Comptes Rendus
Error estimate evaluation in numerical approximations of partial differential equations: A pilot study using data mining methods
Comptes Rendus. Mécanique, Volume 341 (2013) no. 3, pp. 304-313.

In this Note, we propose a new methodology based on exploratory data mining techniques to evaluate the errors due to the description of a given real system. First, we decompose this description error into four types of sources. Then, we construct databases of the entire information produced by different numerical approximation methods, to assess and compare the significant differences between these methods, using techniques like decision trees, Kohonenʼs cards, or neural networks. As an example, we characterize specific states of the real system for which we can locally appreciate the accuracy between two kinds of finite elements methods. In this case, this allowed us to precise the classical Bramble–Hilbert theorem that gives a global error estimate, whereas our approach gives a local error estimate.

Dans cette Note, on propose une nouvelle méthodologie fondée sur les techniques exploratoires du data mining afin dʼévaluer les erreurs suscitées par la description dʼun système physique donné. Pour ce faire, on identifie quatre type de sources dʼerreurs. On constitue alors une base de données regroupant lʼensemble des résultats numériques calculés par différentes méthodes dʼapproximation, afin dʼen comparer les différences significatives, par des techniques telles que les arbres de décision, les cartes de Kohonen, ou encore les réseaux de neurones. À titre dʼexemple, nous caractérisons des états spécifiques du système réel pour lesquels on peut localement estimer la différence de précision entre deux méthodes dʼéléments finis. Il est ainsi possible de préciser les résultats classiques du théorème de Bramble–Hilbert qui procure une estimation globale, alors que notre méthode propose une caractérisation locale des méthodes dʼapproximation considérées.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2013.01.002
Keywords: Data mining, Error estimate, Vlasov–Maxwell equations, Asymptotic analysis, Paraxial model
Mot clés : Data mining, Estimation dʼerreur, Équations de Vlasov–Maxwell, Analyse asymptotique, Modèle paraxial

Franck Assous 1; Joël Chaskalovic 2

1 Ariel University Center & Bar-Ilan University, Ramat Gan, Israel
2 University Pierre-and-Marie-Curie, 4, place Jussieu, 75202 Paris cedex 05, France
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Franck Assous; Joël Chaskalovic. Error estimate evaluation in numerical approximations of partial differential equations: A pilot study using data mining methods. Comptes Rendus. Mécanique, Volume 341 (2013) no. 3, pp. 304-313. doi : 10.1016/j.crme.2013.01.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.01.002/

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