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CERN
Genève
18 December 2013
Scientific collaboration with CERN
F. Assous
J. Chaskalovic
Mathematics Department
Institut Jean le Rond d’Alembert
Bar Ilan Ariel University
University Pierre and Marie Curie
Agenda
The team
Theoretical and numerical approaches for
charged particle beams
Actual items
Our future projects
Data Mining for the CERN
The team
The team
Franck Assous: Academic and industrial experiment
– PhD in Applied Mathematics (Dauphine University, Paris 9).
– Associate Professor in Applied Mathematics, 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 Mathematical Modeling applied to
Engineering Sciences, (University Pierre & Marie Curie).
– Director of Data Mining and Media Research, Publicis Group,
(1993-2007).
Theoretical and numerical
approaches for particles accelerators
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
A new method to evaluate
asymptotic numerical models
by Data Mining techniques
The physical problem
• Physical frameworks: collisionless charged particles
beams (Accelerators, F.E.L, …)
The mathematical model
Approximate models
Exploit given physical/experimental assumptions:
 Neglect the time derivative
Poisson model
 Neglect the time derivative
Magneto-static model
 Neglect the transverse part of
Darwin model
 Use the paraxial property
Paraxial model
How we derive a paraxial model
1. Write the equations in the beam frame.
2. Introduce a scaling of the equations.
3. Define a small parameter.
4. Use expansion techniques and retains the first orders.
5. Build an ad hoc discretization.
6. Simulations with numerical results.
The asymptotic expansions
The first paraxial model
(axisymmetric case)
• Zero order:
• First order:
• Second order:
Numerical Results
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
Data processing
100 time steps
1250 space nodes
125 000 rows
26 columns
The Database
Data Modeling
1,2 is around 1: equivalence of numerical results
obtained between the two models M1 and M2 for
the calculation of X.
1,2 is either very small or very great compared to 1:
the numerical results between M1 and M2 are
significantly different.
Data Mining Exploration
Significant differences between Vr (1) and Vr (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)
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
On a new paraxial model
Revisiting the scaling
Lr
Lz
Z = Lz Z’, r = Lr r’, (Lz Lr)
The characteristic longitudinal dimension Lz is chosen
different from the characteristic transverse dimension Lr.
The new paraxial model
(axisymmetric case)
• Zero order:
• First order:
(F. Assous and J. Chaskalovic, CRAS, 2012)
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.
Data Mining
a tool to evaluate the quality of models
The four sources of error
Error sources

The modeling error

The approximation error

The discretization error

The parameterization error
The famous theorems of calculus
Rolle’s theorem
Lagrange’s theorem
Taylor’s theorem
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:
The discretization error
P1 - P2 finite elements method for numerical
approximation to Vlasov-Maxwell equations
The P1 – P2 finite elements Database
“surprising ” rows w.r.t Bramble Hilbert
theorem
If |Er2-Er1| ≤ 0.65 (5% of Max |Er2-Er1|)
P1 vs P2 = Same order
Same order  14 % of the Dataset
Kohonen’ cards
Kohonen’s Cluster Analysis
Rules of Cluster “P1 – P2 same order”
P1
P1vs
vs P2
P2
An example :
Equivalent results between P1 and P2 finite 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 100 time steps.
P2 finite elements overqualified
at the beginning of the propagation
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
Data Mining for the CERN
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)
Data Mining : les clefs pour une
exploitation pertinente des données
Project Management
•
•
Business
Expertise
•
•
Data Exploration
Software
Engineering
Data Mining and not Data Analysis
The Data Mining is a discovery process
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
Data Mining Principles
Supervised Data Mining : One or more target variables must
be explained in terms of a set of predictor variables.
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.
Outlooks
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.
Data Mining with CERN
Merci !