Bayesian inference of impurity transport coefficient profiles
|
|
- Edwina Paul
- 5 years ago
- Views:
Transcription
1 Bayesian inference of impurity transport coefficient profiles M.A. Chilenski,, M. Greenwald, Y. Marzouk, J.E. Rice, A.E. White Systems and Technology Research MIT Plasma Science and Fusion Center/Alcator C-Mod MIT Aero/Astro, Uncertainty Quantification Group May, 7 This material is based upon work conducted using the Alcator C-Mod tokamak, a DOE Office of Science user facility. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Number DE-FC-99ER55. This material is based upon work supported in part by the U.S. Department of Energy Office of Science Graduate Research Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 9, administered by ORISE-ORAU under contract number DE-AC5-6OR. The XEUS and LoWEUS spectrometers were developed at the LLNL EBIT lab. Work at LLNL was performed under the auspices of the US DOE under contract DE-AC5-7NA-7. Some of the computations using STRAHL were carried out on the MIT PSFC parallel AMD Opteron/Infiniband cluster Loki. /9
2 Bayesian inference of impurity transport coefficient profiles Background, issues with existing approaches Linearized model to verify diagnostic sufficiency Inferring transport coefficient profiles with multimodal nested sampling /9
3 Bayesian inference of impurity transport coefficient profiles Background, issues with existing approaches Linearized model to verify diagnostic sufficiency Inferring transport coefficient profiles with multimodal nested sampling /9
4 Measuring impurity transport and testing simulations: making sure we know what we think we know Impurity transport coefficients D, V are often used in validation metrics n Z t = Γ Z + Q Z Model impurity flux Γ Z with diffusion coefficient D, convective velocity V: Γ Z = D n Z + Vn Z D, V are often used to validate impurity transport simulations: Important to measure D, V properly to have a strong test of the code. /9
5 Measuring impurity transport and testing simulations: making sure we know what we think we know Γ Z = D n Z + Vn Z Current approaches for measuring D, V have considerable shortcomings Error bars not consistent with intuition. Different starting points give different results: Multiple solutions? Broad region of acceptable solutions? New approach fixes these issues Verify diagnostic sufficiency with linearized model. Use advanced inference techniques to find D(r), V(r). Key result: rigorous selection of level of complexity in D(r), V(r) is critical. 5/9
6 Inferring impurity transport coefficients: a nonlinear inverse problem transport coefficient profiles D, V forward model STRAHL The rest of this talk: observed spectrometer signals modeled s(r, t) spectrometer signals s(r, t) Are our diagnostics sufficient? f (D, V s) probability model How to infer D, V with rigorous uncertainty estimates? Inject impurity with laser blow-off Can only observe s, want to know D, V. Only know the forward mapping s = m(d, V), but want D, V = m (s). Key questions: Existence: Is there a D, V such that s s? Uniqueness: How many D, V are there such that s s? Stability: How much do D, V change when I perturb s? 6/9
7 Bayesian inference of impurity transport coefficient profiles Background, issues with existing approaches Linearized model to verify diagnostic sufficiency Inferring transport coefficient profiles with multimodal nested sampling 7/9
8 A simple model to verify diagnostic sufficiency Basic D, V profiles r/a n Z [ m ] n Z [ m ] Temporal evolution of n Z 5 r/a =.5 8 t t inj [ms] Spatial evolution of n Z r/a 6 8 Simple D, V profiles yield representative signals. Three figures of merit capture most of the information: t t inj [ms] Confinement time τ imp Rise time t r Profile broadness b r/a = n Z (r/a)/n Z () 8/9
9 Intersection of the contours of τ imp, t r and b.75 determines D, V 8. Intersection of τ imp, t r and b true /9
10 Intersection of the contours of τ imp, t r and b.75 determines D, V broadness b.75 Intersection of τ imp, t r and b.75 confinement time τ imp core rise time t r true /9
11 Intersection of the contours of τ imp, t r and b.75 determines D, V broadness b.75 Intersection of τ imp, t r and b.75 confinement time τ imp core rise time t r true /9
12 Intersection of the contours of τ imp, t r and b.75 determines D, V broadness b.75 Intersection of τ imp, t r and b.75 confinement time τ imp core rise time t r true 95% region /9
13 Making the picture quantitative Linearize each figure of merit y i = g i (D, V) with respect to D, V. Assume Gaussian noise: y i N (μ yi, σ y i ). Transport coefficient vector T = [D, V] T N (μ T y, Σ T y ): μ T y = (C T Σ y C) (C T Σ y (y a)) Σ T y = (C T Σ y C) μ T y is the actual prediction of T = [D, V] T, Σ T y contains the uncertainties. a and C come from the linearization. y contains the actual observations, Σ y contains the uncertainties. This is exactly the result of weighted least squares regression: the analysis attempts to match the observations in the least squares sense. /9
14 Alcator C-Mod s diagnostics should be sufficient to reconstruct D, V rel. noise u rel. noise u Uncertainty in V point points 5 points time res. Δt [s] points HiReX-SR VUV time res. Δt [s] σ Generated synthetic data, ran through linearized model. σ D /D < σ V / V : just need to focus on V. Dashed lines: contours of constant photon rate. Green contours: σ V /V = ±%. HiReX-SR imaging crystal spectrometer should readily obtain σ V /V < %. /9
