaced by discretized ones and real numbers replaced by finite-precision representations. The use of finite-precision arithmetic generates round-off errors at each arithmetical expression and some mathematical properties are lost. For example, floating point summation is no longer associative. The same numerical code using the same data could produce different results on different computers. For example, Goel & Dash in [1] present the numerical difference obtained by running the same weather prediction code, with the same input data, on three different computer architectures. There is obviously a need to detect the effect of round-off error propagation on the computed results. Several methods have been developed over the years to analyse round-off error propagation. These include direct analysis, inverse analysis, methods based on interval arithmetic, randomised interval arithmetic and the CESTAC method. At EDF R&D, there is a requirement to have a less intrusive tool to avoid having to rewrite the original code to study its numerical accuracy. The CADNA (Control of Accuracy and Debugging for Numerical Applications) library, developed by the Laboratoire d’Informatique de Paris 6 (http://www.lip6.fr/) appears the most promising approach for industrial applications. This paper is divided into two parts. In the first part, the 1 EDF R&D, SINETICS Department, 1, avenue du Général de Gaulle, Clamart, France. 2 EDF R&D, SINETICS Department, 1, avenue du Général de Gaulle, Clamart, France. c © EDP Sciences, SMAI 2011 ha l-0 07 65 53 6, v er si on 1 14 D ec 2 01 2 Author manuscript, published in "ESAIM: Proceedings 35 (2012) 107-113" DOI : 10.1051/proc/201235006