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 and a 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.
Accepted:
Published online:
Franck Assous 1, 2; Joël Chaskalovic 3
@article{CRMECA_2014__342_10-11_560_0, 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}, }
TY - JOUR 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 -
Franck Assous; Joël Chaskalovic. Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining. Comptes Rendus. Mécanique, Theoretical and numerical approaches for Vlasov-maxwell equations, Volume 342 (2014) no. 10-11, pp. 560-569. doi : 10.1016/j.crme.2014.06.010. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.06.010/
[1] Plasmas Physics via Computer Simulation, MacGraw-Hill, New York, 1985
[2] On the paraxial approximation of the stationary Vlasov–Maxwell, Math. Models Methods Appl. Sci., Volume 3 (1993) no. 4, pp. 513-562
[3] The finite Larmor radius approximation, SIAM J. Math. Anal., Volume 32 (2001) no. 6, pp. 1227-1247
[4] Paraxial approximation of ultrarelativistic intense beams, Numer. Math., Volume 69 (1994) no. 1, pp. 33-60
[5] The ARCTIC charged particle beam propagation code, J. Comput. Phys., Volume 128 (1996) no. 2, pp. 489-497
[6] A hierarchy of approximate models for the Maxwell equations, Numer. Math., Volume 73 (1996) no. 3, pp. 329-372
[7] ELBA – a three dimensional particle simulation code for high current beams, Annapolis, MD, USA (1991)
[8] A particle-tracking method for 3D electromagnetic PIC codes on unstructured meshes, Comput. Phys. Commun., Volume 72 (1992) no. 6, pp. 105-114
[9] Numerical paraxial approximation for highly relativistic beams, Comput. Phys. Commun., Volume 180 (2009), pp. 1086-1097
[10] 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] Conservative semi-Lagrangian schemes for the Vlasov equation, J. Comput. Phys., Volume 229 (2010), pp. 1927-1953
[12] 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] From data mining to nanotechnology and back: the new problems of numerical linear algebra, Computer Science & Eng. Colloquium, University of Minnesota, November 2009
[14] 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] et al. Explicative factors for prognostics IIU: exploration on 2089 cycles done with statistical and data mining tools, Palais des Congrès, Paris (2004)
[16] 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] 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] Data Mining – Gestion de la Relation Client, Eyrolles, Paris, 2001
[19] A new approach in media/marketing databases explorations for application in e-business, Paris (1999)
[20] Innovation in estimations: a reliable approach for radio audience indicators, WM3 2007, Dublin, 3–6 June (2007)
[21] 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] The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics, 2009
[23] Self-Organizing Maps, Springer Series in Inf. Sci., vol. 30, Springer, Berlin, 2001
[24] Self-Organizing Neural Networks: Recent Advances and Applications, Physica Verlag, Heidelberg, Germany, 2002
[25] Data Mining with Decision Trees: Theory and Applications, World Scientific Publishing Company, 2001
[26] Seismic Migration, Elsevier, Amsterdam, Oxford, New York, Tokyo, 1984
[27] Imaging the Earth's Interior, Blackwell Scientific Publications, Oxford, 1985
[28] A new paraxial asymptotic model for the relativistic Vlasov–Maxwell equations, C. R. Mecanique, Volume 340 (2012), pp. 706-714
[29] 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] Mathematical and Numerical Methods for Partial Differential Equations, Springer Verlag, 2014
[31] FreeFem++, Numerical mathematics and scientific computation 3.7, Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, Paris, 2010 http://www.freefem.org/ff++/
[32] Computer Simulation Using Particles, Adam Hilger imprint by IOP Publishing Ltd., 1988
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