Page 6 - Achievement of Sustained Net Plasma Heating in a Fusion Experiment with the Optometrist Algorithm

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by three subsystems: the rotating magnetic eld, the preionization, and the fast magnetic bias. Control of these systems is reduced to 3 voltages and 2 timing MPs. e ionized plasma is then linearly accelerated by a series of 17 very fast magnetic elds on each side of the device. e timings are adjustable individually (34 parameters, and another 34 for the activation of the crowbar system for safety), while the voltages are grouped into 16 banks. is complex system is reduced to a single uni ed voltage MP and two timing MPs. One of these is the “acceleration” MP where the timings are expanded or shrunk toward the zero point, for example the subset [0, 2, 4] → [0, 1.9, 3.8] μs.
Once the plasma is formed, it is maintained by the “equilibrium” systems. e dipole magnetic eld is driven by 7 power supplies, which we chose to keep separate. At the ends of the con nement region, powerful mirror elds are speci ed by current and onset time. ese can be speci ed separately for the two ends (4 MPs) or symmetrically (2 MPs). Finally, electrical biasing is typically achieved via plasma guns, which are controlled by gas pressure, timing, and two electrode voltages each. ese can also be speci ed individually (8 MPs) or sym- metrically (4 MPs). With these engineering choices, we reduce the dimensionality of the control space to <30, as summarised in Table 2. Most Optometrist runs adjusted only the symmetric set of 23, and we o en chose to adjust a reduced set of 17 which le out the Mirror Field and Electrical Biasing parameters.

Γ((ν + N)/2) 1 T −(ν+N)/2
p(g) = Γ(ν/2)(πν)N/2 1 + νg g (1)
where v is the degrees of freedom on the t distribution. We choose v = N, as a compromise between a light-tailed Gaussian (v → ∞) and a very-heavy-tailed Cauchy (v = 1) distribution. A moderately heavy-tailed distribution of steps allows us to occasionally take very large steps (relative to the normal distribution) to better sample the unknown character of the performance of the machine. Taking heavy-tailed steps has been shown to be e ec- tive in Monte Carlo optimisation20. Notice, however, that the Optometrist Algorithm does not depend on the Student-t distribution.
Experiment proposals. We choose directions, g , isotropically in the MP-space, drawing from an N-dimensional multivariate Student-t distribution19:
meta-parameters x to the reference shot xt−1:

e random direction at step t, g , is multiplied by a step size, ηt and applied as a relative adjustment of the t
xt =xt−1 (1+ηtgt) (2) where the step size ηt is adapted dynamically (as described, below).
Step size adjustment. We adjust the step size dynamically to ensure that the probability of accepting the new reference shot is roughly controlled. In classical Monte Carlo, there is an optimal acceptance probability of 23.4%21. Accepting too many proposals means that exploration is not aggressive enough. Accepting too few pro- posals means not exploring enough or stepping too far. e step size ηt is adjusted via
ηt ≡ Aexp(γt) (3)
γ
 t−1
+ α , t
− β , t
plicatively by exp(γt) which starts at 1. e choice of αt and βt set the probability of an accepted step to be:
βt
αt + βt (5)
We choose βt = 0.5αt to keep an acceptance probability of approximately 33%.
Practical implementation. As discussed previously, the average shot cadence is eight minutes. Typically several minutes elapse before the rst-line data analysis is complete. Only then can the expert human operator make an evaluation of the shot’s performance. e delay of the data analysis leaves only two or three minutes to generate and input a new proposal into the control system. To remove this bottleneck, we construct a decision tree of proposals, starting from known reference settings. is can be loaded into the control system in advance. e human operator can then simply traverse the decision tree of shot settings, at each step selecting the branch according to the choice made about the previous shot. In practice, we load binary trees 9 levels deep consisting of 512 shots with all settings predetermined, thereby staying over an hour ahead of actual experiments. We also explored higher order trees, requesting that the expert human operator provide more than a single binary deci- sion, but this method did not provide additional bene ts.
References
1. Czitrom, V. One-factor-at-a-time versus designed experiments. e American Statistician 53, 126–131 (1999).
2. Louviere, J. J., Flynn, T. N. & Carson, R. T. Discrete choice experiments are not conjoint analysis. Journal of Choice Modelling 3,
57–72 (2010).
3. Robert, C. & Casella, G. Monte Carlo Statistical Methods (Springer Science & Business Media, 2013).
γt =  γ
if human prefers new shot; otherwise.
( 4 )
 t − 1
e step size is also adjusted fractionally, starting from a baseline A. At each stage the step size is adjusted multi-
P= accepted
Scientific REPORTS | 7: 6425 | DOI:10.1038/s41598-017-06645-7 6