We present the first part of an efficient framework for nonlinear beam dynamics, termed Approximate Invariant Analysis (AIA). The framework is based on the construction of approximate invariants [Y. Li, D. Xu, and Y. Hao, Phys. Rev. Accel. Beams 28, 074001 (2025)] and on the extraction of the betatron frequency with the geometric foundations of Poincaré rotation number [S. Nagaitsev and T. Zolkin, Phys. Rev. Accel. Beams 23, 054001 (2020)]. The method is demonstrated using the National Synchrotron Light Source II (NSLS-II) storage ring as an illustrative example.

We highlight the virtues of 2D elliptic-multipole field expansions in modeling the magnetic fields of narrow-aperture, straight-axis bending magnets with parallel faces, addressing the limitations of the conventional circular multipole series when the beam-orbit sagitta exceeds the magnet's vertical half-gap. The elliptic multipoles provide a convenient way to represent the field in all aspects of the magnet development (design, particle-tracking simulations, measurements). We propose a numerically robust method of data analysis to determine the elliptic (or circular) multipoles from stretched-wire measurements with the wire moving on an arbitrary path.

Advances in fidelity and performance of accelerator modeling tools, in tandem with novel machine learning capabilities, has prompted community initiatives aiming to realize “virtual test stands” that can serve as true analogues to physical machines. Such efforts require integrated, end-to-end modeling capabilities with support for parametric optimization and benchmarking. We present the ongoing development of an integrated Sirepo application to support the holistic modeling of accelerators. Our approach leverages existing modeling workflows, such as the Light Source Unified Modeling Environment (LUME), as well as community I/O frameworks, such as openPMD, to provide a toolbox for constructing and modeling beamlines. Users can build and test simulations using different community modeling tools, as well as connect individual tools to produce end-to-end simulations. Additional workflows have been developed to support machine learning tools that facilitate optimization and the development of surrogate models. We discuss some specific beamline modeling demonstrations as well as ongoing efforts to support code-agnostic design and development.

Particle accelerator modeling is an important field of research and development, essential to investigating, designing and operating some of the most complex scientific devices ever built. Kinetic simulations of relativistic, charged particle beams and advanced plasma accelerator elements are often performed with high-fidelity particle-in-cell simulations, some of which fill the largest GPU supercomputers. Start-to-end modeling of a particle accelerator includes many elements and it is desirable to integrate and model advanced accelerator elements fast, in effective models. Traditionally, analytical and reduced-physics models fill this role. The vast data from high-fidelity simulations and power of GPU-accelerated computation open a new opportunity to complement traditional modeling without approximations: surrogate modeling through machine learning. In this paper, we implement, present and benchmark such a data-driven workflow, synthesising a fully GPU-accelerated, conventional-surrogate simulation for hybrid particle accelerator beamlines.

Laser-driven ion acceleration provides ultrashort, high-charge, low-emittance beams, which are desirable for a wide range of high-impact applications. Yet after decades of research, a significant increase in maximum ion energy is still needed. This paper introduces a quality-preserving staging concept for ultraintense ion bunches that is seamlessly applicable from the nonrelativistic plasma source to the relativistic regime. Full three-dimensional particle-in-cell simulations prove robustness and capture of a high-charge proton bunch, suitable for readily available and near-term laser facilities.

Developing complex, reliable advanced accelerators requires a coordinated, extensible, and comprehensive approach in modeling, from source to the end of beam lifetime. We present highlights in Exascale Computing to scale accelerator modeling software to the requirements set for contemporary science drivers. In particular, we present the first laser-plasma modeling on an exaflop supercomputer using the US DOE Exascale Computing Project WarpX. Leveraging developments for Exascale, the new DOE SCIDAC-5 Consortium for Advanced Modeling of Particle Accelerators (CAMPA) will advance numerical algorithms and accelerate community modeling codes in a cohesive manner: from beam source, over energy boost, transport, injection, storage, to application or interaction. Such start-to-end modeling will enable the exploration of hybrid accelerators, with conventional and advanced elements, as the next step for advanced accelerator modeling. Following open community standards, we seed an open ecosystem of codes that can be readily combined with each other and machine learning frameworks. These will cover ultrafast to ultraprecise modeling for future hybrid accelerator design, even enabling virtual test stands and twins of accelerators that can be used in operations.

SciDAC-5 goals: Deliver particle accelerator and beam simulations tools that go beyond the current state of the art, up to the realization of virtual twins of particle accelerators, enabling design and modeling of particle accelerators at unprecedented speed, levels of accuracy, and realism; and apply these tools to key accelerator facilities relevant to DOE HEP (such as PIP-II/DUNE, FACET-II).