15 Bayesian inference of impurity transport coefficient profiles Background, issues with existing approaches Linearized model to verify diagnostic sufficiency Inferring transport coefficient profiles with multimodal nested sampling /9
16 Inferring impurity transport coefficients: a nonlinear inverse problem inject impurity diffusion, convection move impurity observe signals Finding D, V with Bayesian inference f D,V s (D, V s) posterior distribution = prior likelihood: distribution from STRAHL + data (D >, V() =, etc.) f s D,V (s D, V) f s (s) evidence f D,V (D, V) Parameter estimation: Find D, V: characterize f D,V s (D, V s). f : probability density function s: measured signals Model selection: Find best way of parameterizing D, V: maximize f s (s). M: functional form (model) used to parameterize D(r), V(r). D, V: parameters describing radial profiles of diffusion, convection Use MultiNest [Feroz MNRAS 8, 9]: samples f D,V s (D, V s) and estimates f s (s). /9
17 Inferring impurity transport coefficients: a nonlinear inverse problem inject impurity diffusion, convection move impurity observe signals Finding D, V with Bayesian inference f D,V s,m (D, V s, M) posterior distribution likelihood: from STRAHL + data prior distribution (D >, V() =, etc.) f s D,V,M (s D, V, M) f D,V M (D, V M) = f s M (s M) evidence Parameter estimation: Find D, V: characterize f D,V s,m (D, V s, M). f : probability density function s: measured signals D, V: parameters describing radial profiles of diffusion, convection Model selection: Find best way of parameterizing D, V: maximize f s M (s M). M: functional form (model) used to parameterize D(r), V(r). Use MultiNest [Feroz MNRAS 8, 9]: samples f D,V s,m (D, V s, M) and estimates f s M (s M). /9
18 MultiNest successfully reconstructs simple D, V profiles True and inferred D, V. true 6 inferred r/a Five local measurements Δt = 6 ms, 5% noise Have also tested with HiReX-SR chords (synthetic data) 5 smaller σ D, σ V than linearized analysis, but same correlation f D,V y (D, V y) /9
19 MultiNest successfully reconstructs simple D, V profiles True and inferred D, V. true 6 inferred r/a Five local measurements Δt = 6 ms, 5% noise Have also tested with HiReX-SR chords (synthetic data) 5 smaller σ D, σ V than linearized analysis, but same correlation f D,V y (D, V y) /9
20 MultiNest successfully determines how many spline coefficients to use D, V for various levels of complexity r/a MultiNest estimates the evidence f s c (s c): probability of observing the data given c. Ran with various numbers of free parameters: correctly selected c = case. Bayes factors: BF(c, ) = f s c (s c)/f s (s ) BF(c, ) Probability relative to c = case 6 coefficients per profile, c 5/9
21 Testing with more complicated synthetic data More complicated D, V profiles r/a Need to test with data representative of reality. Used result from [Howard NF, Chilenski NF 5] as true profile. Realistic diagnostic configuration: x-ray spectrometer chords (Ca 8+ ), 6 ms time resolution, 5% noise VUV spectrometer chords (Ca 7+, Ca 6+ ), ms time resolution, 5% noise 6/9
22 More complicated synthetic data pose a challenge BF(c, 5) 56 BF(c, 5) Relative probabilities Full c < too small to compute Zoomed coefficients per profile, c evaluations Model selection is expensive coefficients per profile, c 7 coefficient case took 7 CPU-hours = 5 wall-clock days! Need to speed up model, deploy on cluster to make this practical. (Recall: BF(c, 5) = f s c (s c)/f s 5 (s 5)) 7/9
23 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 8/9
24 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 (too small to see) 6 true r/a 8/9
25 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 8/9
26 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ true r/a 8/9
27 Getting D, V right requires careful statistical analysis Conclusions Need to select right level of complexity: Can appear to match data well while not matching the real D and V at all. New approach rigorously selects the most likely model. Can also estimate/verify diagnostic requirements. Validation of impurity transport simulations is still an open question, but we have a path forward. Additional details are in my PhD thesis: markchil.github.io/pdfs/thesis.pdf Open-source software: github.com/markchil/bayesimp 9/9
28 Backup slides /9
29 Inferring impurity transport coefficients is a difficult inverse problem D(r, [t]) V(r, [t]) spectroscopic observations XICS/VUV: s = (r, t) dl probability model f s D,V (s D, V) Ca source STRAHL n Z,i (r, t) s = (r, t) dl n e (r, [t]) T e (r, [t]) forward model PEC: P ij (T e, n e ) view geometry /9
30 Spectrometer chords on Alcator C-Mod Spectrometer lines of sight Z [m] R [m] HiReX-SR X-ray imaging crystal spectrometer (XICS) with chords split into 8 groups. Views He-like Ca (. nm) with 6 ms time resolution. XEUS Vacuum ultraviolet (VUV) spectrometer with one chord. Views Li-like Ca (.9 nm) with ms time resolution. LoWEUS Vacuum ultraviolet (VUV) spectrometer with one chord. Views Be-like Ca (9 nm) with ms time resolution. /9
31 Model selection using the evidence (a.k.a. the marginal likelihood) evidence f s (s) likelihood prior f s D,V (s D, V) f D,V (D, V) f D,V s (D, V s) = f s (s) posterior evidence f s (s) = f s D,V (s D, V)f D,V (D, V) dd dv.6 simple.5. observation... moderate complex. 6 8 data s Tradeoff between goodness of fit, complexity [Schwarz AS 978]: ln f s (s) ln f s θ (s θ) goodness-of-fit d ln N complexity (d is number of parameters, N number of datapoints) f s (s) is maximized by model with right level of complexity. Simple models can only explain a few data sets, low evidence for most s. Complex models can explain many data sets, any given s has low probability. /9
32 Simple example: fitting noisy data from y = x + x 5x + y y true y 8 5 Cubic data, polynomial fits 5 not clear which fit to prefer based on residuals alone... Residuals 5 Zoomed x BF(d, ) BF(d, ) Relative probability linear fit strongly rejected cubic fit correctly selected 5 polynomial degree, d /9
33 More complex models match the true profiles better Mahalanobis distance: M = (T true μ) T Σ (T true μ), T = [D, V] Mahalanobis distance Distance from true profile coefficients per profile, c 5/9
34 V/D is captured for r/a.7, c 5 V/D [m ] c = Impurity peaking factor r/a 6/9
35 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
36 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 (too small to see) 6 true r/a 7/9
37 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
38 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
39 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
40 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
41 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ 8 6 true r/a 7/9
42 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ true r/a 7/9
43 Results only resemble true profile when a minimum level of complexity is obtained despite good match to data: eyeballing does not work! Relative probabilities: BF(c, 5) c < too small to compute coefficients per profile, c. Core signal [AU] Edge signal [AU] 8 t t inj [ms] shaded region: ±σ true r/a 7/9
44 Rate coefficients have low sensitivity to n e, T e over their uncertainties α i i [m /s] S i i [m /s] Recombination Ionization r/a i samples from n e, T e ±σ band on α, S not even visible! These are the only ways that n e, T e enter the calculation. P ij [ 8 m /s] Photon emission r/a 8/9
45 GPR permits efficient propagation of uncertainty [Chilenski NF 5] Drawing samples of the profile y: λ / [ m ] λ / [kev] y N (μ, Σ), Σ = QΛQ y = QΛ / u + μ, u N (, I) n e T e eigenvalue index λ / q [ m ] λ / q [kev] n e, T e eigenvectors n e T e r/a 9/9
46 D D D D 5 V V V V V f D,V s (D, V s), c = 5.5. D.75. D..8 D..6 D..8 D 5 V V 5 V V 5 V 5 /9
47 D D D D 5 D 6 D 7 V V V V V 5 V 6 V f D,V s (D, V s), c = D D D D V D 5 D D 7 V V 6 V V 5 V 6 V 7 /9
ICRF Induced Argon Pumpout in H-D Plasmas in Alcator C-Mod
ICRF Induced Argon Pumpout in H-D Plasmas in Alcator C-Mod C. Gao, J.E. Rice, M.L. Reinke, Y. Lin, S.J. Wukitch, L. Delgado-Aparicio, E.S. Marmar, and Alcator C-Mod Team MIT-PSFC, University of York, Princeton
More informationReduction of Turbulence and Transport in the Alcator C-Mod Tokamak by Dilution of Deuterium Ions with Nitrogen and Neon Injection
Reduction of Turbulence and Transport in the Alcator C-Mod Tokamak by Dilution of Deuterium Ions with Nitrogen and Neon Injection M. Porkolab, P. C. Ennever, S. G. Baek, E. M. Edlund, J. Hughes, J. E.
More informationICRF Induced Argon Pumpout in H-D Plasmas at Alcator C-Mod
0 ICRF Induced Argon Pumpout in H-D Plasmas at Alcator C-Mod C Gao J.E. Rice, M.L. Reinke, S.J. Wukitch, Y. Lin and Alcator C-Mod Team MIT-PSFC October 29, 2014 C Gao etc., MIT-PSFC 56th APS-DPP New Orleans,
More informationObservations of Rotation Reversal and Fluctuation Hysteresis in Alcator C-Mod Plasmas
Observations of Rotation Reversal and Fluctuation Hysteresis in Alcator C-Mod Plasmas N.M. Cao 1, J.E. Rice 1, A.E. White 1, S.G. Baek 1, M.A. Chilenski 1, P.H. Diamond 2, A.E. Hubbard 1, J.W. Hughes 1,
More informationDependence of non-local effects on plasma parameters during cold-pulse experiments in Alcator C-Mod
Dependence of non-local effects on plasma parameters during cold-pulse experiments in Alcator C-Mod P. Rodriguez-Fernandez 1, N.M. Cao 1, A. Creely 1, M. Greenwald 1, S. Houshmandyar 2, N.T. Howard 1,
More information2017 US/EU Transport Task Force Workshop April 26 th 2017 Williamsburg, VA
Pablo Rodriguez-Fernandez 1, A. E. White 1, N. M. Cao 1, A. J. Creely 1, M. J. Greenwald 1, N. T. Howard 1, A. E. Hubbard 1, J. W. Hughes 1, J. H. Irby 1, C. C. Petty 2, J. E. Rice 1 1 Plasma Science and
More informationValidating Simulations of Multi-Scale Plasma Turbulence in ITER-Relevant, Alcator C-Mod Plasmas
Validating Simulations of Multi-Scale Plasma Turbulence in ITER-Relevant, Alcator C-Mod Plasmas Nathan Howard 1 with C. Holland 2, A.E. White 1, M. Greenwald 1, J. Candy 3, P. Rodriguez- Fernandez 1, and
More informationPlasma impurity composition in Alcator C-Mod tokamak.
Plasma impurity composition in Alcator C-Mod tokamak. I. O. Bespamyatnov a, W. L. Rowan a, K. T. Liao a, M. Brookman a, M. L. Reinke b, E. S. Marmar b, M. J. Greenwald b a Institute for Fusion Studies,
More informationValidation Study of gyrokinetic simulation (GYRO) near the edge in Alcator C-Mod ohmic discharges
Validation Study of gyrokinetic simulation (GYRO) near the edge in Alcator C-Mod ohmic discharges C. Sung, A. E. White, N. T. Howard, D. Mikkelsen, C. Holland, J. Rice, M. Reinke, C. Gao, P. Ennever, M.
