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Udo v. Toussaint, Silvio Gori

2-D Deconvolution using Thin-Plate Splines. Udo v. Toussaint, Silvio Gori. Astrophysics: F. Guglielmetti, R. Fischer, V. Dose, MaxEnt Proceedings 2004, p.111-118. Fusion research and Bayes. Fusion research: Huge number of diagnostics Complementary information

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Udo v. Toussaint, Silvio Gori

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  1. 2-D Deconvolution using Thin-Plate Splines Udo v. Toussaint, Silvio Gori Astrophysics: F. Guglielmetti, R. Fischer, V. Dose, MaxEnt Proceedings 2004, p.111-118

  2. Fusion research and Bayes • Fusion research: • Huge number of diagnostics • Complementary information • Long operation time (decades) Perfectly suited for Bayesian Analysis: Synergism by exploiting full probabilistic correlation structure of diagnostics: Integrated Data Analysis Possibility of Experimental Design

  3. Fusion Research Imaging Flux surface measurements Example: W7-X Thermography Video Integrated data analysis W7-X Diagnostics Software Ha Diagnostics Software (data analysis) Spectroscopy Vis. spectroscopy Zeff BES CXRS General purpose (stellarator) software SPRED CO-Monitor Diagnostic beam Soft-Xray Tomography Modelling Bolometry Infrastructure & management Interferometry Dedicated Diagnostics (misc.) Thomsonscattering ECE Neutron detectors 2-D diagnostics essential! Magnetics Neutral gas Langmuir Probes Calorimetry (by A. Dinklage)

  4. Fusion Research: Tomography Wide variety of tomographic configurations: experimental requirements, costs,… Goal: best possible reconstruction of 2-D profile

  5. Fusion Research: Tomography • Tomographic reconstruction: underdetermined & ill-posed • Prior information available • standard inversion techniques • mostly perform poorly ? ? ? ? ? ? ? ? ?

  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 IBof f(x,y) (can be generalized  ) Fundamental solution[1]: [1] G. Wahba, Spline models for Observational Data, SIAM, 1990

  7. Thin-Plate Splines Interpolating function z=f(x,y): Coefficients wi, a are given by the solution of with Kij=U(||(xi,yi)-(xj,yj)||), Pi=(1,xi,yi) The bending energy is given by

  8. Model Model: , Cij: response matrix gj : emissivity on grid space with Likelihood:

  9. Prior Distributions • Model prior factorizes: • Number of support points n on grid with N places: • Curvature : Testable information • Hyperparameter  :

  10. Posterior • Posterior for Model Mi : • Optimization preferred: Laplace Approximations… • 1-D search in : * • Evidence approximation: • Simple sampling in N,R

  11. Thin-Plate Splines: Results • Reconstruction of challenging (but realistic) 2-d emission profile: • Result of evidence weighted average Key features, shape and absolut intensity recovered Mock profile: Reconstruction: Difference:

  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. 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. Thin-Plate Splines: Conclusion Thank you!

  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

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