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This paper proposes a statistical and probabilistic approach to compare and analyze the errors of two different approximation methods. We introduce the principle of numerical uncertainty in such a process, and we illustrate it by considering the discretization difference between two different approximation orders, e.g., first and second order Lagrangian finite element. Then, we derive a probabilistic approach to define and to qualify equivalent results. We illustrate our approach on a model problem on which we built the two above mentioned finite element approximations. We consider some variables as physical “predictors”, and we characterize how they influence the odds of the approximation methods to be locally “same order accurate”. [ABSTRACT FROM PUBLISHER]

This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov–Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov–Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P 1 and a P 2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov–Maxwell equations take a central part. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: This paper proposed a new approach based on data mining to evaluate the e_ciency of numerical asymptotic models. Indeed, data mining has proved to be an e_cient tool of analysis in several domains. In this work, we first derive an asymptotic paraxial approximation to model ultrarelativistic particles. Then, we use data mining methods that directly deal with numerical results of simulations, to understand what each order of the asymptotic expansion brings to the simulation results. This new approach o_ers the possibility to understand, on the numerical results themselves, the precision level of a numercial asymptotic model. [Copyright &y& Elsevier]

Abstract: We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications. Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial approximation to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less accurate means. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models. [Copyright &y& Elsevier]

Abstract: We propose a novel approach that consists in using data mining techniques to perform a sensitivity analysis of approximate models. We give here the example of asymptotic solutions to Vlasov–Maxwell equations, obtained with a paraxial model for relativistic short beams. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models. [ABSTRACT FROM AUTHOR]