More informationITB Transport Studies in Alcator C-Mod. Catherine Fiore MIT Plasma Science and Fusion Center Transport Task Force March 26th Boulder, Co
Transport Studies in Alcator C-Mod Catherine Fiore MIT Plasma Science and Fusion Center Transport Task Force March 26th Boulder, Co With Contributions from: I. Bespamyatnov, P. T. Bonoli*, D. Ernst*, M.
More informationNon-local Heat Transport in Alcator C-Mod Ohmic L-mode Plasmas
Non-local Heat Transport in Alcator C-Mod Ohmic L-mode Plasmas C. Gao 1, J.E.Rice 1, H.J. Sun 2,3, M.L.Reinke 1, N.T.Howard 1, D. Mikkelson 4, A.E.Hubbard 1, M.Chilenski 1, J.R.Walk 1, J.W.Hughes 1, P.Ennever
More informationInvestigation of Intrinsic Rotation Dependencies in Alcator C-Mod
Investigation of Intrinsic Rotation Dependencies in Alcator C-Mod D. Kwak, A. E. White, J. E. Rice, N. T. Howard, C. Gao, M. L. Reinke, M. Greenwald, C. Angioni, R. M. McDermott, and the C-Mod and ASDEX
More informationValidation of spectral MSE for Alcator C-Mod and for ITER
Validation of spectral MSE for Alcator C-Mod and for ITER K.T. Liao 1, W.L. Rowan 1, R.T. Mumgaard 2, Bob Granetz 2, Steve Scott 3, Fred Levinton 4, Howard Yuh 4, O. Marchuk 5, Y. Ralchenko 6 1 The University
More informationObservation of Co- and Counter Rotation Produced by Lower Hybrid Waves in Alcator C-Mod*
Observation of Co- and Counter Rotation Produced by Lower Hybrid Waves in Alcator C-Mod* R. R. Parker, Y. Podpaly, J. Lee, M. L. Reinke, J. E. Rice, P.T. Bonoli, O. Meneghini, S. Shiraiwa, G. M. Wallace,
More informationEnhanced Energy Confinement Discharges with L-mode-like Edge Particle Transport*
Enhanced Energy Confinement Discharges with L-mode-like Edge Particle Transport* E. Marmar, B. Lipschultz, A. Dominguez, M. Greenwald, N. Howard, A. Hubbard, J. Hughes, B. LaBombard, R. McDermott, M. Reinke,
More informationFormation and stability of impurity snakes in tokamak plasmas
PSFC/JA--9 Formation and stability of impurity snakes in tokamak plasmas L. Delgado-Aparicio,, L. Sugiyama, R. Granetz, J. Rice, Y. Podpaly, M. Reinke, D. Gates, P. Beirsdorfer 4, M. Bitter, S. Wolfe,
More informationTemporal Evolution of Temperature and Argon Impurity Density Profiles Observed by X-ray Imaging Spectrometer Measurements at Wendelstein 7-X
EUROFUSION WPS1-PR(16) 15653 A. Langenberg et al. Temporal Evolution of Temperature and Argon Impurity Density Profiles Observed by X-ray Imaging Spectrometer Measurements at Wendelstein 7-X Preprint of
More informationObservations of Counter-Current Toroidal Rotation in Alcator C-Mod LHCD Plasmas
1 EX/P5-4 Observations of Counter-Current Toroidal Rotation in Alcator C-Mod LHCD Plasmas J.E. Rice 1), A.C. Ince-Cushman 1), P.T. Bonoli 1), M.J. Greenwald 1), J.W. Hughes 1), R.R. Parker 1), M.L. Reinke
More informationStochastic Spectral Approaches to Bayesian Inference
Stochastic Spectral Approaches to Bayesian Inference Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 4, 2011 Prof. Gibson (OSU) Spectral Approaches to
More informationGyrokine.c Analysis of the Linear Ohmic Confinement Regime in Alcator C- Mod *
Gyrokine.c Analysis of the Linear Ohmic Confinement Regime in Alcator C- Mod * Miklos Porkolab in collabora.on with J. Dorris, P. Ennever, D. Ernst, C. Fiore, M. Greenwald, A. Hubbard, E. Marmar, Y. Ma,
More informationThis work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract
This work was performed under the auspices of the U.S. Department of Energy by under contract DE-AC52-7NA27344. Lawrence Livermore National Security, LLC The ITER tokamak Tungsten (W) is attractive as
More informationTungsten impurity transport experiments in Alcator C-Mod to address high priority R&D for ITER
Tungsten impurity transport experiments in Alcator C-Mod to address high priority R&D for ITER M.L. Reinke 1, A. Loarte 2, M. Chilenski 3, N. Howard 3, F. Köchl 4, A. Polevoi 2, A. Hubbard 3, J.W. Hughes
More informationThe Instrumental Function of the X-ray Imaging Crystal Spectrometer on Alcator C-Mod
The Instrumental Function of the X-ray Imaging Crystal Spectrometer on Alcator C-Mod M. Bitter, K. W. Hill, B. Stratton, S. Scott Princeton Plasma Physics Laboratory, Princeton, NJ, USA A. Ince-Cushman,
More informationStudies of Lower Hybrid Range of Frequencies Actuators in the ARC Device
Studies of Lower Hybrid Range of Frequencies Actuators in the ARC Device P. T. Bonoli, Y. Lin. S. Shiraiwa, G. M. Wallace, J. C. Wright, and S. J. Wukitch MIT PSFC, Cambridge, MA 02139 59th Annual Meeting
More informationLearning Gaussian Process Models from Uncertain Data
Learning Gaussian Process Models from Uncertain Data Patrick Dallaire, Camille Besse, and Brahim Chaib-draa DAMAS Laboratory, Computer Science & Software Engineering Department, Laval University, Canada
More informationAdvanced Statistical Methods. Lecture 6
Advanced Statistical Methods Lecture 6 Convergence distribution of M.-H. MCMC We denote the PDF estimated by the MCMC as. It has the property Convergence distribution After some time, the distribution
More informationFitting Narrow Emission Lines in X-ray Spectra
Outline Fitting Narrow Emission Lines in X-ray Spectra Taeyoung Park Department of Statistics, University of Pittsburgh October 11, 2007 Outline of Presentation Outline This talk has three components:
More informationADVANCED MACHINE LEARNING ADVANCED MACHINE LEARNING. Non-linear regression techniques Part - II
1 Non-linear regression techniques Part - II Regression Algorithms in this Course Support Vector Machine Relevance Vector Machine Support vector regression Boosting random projections Relevance vector
More informationStudy of B +1, B +4 and B +5 impurity poloidal rotation in Alcator C-Mod plasmas for 0.75 ρ 1.0.
Study of B +1, B +4 and B +5 impurity poloidal rotation in Alcator C-Mod plasmas for 0.75 ρ 1.0. Igor Bespamyatnov, William Rowan, Ronald Bravenec, and Kenneth Gentle The University of Texas at Austin,
More informationIntroduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma
Introduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma Core diagnostics II: Bolometry and Soft X-rays J. Arturo Alonso Laboratorio Nacional de Fusión EURATOM-CIEMAT E6 P2.10 arturo.alonso@ciemat.es
More informationPlasma Radiation. Ø Free electrons Blackbody emission Bremsstrahlung
Plasma Radiation Ø Free electrons Blackbody emission Bremsstrahlung Ø Bound electrons (Z>2) Unresolved, multi-line emission Resolved line emission -- Single Z +n Objective Infer a thermodynamic quantity
More informationFull-wave Simulations of Lower Hybrid Wave Propagation in the EAST Tokamak
Full-wave Simulations of Lower Hybrid Wave Propagation in the EAST Tokamak P. T. BONOLI, J. P. LEE, S. SHIRAIWA, J. C. WRIGHT, MIT-PSFC, B. DING, C. YANG, CAS-IPP, Hefei 57 th Annual Meeting of the APS
More informationLecture : Probabilistic Machine Learning
Lecture : Probabilistic Machine Learning Riashat Islam Reasoning and Learning Lab McGill University September 11, 2018 ML : Many Methods with Many Links Modelling Views of Machine Learning Machine Learning
More informationOn the ρ Scaling of Intrinsic Rotation in C-Mod Plasmas with Edge Transport Barriers
On the ρ Scaling of Intrinsic Rotation in C-Mod Plasmas with Edge Transport Barriers J.E. Rice, J.W. Hughes, P.H. Diamond, N. Cao, M.A. Chilenski, A.E. Hubbard, J.H. Irby, Y. Kosuga Y. Lin, I.W. Metcalf,
More information1 Using standard errors when comparing estimated values
MLPR Assignment Part : General comments Below are comments on some recurring issues I came across when marking the second part of the assignment, which I thought it would help to explain in more detail
More informationReliability Monitoring Using Log Gaussian Process Regression
COPYRIGHT 013, M. Modarres Reliability Monitoring Using Log Gaussian Process Regression Martin Wayne Mohammad Modarres PSA 013 Center for Risk and Reliability University of Maryland Department of Mechanical
More informationEvaluation of Anomalous Fast-Ion Losses in Alcator C-Mod
Evaluation of Anomalous Fast-Ion Losses in Alcator C-Mod S. D. Scott Princeton Plasma Physics Laboratory In collaboration with R. Granetz, D. Beals, M. Greenwald MIT PLASMA Science and Fusion Center W.
More informationEXONEST The Exoplanetary Explorer. Kevin H. Knuth and Ben Placek Department of Physics University at Albany (SUNY) Albany NY
EXONEST The Exoplanetary Explorer Kevin H. Knuth and Ben Placek Department of Physics University at Albany (SUNY) Albany NY Kepler Mission The Kepler mission, launched in 2009, aims to explore the structure
More informationImpurity transport analysis & preparation of W injection experiments on KSTAR
Impurity transport analysis & preparation of W injection experiments on KSTAR J. H. Hong, H. Y. Lee, S. H. Lee, S. Jang, J. Jang, T. Jeon, H. Lee, and W. Choe ( ) S. G. Lee, C. R. Seon, J. Kim, ( ) 마스터부제목스타일편집
More informationIntegrated Bayesian Experimental Design
Integrated Bayesian Experimental Design R. Fischer, H. Dreier, A. Dinklage, B. Kurzan and E. Pasch Max-Planck-Institut für Plasmaphysik, EURATOM Association, Boltzmannstr. 2, 85 748 Garching, Germany Max-Planck-Institut
More informationStatistical Tools and Techniques for Solar Astronomers
Statistical Tools and Techniques for Solar Astronomers Alexander W Blocker Nathan Stein SolarStat 2012 Outline Outline 1 Introduction & Objectives 2 Statistical issues with astronomical data 3 Example:
More informationPhysics of the detached radiative divertor regime in DIII-D
Plasma Phys. Control. Fusion 41 (1999) A345 A355. Printed in the UK PII: S741-3335(99)97299-8 Physics of the detached radiative divertor regime in DIII-D M E Fenstermacher, J Boedo, R C Isler, A W Leonard,
More informationAn introduction to Markov Chain Monte Carlo techniques
An introduction to Markov Chain Monte Carlo techniques G. J. A. Harker University of Colorado ASTR5550, 19th March 2012 Outline Introduction Bayesian inference: recap MCMC: when to use it and why A simple
More informationFlow measurements in the Scrape-Off Layer of Alcator C-Mod using Impurity Plumes
Flow measurements in the Scrape-Off Layer of Alcator C-Mod using Impurity Plumes S. Gangadhara,. Laombard M.I.T. Plasma Science and Fusion Center, 175 Albany St., Cambridge, MA 2139 USA Abstract Accurate
More informationPhysics 403. Segev BenZvi. Parameter Estimation, Correlations, and Error Bars. Department of Physics and Astronomy University of Rochester
Physics 403 Parameter Estimation, Correlations, and Error Bars Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Review of Last Class Best Estimates and Reliability
More informationLower Hybrid Current Drive Experiments on Alcator C-Mod: Comparison with Theory and Simulation
Lower Hybrid Current Drive Experiments on Alcator C-Mod: Comparison with Theory and Simulation P.T. Bonoli, A. E. Hubbard, J. Ko, R. Parker, A.E. Schmidt, G. Wallace, J. C. Wright, and the Alcator C-Mod
More informationBayesian Regression Linear and Logistic Regression
When we want more than point estimates Bayesian Regression Linear and Logistic Regression Nicole Beckage Ordinary Least Squares Regression and Lasso Regression return only point estimates But what if we
More informationIntroduction to Integrated Data Analysis
7th Workshop on Fusion Data Processing, Validation and Analysis Introduction to Integrated Data Analysis R. Fischer Max-Planck-Institut für Plasmaphysik, Garching EURATOM Association Frascati, Mar 26-28,
More informationHard Xray Diagnostic for Lower Hybrid Current Drive on Alcator C- Mod
Hard Xray Diagnostic for Lower Hybrid Current Drive on Alcator C- Mod J. Liptac, J. Decker, R. Parker, V. Tang, P. Bonoli MIT PSFC Y. Peysson CEA Cadarache APS 3 Albuquerque, NM Abstract A Lower Hybrid
More informationRadial impurity transport in the H mode transport barrier region in Alcator C-Mod
Radial impurity transport in the H mode transport barrier region in Alcator C-Mod T. Sunn Pedersen, R.S. Granetz, A.E. Hubbard, I.H. Hutchinson, E.S. Marmar, J.E. Rice, J. Terry Plasma Science and Fusion
More informationPATTERN RECOGNITION AND MACHINE LEARNING
PATTERN RECOGNITION AND MACHINE LEARNING Chapter 1. Introduction Shuai Huang April 21, 2014 Outline 1 What is Machine Learning? 2 Curve Fitting 3 Probability Theory 4 Model Selection 5 The curse of dimensionality
More informationOutline lecture 2 2(30)
Outline lecture 2 2(3), Lecture 2 Linear Regression it is our firm belief that an understanding of linear models is essential for understanding nonlinear ones Thomas Schön Division of Automatic Control
More informationNonparametric Bayesian Methods (Gaussian Processes)
[70240413 Statistical Machine Learning, Spring, 2015] Nonparametric Bayesian Methods (Gaussian Processes) Jun Zhu dcszj@mail.tsinghua.edu.cn http://bigml.cs.tsinghua.edu.cn/~jun State Key Lab of Intelligent
More informationOhmic and RF Heated ITBs in Alcator C-Mod
Ohmic and RF Heated s in Alcator C-Mod William L. Rowan, Igor O. Bespamyatnov Fusion Research Center, The University of Texas at Austin C.L. Fiore, A. Dominguez, A.E. Hubbard, A. Ince-Cushman, M.J. Greenwald,
More informationNIMEQ: MHD Equilibrium Solver for NIMROD
NIMEQ: MHD Equilibrium Solver for NIMOD E.C.Howell, C..Sovinec University of Wisconsin-Madison 5 th Annual Meeting of Division of Plasma Physics Dallas, Texas, Nov. 17-Nov. 1,8 1 Abstract A Grad-Shafranov
More informationLecture 23. Lidar Error and Sensitivity Analysis (2)
Lecture 3. Lidar Error and Sensitivity Analysis ) q Derivation of Errors q Background vs. Noise q Sensitivity Analysis q Summary 1 Accuracy vs. Precision in Lidar Measurements q The precision errors caused
More informationNeutral Density Profiles From a Diverted Plasma in Alcator C-Mod. Christopher Paul Barrington-Leigh
Neutral Density Profiles From a Diverted Plasma in Alcator C-Mod by Christopher Paul Barrington-Leigh Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of
More informationICRF Mode Conversion Flow Drive on the Alcator C Mod Tokamak
23 rd IAEA Fusion Energy Conference, EXW/4 1 ICRF Mode Conversion Flow Drive on the Alcator C Mod Tokamak Yijun Lin, J.E. Rice, S.J. Wukitch, M.L. Reinke, M. Greenwald, A. E. Hubbard, E.S. Marmar, Y. Podpaly,
More informationDevelopment of LH wave fullwave simulation based on FEM
Development of LH wave fullwave simulation based on FEM S. Shiraiwa and O. Meneghini on behalf of LHCD group of Alacator C-Mod PSFC, MIT 2010/03/10 San-Diego, CA Special acknowledgements : R. Parker, P.
