Scientific Collaboration with CERN: Theoretical and Numerical Approaches for Charged Particle Beams

1. CERN 18 December 2013 Genève Scientific collaboration with CERN F. Assous Mathematics Department Bar Ilan Ariel University J. Chaskalovic Institut Jean le Rond d’Alembert University Pierre and Marie Curie

2. Agenda • The team • Theoretical and numerical approaches for charged particle beams • Actual items • Our future projects • Data Mining for the CERN

3. The team

4. The team • Franck Assous: Academic and industrial experiment • PhD in Applied Mathematics (Dauphine University, Paris 9). • Associate Professor in AppliedMathematics, Bar Ilan Ariel University, (Israel). • Scientific Consultant, CEA, (France), (1990-2002). • Joel Chaskalovic: Dual expertise • PhD in Theoretical Mechanics (University Pierre & Marie Curie) and Engineer of « Ecole Nationale des Ponts & Chaussées ». • Associate Professor in MathematicalModelingapplied to Engineering Sciences, (University Pierre & Marie Curie). • Director of Data Mining and Media Research, Publicis Group, (1993-2007).

5. Theoretical and numerical approaches for particles accelerators

6. Actual items A new method to evaluate asymptotic numerical models by Data Mining techniques On a new paraxial model Data Mining: a tool to evaluate the quality of models

7. A new method to evaluate asymptotic numerical models by Data Mining techniques

8. Physical frameworks: collisionless charged particles beams (Accelerators, F.E.L, …) The physical problem

9. The mathematical model

10. Poisson model • Neglect the time derivative Magneto-static model • Neglect the time derivative • Neglect the transverse part of Darwin model • Use the paraxial property Paraxial model Approximate models Exploit given physical/experimental assumptions:

11. How we derive a paraxial model Write the equations in the beam frame. Introduce a scaling of the equations. Define a small parameter. Use expansion techniques and retains the first orders. Build an ad hoc discretization. Simulations with numerical results.

12. The asymptotic expansions

13. The first paraxial model (axisymmetric case) • Zero order: • First order: • Second order:

14. Numerical Results

15. But…fundamental questions Despite a theoretical result (controlled accuracy)… How many terms to retain in the asymptotic expansion to get a “precise” model ? How to compare the different orders of approximation: What each order of the asymptotic expansion brings to the numerical results ? Which variables are responsible of the improvement between models Mi and Mi+1 ? Use of Data Mining Methodology

16. Data processing 100 time steps 1250 space nodes 125 000 rows 26 columns

17. The Database

18. 1,2is around 1: equivalence of numerical results obtained between the two models M1 and M2 for the calculation of X. 1,2is either very small or very great compared to 1: the numerical results between M1 and M2 are significantly different. Data Modeling

19. Data Mining Exploration Significant differences between Vr(1) andVr(2) • Ez(2) is the most discriminate predictor. (Expected because Ez(1) = 0). • The second most important predictor is Er(2). (Non expected because Er(1)  0). • Bz(2) appears as a non significant predictor. (Non expected because Bz(1) = 0). (F. Assous and J. Chaskalovic, J. Comput. Phys., 2011)

20. Future developments • Which is the best asymptotic expansion? • Globally the second order is better than the first order. But locally, could we status when and where the first one could be better ? Data Experiments and Data Mining

21. On a new paraxial model

22. Revisiting the scaling The characteristic longitudinal dimension Lzis chosen different from the characteristic transverse dimension Lr. Lr Lz Z = Lz Z’, r = Lr r’, (Lz Lr)

23. The new paraxial model(axisymmetric case) • Zero order: • First order: (F. Assous and J. Chaskalovic, CRAS, 2012)

24. Future developments Numerical simulations. Validation and characterization by Data Mining techniques of significant differences between the two asymptotic models(Lz = Lr) and (Lz Lr). Comparison with experimental data.

25. Data Mining a tool to evaluate the quality of models

26. The four sources of error Error sources • The modelingerror • The approximation error • The discretization error • The parameterization error

27. Rolle’s theorem Lagrange’s theorem Taylor’s theorem The famous theorems of calculus

28. The discretization error • The discretization error is the error which corresponds to the difference of order between two numerical models (MN1) and (MN2) from a given family of approximations methods. • Suppose we solve a given mathematical model (E) with finite elements P1 and P2. Bramble-Hilbert theorem claims:

29. The discretization error P1 - P2 finite elements method for numerical approximation to Vlasov-Maxwell equations

30. The P1 – P2 finite elements Database

31. “surprising ” rows w.r.t Bramble Hilbert theorem If |Er2-Er1| ≤ 0.65 (5% of Max |Er2-Er1|) P1vs P2 = Same order Same order  14 % of the Dataset

32. Kohonen’ cards

33. Kohonen’s Cluster Analysis

34. P1 vs P2 P1 vs P2 Rules of Cluster “P1 – P2 same order”

35. An example : Equivalent results between P1andP2finite elements Er(1) and Er(2) are equivalent on 14% elements of the data. Data Mining techniques identified the number of time steps tn as the most discriminate predictor. The critical computed threshold of tn is equal to 42 on 100time steps. P2 finite elements overqualified at the beginning of the propagation

36. Future developments Physical interpretations of the above results : The threshold tn = 42. Robustness of the results: comparison with other data technologies, (Neural Networks, Kohonen Cards, etc.). Extensions to other physical unknowns. Sensibility regarding the Data. Coupling errors. Data Mining

37. Data Mining for the CERN

38. The CERN and the Data Mining « Les expériences du Large Hadron Collider représentent environ 150 millions de capteurs délivrant des données 40 millions de fois par seconde. Il y a autour de 600 millions de collisions par seconde, et après filtrage, il reste 100 collisions d’intérêt par seconde. En conséquence, il y a 25 Po de données à stocker chaque année. » (source : Wikipédia)

39. Data Mining : les clefs pour une exploitation pertinente des données Project Management Software Engineering Business Expertise Data Exploration

40. Data Mining and not Data Analysis • Data Scan : inventory of potential and explicative variables. • Data Management : collection, arrangement and presentation of the Data in the right way for mining. • Data Modeling : • Learning • Clustering • Forecasting The Data Mining is a discovery process

41. Supervised Data Mining: One or more targetvariables must be explained in terms of a set of predictor variables. Data Mining Principles • Segmentation by Decision Tree, Neural Networks, etc. • Non supervised Data Mining: No variable to explain, all available variables are considered to create groups of individuals with homogeneous behavior. • Typology by Kohonen’s cards, Clustering, etc.

42. Future developments Accuracy comparison of asymptotic models. Choice of a given order accuracy. Accuracy comparison of numerical methods. Curvature of the trajectories. Non relativistic beams. Etc. Outlooks Data Mining with CERN

43. Merci !