When a charged-particle bunch in a storage ring is kicked to a large transverse offset, the time series describing the dynamics of the bunch centroid is determined both by the lattice focusing and by the Fourier transform of the 1D density profile of the bunch projected along the angle of the kick. In the presence of nonlinear focusing, we show that this fact can be exploited to enable 2D phase space reconstruction of the bunch (computational tomography) based only on turn-by-turn beam position monitor data. We demonstrate various tomography methods based on this principle, including machine learning methods, and discuss their advantages and disadvantages, and measure of reliability. We also mention a possible extension to 4D phase space computational tomography.

Supplementary materials (aka data artifact or data archive) for our HB2023 publication: " ImpactX Modeling of Benchmark Tests for Space Charge Validation" (Paper ID: THBP44). This work was supported by the Director, Office of Science of the U.S. Department of Energy under Contracts No. DE-AC02-05CH11231 and DE-AC02-07CH11359. This material is based upon work supported by the CAMPA collaboration, a project of the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of High Energy Physics, Scientific Discovery through Advanced Computing (SciDAC) program. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAP0023719.

Data archive for stage 1 of manuscript for PASC24. See readme.txt for more details. This work was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DE-AC02-05CH11231 and by LLNL under Contract DE-AC52-07NA27344. This material is based upon work supported by the CAMPA collaboration, a project of the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of High Energy Physics, Scientific Discovery through Advanced Computing (SciDAC) program. This research was supported by the Exascale Computing Project (17-SC-20-SC), a joint project of the U.S. Department of Energy's Office of Science and National Nuclear Security Administration, responsible for delivering a capable exascale ecosystem, including software, applications, and hardware technology, to support the nation's exascale computing imperative.This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAP0023719.

Measures of discrepancy between probability distributions (statistical distance) are widely used in the fields of artificial intelligence and machine learning. We describe how certain measures of statistical distance can be implemented as numerical diagnostics for simulations involving charged-particle beams. Related measures of statistical dependence are also described. The resulting diagnostics provide sensitive measures of dynamical processes important for beams in nonlinear or high-intensity systems, which are otherwise difficult to characterize. The focus is on kernel-based methods such as Maximum Mean Discrepancy, which have a well-developed mathematical foundation and reasonable computational complexity. Several benchmark problems and examples involving intense beams are discussed. While the focus is on charged-particle beams, these methods may also be applied to other many-body systems such as plasmas or gravitational systems.

Particle accelerators are among the largest, most complex devices. To meet the challenges of increasing energy, intensity, accuracy, compactness, complexity and efficiency, increasingly sophisticated computational tools are required for their design and optimization. It is key that contemporary software take advantage of the latest advances in computer hardware and scientific software engineering practices, delivering speed, reproducibility and feature composability for the aforementioned challenges. A new open source software stack is being developed at the heart of the Beam pLasma Accelerator Simulation Toolkit (BLAST) by LBNL and collaborators, providing new particle-in-cell modeling codes capable of exploiting the power of GPUs on Exascale supercomputers. Combined with advanced numerical techniques, such as mesh-refinement, and intrinsic support for machine learning, these codes are primed to provide ultrafast to ultraprecise modeling for future accelerator design and operations.

Beamlines capable of merging beams with different energies are critical to many applications related to advanced accelerator concepts and energy-recovery linacs (ERLs). In an ERL, a low-energy “fresh” bright bunch is generally injected into a superconducting linac for acceleration using the fields established by a decelerated “spent” beam traveling on the same axis. A straight-merger system composed of a selecting cavity with a superimposed dipole magnet was proposed and recently tested at AWA. This paper reports on the experimental results obtained so far along with detailed beam dynamics investigations of the merger concept and its ability to conserve the beam brightness associated with the fresh bunch.

For modeling the dynamics within a dipole of a bunch whose length is much larger than the vacuum pipe radius, it is typical to use a 2D (or 2.5D) Poisson solver, with arc length taken as the independent variable. However, sampled at a fixed time, the beam is curved, space charge is not truly 2D, and the usual cancellation between E and B contributions to the Lorentz force need not exactly hold. The size of these effects is estimated using an idealized model of a uniform torus of charge rotating inside a toroidal conducting pipe. Simple expressions are provided for the correction of the electric and magnetic fields to first order in the reciprocal of the curvature radius.

