Innovative applications of genetic algorithms to problems in accelerator physics

The genetic algorithm (GA) is a powerful technique that implements the principles nature uses in biological evolution to optimize a multidimensional nonlinear problem. The GA works especially well for problems with a large number of local extrema, where traditional methods (such as conjugate gradient, steepest descent, and others) fail or, at best, underperform. The field of accelerator physics, among others, abounds with problems which lend themselves to optimization via GAs. In this paper, we report on the successful application of GAs in several problems related to the existing Continuous Electron Beam Accelerator Facility nuclear physics machine, the proposed Medium-energy Electron-Ion Collider at Jefferson Lab, and a radio frequency gun-based injector. These encouraging results are a step forward in optimizing accelerator design and provide an impetus for application of GAs to other problems in the field. To that end, we discuss the details of the GAs used, include a newly devised enhancement which leads to improved convergence to the optimum, and make recommendations for future GA developments and accelerator applications.

Figure 1

The search space (red and green) and Pareto-optimal front (green) for the minimization of the system in Eq. (2) [19]. Pareto-optimal front estimates found by the Strength Pareto Evolutionary Algorithm (SPEA2) [20] after 1, 10, and 20 generations for 16 individuals are marked. The overlap between the search space and the rectangles with individuals $A$, $B$, and $C$ at the top right vertices contain the points, if any, in the search space that dominate the respective individuals.

Figure 2

Layout of the MEIC.

Figure 3

A grid of sum (black) and difference (green) resonances up to 7th order. Shaded regions mark restricted search space for the GA, which is completely devoid of black resonance lines. The dots denote the optimal working point: red represents the betatron tunes for the proton beam, and blue for the electron beam.

Figure 4

GA at work: beam-beam simulation of the MEIC with five generations of 64 individuals each, sampling the 4D tune space $[0.5,0.55{]}^4$. The green line represents the design luminosity. The optimization locates a near-optimal working point in fewer than 300 simulations (blue $\times$).

Figure 5

Tune footprint for 4000 randomly selected representative particles from each beam for the near-optimal working point denoted by a blue $\times$ in Fig. 4, corresponding to $({\nu }_x,{\nu }_y)=(0.53,0.548)$ for the electron beam and (0.501, 0.527) for the proton beam. The electron beam is shown in blue, and the proton in red. The location of the near-optimal working point is denoted with two crosses, one for each tune.

Figure 6

Top panel: Improvement over the design luminosity after each generation for a GA-based optimization with 20 generations of 128 individuals in each. The improved working point is about 9% better than the one found in Fig. 4. Bottom panel: Improvement over design luminosity after a systematic parameter scan with resolution $k$ in each parameter.

Figure 7

Linear optics of the ion ring’s IR.

Figure 8

Dynamic aperture after the linear chromaticity compensation for the initially chosen working point of $({\nu }_x,{\nu }_y)=(0.31,0.32)$.

Figure 9

Maximization of the dynamic aperture for the MEIC ion collider ring: 64 individuals sampling the $[0,1{]}^2$ domain, evolved using GA over 20 generations. The blue $\times$ denotes the maximum value of the dynamic aperture.

Figure 10

Dynamic aperture versus the fractional betatron tunes for the simulation in Fig. 9.

Figure 11

Comparison of the dynamic aperture for the initial $({\nu }_x,{\nu }_y)=(0.31,0.32)$ working point (red line) to that for the optimal (0.994, 0.001) working point marked with the blue $\times$’s in Figs. 9, 10 (blue line).

Figure 12

Schematic of the CEBAF accelerator.

Figure 13

Spread of all the individuals in the simulation in the $(x_1,x_2)$ subdimension of the entire search space (blue points), and those with the value of the objective functions below ${10}^{-10}$ (red). The simulation shows 25 generations with 24 individuals apiece.

Figure 14

Simulations over the entire search space (blue) and the reduced search space (red).

Figure 15

A representative comparison between the GA simulation of 20 generations and 24 individuals without shrinking enhancement (blue) and with (red). Also plotted is the simulation with the elegant simplex optimizer (black). The number of iterations needed for each method to converge to within the predefined threshold $S$ is marked with an $\times$ of corresponding color.

Figure 16

${\sigma }_{xy}$ correlations at the end of the injector line before and after the decoupling procedure. Not shown are the other off-diagonal terms that are nulled as well.

Figure 17

Pareto front after 24 generations of 64 individuals. The large $\times$’s denote representative points A, B, and C.

Figure 18

Dynamic aperture plots corresponding to points A, B, and C in Fig. 17, as indicated by the color.

Figure 19

Fractional betatron tunes ${\nu }_x$ and ${\nu }_y$ as a function of momentum offset $\Delta p/p$ for points A, B, and C in Fig. 17, as indicated by the color.

Figure 20

Optimized straight line model of the cylindrically symmetric Photo Injector Test Facility Zeuthen (PITZ) rf gun geometry. Total length is 26.191 85 cm, and the horizontal axis is the symmetry axis. The isolines (magenta) computed by Poisson Superfish show the magnetic ($R{H}_{\varphi }$ component) field pattern for an accelerating mode.

Figure 21

PITZ on-axis balanced field profiles for the PITZ curvilinear geometry (black) and model straight line geometry (blue crosses). The optimized straight line geometry (red) is unbalanced.

Figure 22

Beam tube and cell parameters in general cavity description [25]: (a) beam tube or iris; (b) cell. The beam enters each element on the left.

Figure 23

Straight line approximations for elliptical (far left), reentrant (three middle), and pillbox (far right) cell geometries. Each cell radius is 6 cm, and the total length is 31.2 cm.

Figure 24

Schematic of PITZ injector used in the optimization.

Figure 25

Average signed field flatness for the Pareto-optimal front and offspring for each generation.

Figure 26

Average frequency for the Pareto-optimal front and offspring for each generation.

Figure 27

Average longitudinal emittance for the Pareto-optimal front and offspring for each generation.

Figure 28

Probability density functions used in polynomial mutation: (a) when the independent variable domain is unconstrained for three values of ${\eta }_{\mathrm{mut}}$ [Eq. (a1)]; (b) when the independent variable domain is restricted to $x^{\mathrm{L}}\le x\le x^{\mathrm{U}}$ [Eq. (a3)]. In (b), ${\eta }_{\mathrm{mut}}$ is fixed, and $\Delta$ is varied.

Figure 29

Probability density functions used in simulated binary crossover recombination: (a) when the independent variable domain is unconstrained for three values of ${\eta }_{\mathrm{rec}}$ [Eq. (a6)]; (b) when the independent variable domain is restricted to $x^{\mathrm{L}}\le x\le x^{\mathrm{U}}$ [Eq. (a9)]. In (b), ${\eta }_{\mathrm{rec}}$ is fixed, and B is varied.