## Application of Monte Carlo techniques to optimization of high-energy beam transport in a stochastic environmentRussell V. Parrish, James E. Dieudonne, Tassos A. Filippas, Lewis Research Center, United States. National Aeronautics and Space Administration An algorithm employing a modified sequential random perturbation, or creeping random search, was applied to the problem of optimizing the parameters of a high-energy beam transport system. The stochastic solution of the mathematical model for first-order magnetic-field expansion allows the inclusion of state-variable constraints, and the inclusion of parameter constraints allowed by the method of algorithm application eliminates the possibility of infeasible solutions. The mathematical model and the algorithm were programmed for a real-time simulation facility; thus, two important features are provided to the beam designer: (1) a strong degree of man-machine communication (even to the extent of bypassing the algorithm and applying analog-matching techniques), and (2) extensive graphics for displaying information concerning both algorithm operation and transport-system behavior. Chromatic aberration was also included in the mathematical model and in the optimization process. Results presented show this method as yielding better solutions (in terms of resolutions) to the particular problem than those of a standard analog program as well as demonstrating flexibility, in terms of elements, constraints, and chromatic aberration, allowed by user interaction with both the algorithm and the stochastic model. Example of slit usage and a limited comparison of predicted results and actual results obtained with a 600 MeV cyclotron are given. |

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600 MeV cyclotron algorithm operation analog computer analog matching applying analog-matching techniques associated with type basic algorithm beam transport system beam-transport design chromatic aberration classes of techniques cost function creeping random search desired end conditions determined digital program dimensionless doublet end point equations of motion extremum Figure 17 focal axis focal-point constraints HIGH-ENERGY BEAM TRANSPORT horizontal focal point inclusion of state-variable initial conditions Langley Research Center last success magnetic focusing strengths man-machine communication mathematical model meters sh method momentum perturbation Monte Carlo Monte Carlo Method NASA Space Radiation null hypothesis number of parameters number of rays optimization algorithm optimization process oscilloscope parallel-to-parallel parameter constraints performance measure point-to-parallel position along focal possible directions present parameter values problem Quadrupole magnetic Radiation Effects Laboratory results obtained separation distance set of system Space Radiation Effects Spark Chamber standard analog program system parameters type I error variables vertical focal point Wdif x(sn x(sout zs-plane