The time-series Beam Position Monitor (BPM) data of kicked beam is a function of lattice parameters and beam parameters including phase-space density. The decoherence model using the first-order detuning parameter has an exact solution when the beam is Gaussian. We parameterize the beam phase-space density by multiple Gaussian kernels of different weights, means, and sizes to formulate the inverse problem for 2D phase-space tomography. Numerical optimization and Bayesian inference are used to infer the beam density and uncertainty. INTRODUCTION The BPM data of a kicked beam is a function of the linear and nonlinear optics parameters [1], and the beam phasespace density. Ignoring nonlinear normalizing map, we model BPM data by ⟨X⟩t = R ⟨X − iP⟩t (1) = R∫(X − iP) eρX,P(X − X0, P − P0) dXdP, where X and P are normal conjugate variables X(t) = √2βI cos (ωt + φ) , P(t) = −√2βI sin (ωt + φ) , and X0 and P0 are initial kicks (or offsets). Figures 1 and 2 illusrate the sensitivity of the BPM data on initial beam phase-space density. Figure 1: Kicked beam centroid data of initial Gaussian beam. It is convenient to introduce θ such that X0 − iP0 = √2I0β exp (−iθ) , (2) where I0 = (X2 0 + P2 0)/2 is the action corresponding to the initial beam centroid offset. Assuming slowly varying ∗ Work supported by the Director of the Office of Science of the US Department of Energy under Contract no. DE-AC02-05CH11231 † kilean@lbl.gov Figure 2: Kicked beam centroid data of initial uniform disk beam. frequency in the scale of the beam size, the frequency can be modeled by ω(ΔI) = μ0 + μ1ΔI, (3) where ΔI ≡ I − I0. For Gaussian initial beam distribution, an exact solution exists: ⟨X⟩t = X0 (1 − τ2) + 2P0τ (1 + τ2) exp [−I0 ε τ2 1 + τ2 ] cos Ψ (4) − 2X0τ − P0 (1 − τ2) (1 + τ2) exp [−I0 ε τ2 1 + τ2 ] sin Ψ, where τ ≡ εμ1t and Ψ ≡ μ0t − (I0/ε) τ3/ (1 + τ2). 2D PHASE-SPACE TOMOGRAPHY STRATEGY Marginal Distribution For general initial beam distribution, one needs an approximation to extract meaningful expression for the centroid decoherence motion. As the centroid decoherence is due to the phase-mixing, it is also advantageous to work on the frequency domain. We define the following function in the frequency domain: G(k) = 2 √2βI0 T ∑ t=0 ⟨X⟩t [e−ikt] − cos θ. (5) In the limit of large initial offset compared to the initial beam size I0 ≫ ε, it can be shown that [2] ρθ (x) ≃ √2βI0 ∣μ1∣ π R [G (xμ1√2βI0 + μ0) e iθ] . (6) This suggests that if we have multiple kicks of different angles θ, we can reconstruct the 2D phase-space. And more kicks of different angles increase angular resolution. 12th Int. Particle Acc. Conf. IPAC2021, Campinas, SP, Brazil JACoW Publishing ISBN: 978-3-95450-214-1 ISSN: 2673-5490 doi:10.18429/JACoW-IPAC2021-MOPAB235 MC5: Beam Dynamics and EM Fields D02 Non-linear Single Particle Dynamics MOPAB235 763 C on te nt fr om th is w or k m ay be us ed un de rt he te rm s of th e C C B Y 3. 0 lic en ce (© 20 21 ). A ny di st ri bu tio n of th is w or k m us tm ai nt ai n at tr ib ut io n to th e au th or (s ), tit le of th e w or k, pu bl is he r, an d D O I

Integrable dynamical systems play an important role in many areas of science, including accelerator and plasma physics. An integrable dynamical system with $n$ degrees of freedom possesses $n$ nontrivial integrals of motion, and can be solved, in principle, by covering the phase space with one or more charts in which the dynamics can be described using action-angle coordinates. To obtain the frequencies of motion, both the transformation to action-angle coordinates and its inverse must be known in explicit form. However, no general algorithm exists for constructing this transformation explicitly from a set of $n$ known (and generally coupled) integrals of motion. In this paper we describe how one can determine the dynamical frequencies of the motion as functions of these $n$ integrals in the absence of explicitly known action-angle variables, and we provide several examples.

There have been questions regarding the impact of the dipole fringe field models (used by accelerator codes including ELEGANT and MADX) on vertical chromaticity. Here, we analyze the cause of the disagreement among codes and suggest a correction. INTRODUCTION An accurate and realistic way of modeling the dipole fringe field effects requires 3D surface field data, or on-axis vertical field shape modeling [1, 2]. However, in most cases where the bending radius is large or the vertical gap between magnet pole faces is small, such methods are numerically heavy in view of the relative importance of the fringe field. On the other hand, the analytically expressed dipole-fringefield thin map is a computationally efficient way of modeling the dipole-fringe-field effects [3–5]. Many accelerator simulation codes adopt the thin map for dipole-fringe-field modeling. However, we observed some discrepancies in the numerically calculated vertical chromaticities of a few storage rings when different thin map models are used. In a comparison of the reference [3–5], we found that the vertical chromatic focusing terms of Ref. [3] do not agree with the Refs. [4, 5]. In this paper, we discuss the disagreement in detail and try to verify using simulations. We also illustrate the effect on the vertical chromaticity of the Integrable Optics Test Accelerator (IOTA) ring. Finally, we discuss the dipole-fringe-field effect on dispersion. REVIEW In this section, we review the chromatic vertical focusing term of Refs. [3–5]. Notation For consistent comparision, we use the following notation: • edge angle: θ • bending radius: ρ • vertical gap between magnet pole faces: g • field integration parameter: