Page 1 - FRC equilibrium reconstruction by Bayesian evaluation\\ of Monte Carlo transport simulations

FRC equilibrium reconstruction by matching experimental
measurements to snapshots of Monte-Carlo transport simulations
Nikolaus Rath, Marco Onofri, Loren Steinhauer and the TAE Team
1. Prior Work 1.1 Approaches
• L. Steinhauer’s 2D Equilibrium Interpreter [1] matches 3 experimental parameters to a pool of Grad-Shafranov equilibria.
• HyEq extends pool states beyond Grad-Shafranov constraints (cf. poster CP10.00090, this section):
– Pool states solve Fokker-Planck equation
– Free functions determined by 8 parameters – Challenge: finding good parametrizations – Work in progress
• Current Tomography assumes smoothness to derive current distribution (cf. talk TO7.00013 on Thu by J. Romero):
– Considers all possible distributions on finite mesh and orders by probability
– Criteria: smoothness of distribution + reproduction of sensor signals
• M. Onofri’s Q2D simulations approximate full shots by tuning transport coefficents (cf. poster CP10.00088, this session) .
3. Pool generation
Pool is generated by running many Q2D simulations and treating time-slices as independent states:
• Scanned over 8 parameters:
– neutral beam power
– wall current profile
– initial separatrix dimensions
– parallel/perpendicular thermal electron conductivity – resistivity
– wall temperature boundary conditions
– electron/fast-ion heat transfer efficiency
• Ran each simulation for 10 ms
• Then chopped simulation into 25 μs slices to form pool, disregarding all time information
3.1 Pool size
• Originally planned to run 5000 simulations with different combinations of parameters
• For initial test, did scanned just 1000
• Consumed roughly 100,000 core hours of CPU time • Final result: about 221,000 plasma snapshots (1 TB). • Post-processing reduces disk space to 200 MB.
3.2 Plasma Property Distribution
• Individually, plasma properties are nicely distributed
• Many parameters are correlated – but source of correlation is not clear
• Correlations due to physics are good and desired.
• Correlations due to unfortunate sampling of parameter space are bad and should be removed.
• Not clear how to do that, so matching procedure is designed to be robust in presence of correlations
4. Matching Procedure 4.1 Measurements & Likelihood
Matching uses synthetic measurements with errors:
4.2 Derived Measurements
• Sometimes, want to match “higher-level” observations, e.g. excluded flux radius instead of magnetic field strength.
• Such derived quantities are treated like direct measurements, with errors properly propagated:
2
∆f(⃗x)= f(⃗x)−f(⃗x+∆xi) (4)
i
• Derived quantities are always calculated from raw measurements (both synthetic and experimental)
• Examples: excluded flux radius, total temperature (from force balance), FRC length
4.3 Uncertainty Estimation
• A prediction without error bars is worse than no prediction. • Ideally, error bars indicate “probable interval” – probability
of the true value being in the range is at least x. • Can’t do that here, since we don’t know absolute
probabilities.
• Can’t establish upper bound either: may change with bigger pool
• Therefore, use errors bars to indicate lower bound on uncertainty
1. Consider all states with measurements within uncertainty of experimental values
2. Use spread between associated plasma properties as uncertainty of predictions
• Increasing pool size can increase spread, but never decrease it
• Quality of lower bound depends on how well the synthetic measurements span the experimental uncertainty: the wider the coverage, the tighter the bound.
• Calculating mean and average spread over all measurements gives overall indicator of error bar quality
5. Results 5.1 Test Data
Self consistency test matches pool data against another simulation
• Pick one state from pool and treat as experiment
• Drop all states from pool that don’t differ in at least two parameters
• Matched quantities:
– Excluded flux radius centroid (r and z)
– Excluded flux radius envelope ("Plasma length")
– Total field and excluded flux radius at three points – Midplane interferometer
– Neutral beam shinethrough
Observations:
• Most likely state has similar likelihood at all times
• Derived properties are mostly continuous (they don’t need to be!)
• Derived properties match well with actual values
• Poor resolution of temperature – expected
• Error bars indicate fundamental limits of any available reconstruction technique that uses the same measurements
5.2 Experimental Data
• Several reconstructions were attempted using experimental data from C-2U.
• Most of the time, there are only very few states in the pool whose measurements all fall within the experimental uncertainty. This means that error bars cannot be computed at all, or are very loose.
• We hope that this is a consequence of the state pool still being too small and that results will improve as we increase the size of the state pool.
Summary
• Many Q2D simulations are chopped into even more independent snapshots, forming a “pool” of “states”.
• Experimental measurements are compared with the measurements that each state would produce.
• The most similar states are used to estimate otherwise unknown plasma properties
• States whose synthetic measurements fall within experimental errors are used to derive error bars on plasma properties
• Splitting pool into training/testing data gives accurate predictions and indicates fundamental limits of experimental measurements.
• At the moment, pool size is still too small to accurately reconstruct experiments - it is now being increased.
References
[1] L. Steinhauer, An interpreter tool based on the Grad-Shafranov paradigm, Physics of Plasmas, 21 (2014).
For each pool state, compute synthetic measurements ⃗y, e.g. magnetic field at point, line integrated density, neutral beam shinethrough fraction, electron temperature at point.
Assign uncertainties ⃗σ as estimated by diagnostic physicists For each time-slice of a shot, get experimental
measurements ⃗x
Probability density of pool state H to produce
experimental measurements ⃗x:
P(⃗x|H)= exp −(xi −yi) (1)
Problem: We don’t want P(⃗x|H), but P(Hj |⃗x): probability of each pool state Hj to coincide with experimental state, given the measurements ⃗x.
(2)
(3)
•
• •
•
2 i 2σ2i
P(⃗x|Hj)·P(Hj) P(⃗x)
1.2 Differences
• Pool size (if any)
• Included physics and conservation laws
• Matched parameters
• Predictable plasma properties
2. The Idea
• Time (reconstruction and preparation)
• Required operator expertise
• Automation
• Ability to estimate uncertainties
P(Hj |⃗x) =
• P(Hj)(“prior”)isonedividedbypoolsize
• P(⃗x|Hj (“likelihood”) is given by measurements
• If the pool contains all possible states, j P(Hj |⃗x) = 1, so

P(⃗x)= P(⃗x|Hj)·P(Hj) j
Unfortunately, any finite pool is incomplete.
Therefore, compare relative likelihood instead:
Ulysses combines pool matching with Q2D physics: Use Q2D to generate pool of plasma states then match timeslice-by-timeslice to experiment.
Advantages:
• Can estimate uncertainties
• Does not depend on knowing transport parameters • No expert knowledge of free parameters required • Automatically gives physically sensible profiles
• Can be run automatically for each shot
• Calculate probability of random errors moving all synthetic measurement within uncertainties of corrsponding experimental measurements.
• Most likely state is used to predict plasma properties
• Comparing max likelihood over time indicates quality of fit