Comptes Rendus
Theoretical and numerical approaches for Vlasov–Maxwell equations
Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining
Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 560-569.

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 P1 and a P2 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.

Published online:
DOI: 10.1016/j.crme.2014.06.010
Keywords: Data mining, Error estimate, Vlasov–Maxwell equations, Asymptotic analysis, Paraxial model

Franck Assous 1, 2; Joël Chaskalovic 3

1 Ariel University, 40700 Ariel, Israel
2 Bar-Ilan University, 52900 Ramat-Gan, Israel
3 D'Alembert, University Pierre and Marie Curie, Paris, France
     author = {Franck Assous and Jo\"el Chaskalovic},
     title = {Mathematical and numerical methods for {Vlasov{\textendash}Maxwell} equations: {The} contribution of data mining},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {560--569},
     publisher = {Elsevier},
     volume = {342},
     number = {10-11},
     year = {2014},
     doi = {10.1016/j.crme.2014.06.010},
     language = {en},
AU  - Franck Assous
AU  - Joël Chaskalovic
TI  - Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining
JO  - Comptes Rendus. Mécanique
PY  - 2014
SP  - 560
EP  - 569
VL  - 342
IS  - 10-11
PB  - Elsevier
DO  - 10.1016/j.crme.2014.06.010
LA  - en
ID  - CRMECA_2014__342_10-11_560_0
ER  - 
%0 Journal Article
%A Franck Assous
%A Joël Chaskalovic
%T Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining
%J Comptes Rendus. Mécanique
%D 2014
%P 560-569
%V 342
%N 10-11
%I Elsevier
%R 10.1016/j.crme.2014.06.010
%G en
%F CRMECA_2014__342_10-11_560_0
Franck Assous; Joël Chaskalovic. Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining. Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 560-569. doi : 10.1016/j.crme.2014.06.010.

[1] C.K. Birdsall; A.B. Langdon Plasmas Physics via Computer Simulation, MacGraw-Hill, New York, 1985

[2] P. Degond; P.-A. Raviart On the paraxial approximation of the stationary Vlasov–Maxwell, Math. Models Methods Appl. Sci., Volume 3 (1993) no. 4, pp. 513-562

[3] E. Frénod; E. Sonnendrücker The finite Larmor radius approximation, SIAM J. Math. Anal., Volume 32 (2001) no. 6, pp. 1227-1247

[4] G. Laval; S. Mas-Gallic; P.-A. Raviart Paraxial approximation of ultrarelativistic intense beams, Numer. Math., Volume 69 (1994) no. 1, pp. 33-60

[5] M.A. Mostrom; D.I. Mitrovich; D.I.R. Welch The ARCTIC charged particle beam propagation code, J. Comput. Phys., Volume 128 (1996) no. 2, pp. 489-497

[6] P.A. Raviart; E. Sonnendrücker A hierarchy of approximate models for the Maxwell equations, Numer. Math., Volume 73 (1996) no. 3, pp. 329-372

[7] S. Slinker; G. Joyce; J. Krall; R.F. Hubbard ELBA – a three dimensional particle simulation code for high current beams, Annapolis, MD, USA (1991)

[8] F. Assous; P. Degond; J. Segré A particle-tracking method for 3D electromagnetic PIC codes on unstructured meshes, Comput. Phys. Commun., Volume 72 (1992) no. 6, pp. 105-114

[9] F. Assous; F. Tsipis Numerical paraxial approximation for highly relativistic beams, Comput. Phys. Commun., Volume 180 (2009), pp. 1086-1097

[10] N. Besse; G. Latu; A. Ghizzo; E. Sonnendrücker; P. Bertrand A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov–Maxwell system, J. Comput. Phys., Volume 227 (2008) no. 16, pp. 7889-7916

[11] N. Crouseilles; M. Mehrenberger; E. Sonnendrücker Conservative semi-Lagrangian schemes for the Vlasov equation, J. Comput. Phys., Volume 229 (2010), pp. 1927-1953

[12] F. Filbet Convergence of a finite volume scheme for the one dimensional Vlasov–Poisson system, SIAM J. Numer. Anal., Volume 39 (2001) no. 4, pp. 1146-1169

[13] Y. Saad From data mining to nanotechnology and back: the new problems of numerical linear algebra, Computer Science & Eng. Colloquium, University of Minnesota, November 2009

[14] F. Assous; J. Chaskalovic Data mining techniques for scientific computing: application to asymptotic paraxial approximations to model ultra-relativistic particles, J. Comput. Phys., Volume 230 (2011), pp. 4811-4827

[15] O. Kulski; J. Chaskalovic et al. Explicative factors for prognostics IIU: exploration on 2089 cycles done with statistical and data mining tools, Palais des Congrès, Paris (2004)

[16] X.L. Nguyên; J. Chaskalovic et al. Residual subjective daytime sleepiness under CPAP treatment in initially somnolent apnea patients: a pilot study using data mining methods, Sleep Med., Volume 9 (2007) no. 5, pp. 511-516

[17] X.L. Nguyên; J. Chaskalovic et al. Insomnia symptoms and CPAP compliance in OSAS patients: a descriptive study using data mining methods, Sleep Med., Volume 11 (2010) no. 8, pp. 777-784

[18] R. Lefébure; G. Venturi Data Mining – Gestion de la Relation Client, Eyrolles, Paris, 2001

[19] J. Chaskalovic A new approach in media/marketing databases explorations for application in e-business, Paris (1999)

[20] J. Chaskalovic; A. Vanheuverzwyn Innovation in estimations: a reliable approach for radio audience indicators, WM3 2007, Dublin, 3–6 June (2007)

[21] F. Assous; J. Chaskalovic Error estimate evaluation in numerical approximations of partial differential equations: a pilot study using data mining methods, C. R. Mecanique, Volume 341 (2013), pp. 304-313

[22] T. Hastie; R. Tibshirani; J. Friedman The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics, 2009

[23] T. Kohonen Self-Organizing Maps, Springer Series in Inf. Sci., vol. 30, Springer, Berlin, 2001

[24] U. Seiffert; L. Jain Self-Organizing Neural Networks: Recent Advances and Applications, Physica Verlag, Heidelberg, Germany, 2002

[25] L. Rokach; O. Maimon Data Mining with Decision Trees: Theory and Applications, World Scientific Publishing Company, 2001

[26] A.J. Berkhout Seismic Migration, Elsevier, Amsterdam, Oxford, New York, Tokyo, 1984

[27] J.F. Claerbout Imaging the Earth's Interior, Blackwell Scientific Publications, Oxford, 1985

[28] F. Assous; J. Chaskalovic A new paraxial asymptotic model for the relativistic Vlasov–Maxwell equations, C. R. Mecanique, Volume 340 (2012), pp. 706-714

[29] F. Assous; F. Tsipis A PIC method for solving a paraxial model of highly relativistic beams, J. Comput. Appl. Math, Volume 227 (2009) no. 1, pp. 136-146

[30] J. Chaskalovic Mathematical and Numerical Methods for Partial Differential Equations, Springer Verlag, 2014

[31] F. Hecht FreeFem++, Numerical mathematics and scientific computation 3.7, Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, Paris, 2010

[32] R.W. Hockney; J.W. Eastwood Computer Simulation Using Particles, Adam Hilger imprint by IOP Publishing Ltd., 1988

Cited by Sources:

Comments - Policy