Online storage ring optimization using dimension-reduction and genetic algorithms

Particle storage rings are a rich application domain for online optimization algorithms. The Cornell Electron Storage Ring (CESR) has hundreds of independently powered magnets, making it a high-dimensional test-problem for algorithmic tuning. We investigate algorithms that restrict the search space to a small number of linear combinations of parameters (“knobs”) which contain most of the effect on our chosen objective (the vertical emittance), thus enabling efficient tuning. We report experimental tests at CESR that use dimension-reduction techniques to transform an 81-dimensional space to an 8-dimensional one which may be efficiently minimized using one-dimensional parameter scans. We also report an experimental test of a multiobjective genetic algorithm using these knobs that results in emittance improvements comparable to state-of-the-art algorithms, but with increased control over orbit errors.

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Figure 1

The spectrum of eigenvalues obtained from the Hessian matrix of the emittance, normalized so that the first eigenvalue is equal to one. Note the logarithmic vertical scale. The Hessian matrix and its spectrum are available as Supplementary Material [27].

Figure 2

The relative strengths of the kickers used in the first eigenvector as a function of their vertical betatron phase in CESR. Their arrangement in the shape of a betatron wave is readily apparent. The vertical tune is 8.79, nearly matching the 9 crests of the wave.

Figure 3

The average simulated beam size over 1000 instances of a lattice with random misalignments after minimizing the beam size using our first N eigenparameters. The average initial beam size is shown by the point lying on the vertical axis. Note the rapid decrease in beam size from the first few eigenparameters.

Figure 4

The average simulated beam size over 1000 instances of a lattice with random misalignments after correcting dispersion and coupling using our Levenberg-Marquardt-based tuning and then minimizing the beam size using our first N eigenparameters. The average initial beam size is shown by the point lying on the vertical axis. Although not as stark as when starting from an uncorrected lattice, we still see that the first few eigenvectors contribute a disproportionate amount to the reduction in beam size.

Figure 5

Histogram of hundreds of experimental vertical beam size measurements both using the standard lattice for light-source operations with no additional tuning and after tuning with the RCDS algorithm using the best 8 eigenparameters. A clear improvement is observed. Note also the high-side tail of the low-beam-size distribution.

Figure 6

Histogram of hundreds of experimental vertical beam size measurements both after applying just the standard Levenberg-Marquardt minimization of dispersion and coupling and after additional tuning with the RCDS algorithm using the best 8 eigenparameters. A modest but unambiguous improvement is observed. Note also the high-side tails of the measurement distributions.

Figure 7

The improvement due to each knob when tuning the real machine starting from a lattice without additional corrections. Knob 5 contributes a surprisingly large amount. However, we note that it needed to be turned more than twice as far as any other knob, and more than four times as far as the first knob, which suggests that its utility stems from the fact that, when tuning the lattice for use in light-source operations, the first few knobs had already been used, and so the fifth knob is the strongest knob to have not been well-tuned. Note that, when comparing to Fig. 3 or Fig. 4, the latter two plots show the beam size after tuning N knobs while this plot shows the change in beam size from tuning the Nth knob.

Figure 8

The final generation of our genetic algorithm as applied to CESR, with the nondominated individuals differentiated from the rest. Also plotted are the initial beam size and orbit and those obtained after the 8-knob RCDS tuning, as in Fig. 5. The BPMs near the undulator had some offsets, so we are in fact attempting to steer the beam onto an arbitrary off-axis trajectory. Given that steering the beam onto a specific nonzero orbit also represents a trade-off with respect to minimizing emittance, our conclusions are not affected.

Figure 9

Experimental progress of the genetic algorithm with time. Each square marks a generation of the algorithm, showing the location of the mean individual in the trade-off frontier. Arrows indicate the sequence of generations, with one generation taking 10 minutes of machine time. The $x$ axis shows the square root of the ratio of the RMS beam size $\epsilon$ and RMS orbit error $\sigma$, while the $y$ axis shows the square root of their product. For any given $x$, smaller $y$ values are better. Solid lines show the best power-law fit $(y=A{x}^b)$ to the trade-off frontier at each generation. Much of the improvement provided by the genetic algorithm is due to reduction of the orbit error, as evidenced by the trend of the algorithm toward larger values of $\sqrt{\epsilon /\sigma }$.

Figure 10

The epsilon test of convergence in our online study: $\epsilon$ is the scale factor such that the $i$th generation frontier set $X_i$ dominates the rescaled frontier set $\epsilon X_{i+1}$. The algorithm converges as $\epsilon$ approaches unity. The figure shows that the algorithm is still providing improvement at the end of the allotted time. Comparison with Fig. 9 shows that this improvement is mainly due to reduction in the orbit error.