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MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 1
Udo v. Toussaint, Silvio Gori
2-D Deconvolution
using
Thin-Plate Splines
Astrophysics: F. Guglielmetti, R. Fischer, V. Dose, MaxEnt Proceedings 2004, p.111-118
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 2
Fusion research and Bayes
Fusion research:
• Huge number of diagnostics
• Complementary information
• Long operation time (decades)
Synergism by exploiting full probabilistic correlation structure of
diagnostics: Integrated Data Analysis
Possibility of Experimental Design
Perfectly suited for Bayesian Analysis:
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 3
Fusion Research
Example: W7-X
Thomson scattering
ECE
Magnetics
Langmuir Probes
Interferometry
Bolometry
Video
H
Calorimetry
Neutral gas
Vis. spectroscopy
CXRS
SPRED
Soft-X ray
Thermography
Flux surface measurements
Diagnostic beam
Neutron detectors
Diagnostics Software(data analysis)
General purpose (stellarator) software
Integrated data analysis
Modelling
Infrastructure &management
Spectroscopy
Imaging
Tomography
DedicatedDiagnostics(misc.)
W7-X DiagnosticsSoftware
2-D diagnostics essential!
BES
Zeff
CO-Monitor
(by A. Dinklage)
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 4
Fusion Research: Tomography
Wide variety of tomographic configurations: experimental requirements, costs,…
Goal: best possible reconstruction of 2-D profile
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 5
Fusion Research: Tomography
•Tomographic reconstruction: underdetermined & ill-posed
• Prior information available
standard inversion techniques
mostly perform poorly
??
?
??
? ??
?
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 6
Thin-Plate Splines
-Transport along magnetic field lines very fast: smooth profiles usually favored:
curvature only when enforced by the data
- 1D- case: cubic splines
- 2D- case: Minimize curvature IB of f(x,y) (can be generalized )
222 2 yyxyxxB fffdydxI
Fundamental solution[1]:
22 ln rrrU
[1] G. Wahba, Spline models for Observational Data, SIAM, 1990
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 7
Thin-Plate Splines
Interpolating function z=f(x,y):
3,||,,||, 01
nyaxaayxyxUwyxfz yxii
n
ii
Coefficients wi, a are given by the solution of
00
z
a
w
P
PKT
with Kij=U(||(xi,yi)-(xj,yj)||), Pi=(1,xi,yi)
The bending energy is given by zrwwKwwI TB ,,
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 8
Model
Model: jj
iji gCm , Cij: response matrix
gj : emissivity on grid space
Likelihood: 2
2
2,,|
i
iiii
mdIgdp
with ),( yxfdxdygjA
j
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 9
)|(,|),,|()|( InpInRpInRzpIMp
)),(),(exp(1
,,| zRwKzRwZ
nRzp T
!)3(
!)!()|,(
NN
nnNInRp
Prior Distributions
• Number of support points n on grid with N places:
• Curvature : Testable information
• Hyperparameter :
MaxMinIp ,/1)|(
• Model prior factorizes:
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 10
IpIRzpdIRzDpzdIMp
IRzpIRzDpzdIMpIMDpIMpIDMp
|,,|,,||
,|,,||,||,|
Posterior
• Posterior for Model Mi :
• Evidence approximation:
• Simple sampling in N,R
• 1-D search in : *
2/1**** )det(,|,,,,|,| HIMzpIMzDpIMDp
)det(
)2(,,|,,,||
,,|,,||,,|
0
**
HIMzpIMzDpIp
IMzpIMDpzdIpIMDp
n
• Optimization preferred: Laplace Approximations…
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 11
Thin-Plate Splines: Results
• Reconstruction of challenging (but realistic) 2-d emission profile:
• Result of evidence weighted average
Mock profile: Reconstruction: Difference:
Key features, shape and absolut intensity recovered
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 12
Thin-Plate Splines: Results
Location of z-optimized support points:
Open issue: Support points outside of region of interest beneficial?
Reconstruction within the uncertainty
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 13
Thin-Plate Splines: Outlook
• Extension to f(x,y,z): Viewing cone
• Extension to f(x,y,(z),t):
time correlated data
Connection to morphing applications
Drawback: Huge amounts of data to be processed
• Online (monitoring) requirements:Use as input for Bayesian Neural Networks
• Check approximations with MCMC
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 14
Thin-Plate Splines: Conclusion
Thank you!
MaxEnt 2007, Udo v. Toussaint, July 2007, [email protected] 15
W7-X: Design
• 50+36SL-Coils
• Optimized shape
• stability without current-drive
• Size:
major: 5.5m
minor: 0.53m
• Magnetic field:
3.3T (2.5T reg)
12kA, 38MW/m2
• Heating:
10 MW ECR
4 MW ICR
5 MW NBI