More informationBased on slides by Richard Zemel
CSC 412/2506 Winter 2018 Probabilistic Learning and Reasoning Lecture 3: Directed Graphical Models and Latent Variables Based on slides by Richard Zemel Learning outcomes What aspects of a model can we
More informationProbing the covariance matrix
Probing the covariance matrix Kenneth M. Hanson Los Alamos National Laboratory (ret.) BIE Users Group Meeting, September 24, 2013 This presentation available at http://kmh-lanl.hansonhub.com/ LA-UR-06-5241
More informationEdge Rotational Shear Requirements for the Edge Harmonic Oscillation in DIII D Quiescent H mode Plasmas
Edge Rotational Shear Requirements for the Edge Harmonic Oscillation in DIII D Quiescent H mode Plasmas by T.M. Wilks 1 with A. Garofalo 2, K.H. Burrell 2, Xi. Chen 2, P.H. Diamond 3, Z.B. Guo 3, X. Xu
More informationAlcator C-Mod. Particle Transport in the Scrape-off Layer and Relationship to Discharge Density Limit in Alcator C-Mod
Alcator C-Mod Particle Transport in the Scrape-off Layer and Relationship to Discharge Density Limit in Alcator C-Mod B. LaBombard, R.L. Boivin, M. Greenwald, J. Hughes, B. Lipschultz, D. Mossessian, C.S.
More informationSTA 4273H: Sta-s-cal Machine Learning
STA 4273H: Sta-s-cal Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 2 In our
More information+ + ( + ) = Linear recurrent networks. Simpler, much more amenable to analytic treatment E.g. by choosing
Linear recurrent networks Simpler, much more amenable to analytic treatment E.g. by choosing + ( + ) = Firing rates can be negative Approximates dynamics around fixed point Approximation often reasonable
More informationUsing the X-FEL to understand X-ray Thomson scattering for partially ionized plasmas
LLNL-PROC-564720 Using the X-FEL to understand X-ray Thomson scattering for partially ionized plasmas J. Nilsen, W. R. Johnson, K. T. Cheng July 17, 2012 13th International Conference on X-ray Lasers Paris,
More informationGaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008
Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:
More informationGaussian Processes (10/16/13)
STA561: Probabilistic machine learning Gaussian Processes (10/16/13) Lecturer: Barbara Engelhardt Scribes: Changwei Hu, Di Jin, Mengdi Wang 1 Introduction In supervised learning, we observe some inputs
More informationLarge Scale Bayesian Inference
Large Scale Bayesian I in Cosmology Jens Jasche Garching, 11 September 2012 Introduction Cosmography 3D density and velocity fields Power-spectra, bi-spectra Dark Energy, Dark Matter, Gravity Cosmological
More informationGaussian Process Regression
Gaussian Process Regression 4F1 Pattern Recognition, 21 Carl Edward Rasmussen Department of Engineering, University of Cambridge November 11th - 16th, 21 Rasmussen (Engineering, Cambridge) Gaussian Process
More informationCIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions
CIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions December 14, 2016 Questions Throughout the following questions we will assume that x t is the state vector at time t, z t is the
More informationProbabilistic Machine Learning. Industrial AI Lab.
Probabilistic Machine Learning Industrial AI Lab. Probabilistic Linear Regression Outline Probabilistic Classification Probabilistic Clustering Probabilistic Dimension Reduction 2 Probabilistic Linear
More informationUsing Bayesian Analysis to Study Tokamak Plasma Equilibrium
Using Bayesian Analysis to Study Tokamak Plasma Equilibrium Greg T. von Nessi (joint work with Matthew Hole, Jakob Svensson, Lynton Appel, Boyd Blackwell, John Howard, David Pretty and Jason Bertram) Outline
More informationCOMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017
COMS 4721: Machine Learning for Data Science Lecture 19, 4/6/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University PRINCIPAL COMPONENT ANALYSIS DIMENSIONALITY
More informationσ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =
Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,
More informationCharacterization of neo-classical tearing modes in high-performance I- mode plasmas with ICRF mode conversion flow drive on Alcator C-Mod
1 EX/P4-22 Characterization of neo-classical tearing modes in high-performance I- mode plasmas with ICRF mode conversion flow drive on Alcator C-Mod Y. Lin, R.S. Granetz, A.E. Hubbard, M.L. Reinke, J.E.
More informationStudies of Turbulence and Transport in Alcator C- Mod H-Mode Plasmas with Phase Contrast Imaging and Comparisons with GYRO*
Studies of Turbulence and Transport in C- Mod H-Mode Plasmas with Phase Contrast Imaging and Comparisons with GYRO* M. Porkolab 1, L. Lin 1, E.M. Edlund 1, J.C. Rost 1, C.L. Fiore 1, M. Greenwald 1, Y.
More informationNoninductive Formation of Spherical Tokamak at 7 Times the Plasma Cutoff Density by Electron Bernstein Wave Heating and Current Drive on LATE
1 EX/P6-18 Noninductive Formation of Spherical Tokamak at 7 Times the Plasma Cutoff Density by Electron Bernstein Wave Heating and Current Drive on LATE M. Uchida, T. Maekawa, H. Tanaka, F. Watanabe, Y.
More informationBrandon C. Kelly (Harvard Smithsonian Center for Astrophysics)
Brandon C. Kelly (Harvard Smithsonian Center for Astrophysics) Probability quantifies randomness and uncertainty How do I estimate the normalization and logarithmic slope of a X ray continuum, assuming
More informationPlasma Physics and Fusion Energy Research
Coláiste na hollscoile Corcaigh University College Cork Plasma Physics and Fusion Energy Research Plasma Data Analysis Group: Graduate Student: Paddy Mc Carthy Sean Knott (PhD) (Group Leader) (Co-supervisor:
More informationINTRODUCTION TO BAYESIAN INFERENCE PART 2 CHRIS BISHOP
INTRODUCTION TO BAYESIAN INFERENCE PART 2 CHRIS BISHOP Personal Healthcare Revolution Electronic health records (CFH) Personal genomics (DeCode, Navigenics, 23andMe) X-prize: first $10k human genome technology
More informationDevelopment of a High-Speed VUV Camera System for 2-Dimensional Imaging of Edge Turbulent Structure in the LHD
Development of a High-Speed VUV Camera System for 2-Dimensional Imaging of Edge Turbulent Structure in the LHD Masaki TAKEUCHI, Satoshi OHDACHI and LHD experimental group National Institute for Fusion
More informationWhy Bayesian? Rigorous approach to address statistical estimation problems. The Bayesian philosophy is mature and powerful.
