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II. Maximum Entropy Tomography
V. Acknowledgments and References
Particle accelerators generate charged particle beams with tailored distributions in six-dimensional position-momentum space (phase space). Knowledge of the phase space distribution enables modelbased beam optimization and control. In the absence of direct measurements, the distribution must be tomographically reconstructed from its projections. In this paper, we highlight that such problems can be severely underdetermined and that entropy maximization is the most conservative solution strategy. We leverage normalizing flows—invertible generative models—to extend maximum-entropy tomography to six-dimensional phase space and perform numerical experiments to validate the model’s performance. Our numerical experiments demonstrate consistency with exact two-dimensional maximum-entropy solutions and the ability to fit complicated six-dimensional distributions to large measurement sets in reasonable time.
I. INTRODUCTION
Particle accelerators generate charged particle beams with tailored distributions in position-momentum space (phase space). Measuring the phase space distribution in the accelerator enables model-based beam optimization and control and provides a valuable benchmark for simulation codes. In the absence of direct measurements [1–3], the distribution must be reconstructed from its projections.[1] Fig. 1 illustrates a generic setup in which the
beam is measured under varying accelerator conditions and reconstructed at a location upstream of the measurement device.
If the accelerator linearly transforms the phase space coordinates and does not couple the three planes of motion, one can reconstruct the 2D phase space distribution using conventional tomography algorithms. It is more challenging to reconstruct the 4D or 6D phase space distribution. Many conventional algorithms represent the distribution on a grid and face massive storage requirements as the phase space dimension scales [5]. Several authors have developed new algorithms and diagnostics to sidestep this issue and fit 4D phase space distributions to 2D projections [5–10]. Recent work has also explored extensions to 5D and 6D phase space [11–13].
An additional challenge is that high-dimensional reconstructions may be ill-posed; since the measured dimension is fixed, the set of feasible distributions (those consistent with the measurements) may proliferate with the phase space dimension. It is usually infeasible to compensate by exponentially increasing the number of measurements, as one is typically limited to tens of views because of slow diagnostic devices and limited beam time. Additionally, it is not yet clear how to derive the information-maximizing set of high-dimensional phase space transformations under given measurement conditions—and in any case, accelerator constraints place many transformations out of reach.
To select a single solution from the feasible set, our strategy is to define a prior probability distribution over the phase space coordinates and update the prior to a posterior by incorporating the information in the measurements. Our information comes in the form of constraints, and we perform the update by maximizing a convex functional subject to these constraints. Under basic self-consistency requirements, the functional must be the relative entropy [14–17]. Entropy maximization ensures that the posterior does not deviate from the prior unless forced to by the data. This is a conservative strategy that eliminates all spurious features from the reconstructed distribution.
Entropy maximization is not always feasible, especially in high dimensions, because it entails a highly nonlinear constrained optimization. Although a reliable exact maximum-entropy algorithm exists for 2D tomography, its computational complexity scales exponentially with the phase space dimension, rendering its extension to 6D prohibitively expensive at this time. In this paper, we leverage normalizing flows—invertible generative models—to find approximate 6D maximum-entropy solutions. Our approach is a straightforward extension of two previous studies. Loaiza-Ganem, Gao, and Cunningham [18] first proposed the use of normalizing flows for entropy maximization subject to statistical moment constraints; we incorporate projection constraints using the differentiable physics simulations and projected density estimation proposed by Roussel et al. [10] in the Generative Phase Space Reconstruction (GPSR) framework. We refer to the resulting approach as MENT-Flow.
We begin by deriving the form of the n-dimensional maximum-entropy distribution subject to m-dimensional projection constraints, following the analysis in [19]. We then discuss the shortcomings of existing maximumentropy tomography algorithms when n = 6 and describe the flow-based solution. Finally, we perform numerical experiments to validate the model’s reliability in 2D settings and examine the effects of entropic regularization in 6D tomography.
This paper is
[1] The 1D beam density can be measured by recording the secondary electron emission from a wire swept across the beam. Scintillating screens provide 2D projections of electron beams or low-intensity, low-energy hadron beams. 2D projections of higher energy hadron beams are only available from specialized diagnostics such as laser wires [4].
Authors:
(1) Austin Hoover, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA ([email protected]);
(2) Jonathan C. Wong, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China.