1. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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

2. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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:

3. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 3 Fusion Research Example: W7-X Thomson scattering ECE Magnetics Langmuir Probes Interferometry Bolometry Video H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 Dedicated Diagnostics (misc.) W7-X Diagnostics Software 2-D diagnostics essential! BES Z eff CO-Monitor (by A. Dinklage)

4. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 4 Fusion Research: Tomography Wide variety of tomographic configurations: experimental requirements, costs,… Goal: best possible reconstruction of 2-D profile

5. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 5 Fusion Research: Tomography Tomographic reconstruction: underdetermined & ill-posed Prior information available standard inversion techniques mostly perform poorly ? ? ? ? ? ? ? ? ?

6. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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 I B of f(x,y) (can be generalized ) Fundamental solution [1] : [1] G. Wahba, Spline models for Observational Data, SIAM, 1990

7. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 7 Thin-Plate Splines Interpolating function z=f(x,y): Coefficients w i, a are given by the solution of with K ij =U(||(x i,y i )-(x j,y j )||), P i =(1,x i,y i ) The bending energy is given by

8. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 8 Model Model:, C ij : response matrix g j : emissivity on grid space Likelihood: with

9. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 9 Prior Distributions Number of support points n on grid with N places: Curvature : Testable information Hyperparameter  : Model prior factorizes:

10. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 10 Posterior Posterior for Model M i : Evidence approximation: Simple sampling in N,R 1-D search in  :  * Optimization preferred: Laplace Approximations…

11. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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

12. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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

13. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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

14. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 14 Thin-Plate Splines: Conclusion Thank you!

15. MaxEnt 2007, Udo v. Toussaint, July 2007, udo.v.toussaint@ipp.mpg.de 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/m 2 Heating: 10 MW ECR 4 MW ICR 5 MW NBI