Why Bayesian? Rigorous approach to address statistical estimation problems. The Bayesian philosophy is mature and powerful. Even if you aren t Bayesian, you can define an uninformative prior and everything
More informationDIAGNOSTICS FOR ADVANCED TOKAMAK RESEARCH
DIAGNOSTICS FOR ADVANCED TOKAMAK RESEARCH by K.H. Burrell Presented at High Temperature Plasma Diagnostics 2 Conference Tucson, Arizona June 19 22, 2 134 /KHB/wj ROLE OF DIAGNOSTICS IN ADVANCED TOKAMAK
More informationLinear Dynamical Systems
Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations
More informationMeasurements of relativistic emission from runaway electrons in Alcator C-Mod: spectrum, polarization, and spatial structure
Measurements of relativistic emission from runaway electrons in Alcator C-Mod: spectrum, polarization, and spatial structure R. Granetz, R. Mumgaard MIT PSFC APS-DPP New Orleans 2014/10/30 Motivation Modeling
More informationThe First Stars John Wise, Georgia Tech
z=1100 The First Stars John Wise, Georgia Tech z~20-30 z~6 > (P=kT b Δν) Courtesy of J. Pritchard Adapted from Pritchard & Loeb, 2010, Phys. Rev. D, 82, 023006 A great mystery now confronts us: When and
More informationC-Mod Core Transport Program. Presented by Martin Greenwald C-Mod PAC Feb. 6-8, 2008 MIT Plasma Science & Fusion Center
C-Mod Core Transport Program Presented by Martin Greenwald C-Mod PAC Feb. 6-8, 2008 MIT Plasma Science & Fusion Center Practical Motivations for Transport Research Overall plasma behavior must be robustly
More informationAlcator C-Mod. Double Transport Barrier Plasmas. in Alcator C-Mod. J.E. Rice for the C-Mod Group. MIT PSFC, Cambridge, MA 02139
Alcator C-Mod Double Transport Barrier Plasmas in Alcator C-Mod J.E. Rice for the C-Mod Group MIT PSFC, Cambridge, MA 139 IAEA Lyon, Oct. 17, Outline Double Barrier Plasma Profiles and Modeling Conditions
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology 6.867 Machine Learning, Fall 2006 Problem Set 5 Due Date: Thursday, Nov 30, 12:00 noon You may submit your solutions in class or in the box. 1. Wilhelm and Klaus are
More informationCalibrating Environmental Engineering Models and Uncertainty Analysis
Models and Cornell University Oct 14, 2008 Project Team Christine Shoemaker, co-pi, Professor of Civil and works in applied optimization, co-pi Nikolai Blizniouk, PhD student in Operations Research now
More informationMultimodal Nested Sampling
Multimodal Nested Sampling Farhan Feroz Astrophysics Group, Cavendish Lab, Cambridge Inverse Problems & Cosmology Most obvious example: standard CMB data analysis pipeline But many others: object detection,
More informationPlasma Spectroscopy Inferences from Line Emission
Plasma Spectroscopy Inferences from Line Emission Ø From line λ, can determine element, ionization state, and energy levels involved Ø From line shape, can determine bulk and thermal velocity and often
More informationRotation Speed Differences of Impurity Species in the DIII-D Tokamak and Comparison with Neoclassical Theory
Rotation Speed Differences of Impurity Species in the DIII-D Tokamak and Comparison with Neoclassical Theory L.R. Baylor, K.H. Burrell*, R.J. Groebner*, D.R. Ernst #, W.A. Houlberg, M. Murakami, and The
More informationPhysics 403. Segev BenZvi. Propagation of Uncertainties. Department of Physics and Astronomy University of Rochester
Physics 403 Propagation of Uncertainties Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Maximum Likelihood and Minimum Least Squares Uncertainty Intervals
More informationIntroduction to Gaussian Processes
Introduction to Gaussian Processes 1 Objectives to express prior knowledge/beliefs about model outputs using Gaussian process (GP) to sample functions from the probability measure defined by GP to build
More informationModeling Laser-Plasma Interactions in MagLIF Experiment on NIF
Modeling Laser-Plasma Interactions in MagLIF Experiment on NIF Anomalous Absorption Meeting 5 May 2016 D. J. Strozzi, R. L. Berger, A. B. Sefkow, S. H. Langer, T. Chapman, B. Pollock, C. Goyon, J. Moody
More informationPlasma Response Control Using Advanced Feedback Techniques
Plasma Response Control Using Advanced Feedback Techniques by M. Clement 1 with J. M. Hanson 1, J. Bialek 1 and G. A. Navratil 1 1 Columbia University Presented at 59 th Annual APS Meeting Division of
More informationImpurity accumulation in the main plasma and radiation processes in the divetor plasma of JT-60U
1 EX/P4-25 Impurity accumulation in the main plasma and radiation processes in the divetor plasma of JT-6U T. Nakano, H. Kubo, N. Asakura, K. Shimizu and S. Higashijima Japan Atomic Energy Agency, Naka,
More informationPFC/JA Precision Measurements of the Wavelengths of Emission Lines of Mg-like and Na-like Kv in Alcator C Plasmas
PFC/JA-86-43 Precision Measurements of the Wavelengths of Emission Lines of Mg-like and Na-like Kv in Alcator C Plasmas K. Kondo, J.L. Terry, J. E. Rice, E. S. Marmar Plasma Fusion Center Massachusetts
More information