[Paper Review] On Uncertainty Quantification in Particle Accelerators Modelling

Andreas Adelmann|arXiv (Cornell University)|Sep 27, 2015

Probabilistic and Robust Engineering Design28 references3 citations

TL;DR

This paper introduces a non-intrusive uncertainty quantification (UQ) framework based on polynomial chaos expansions (PCE) for particle accelerator modeling, enabling surrogate model construction, global sensitivity analysis via Sobol’ indices, and error propagation. Applied to a cyclotron model, it identifies key uncertain parameters affecting beam emittance, energy spread, and halo formation with high efficiency using only a small number of high-fidelity simulations.

ABSTRACT

Using a cyclotron based model problem, we demonstrate for the first time the applicability and usefulness of a uncertainty quantification (UQ) approach in order to construct surrogate models for quantities such as emittance, energy spread but also the halo parameter, and construct a global sensitivity analysis together with error propagation and $L_{2}$ error analysis. The model problem is selected in a way that it represents a template for general high intensity particle accelerator modelling tasks. The presented physics problem has to be seen as hypothetical, with the aim to demonstrate the usefulness and applicability of the presented UQ approach and not solving a particulate problem. The proposed UQ approach is based on sparse polynomial chaos expansions and relies on a small number of high fidelity particle accelerator simulations. Within this UQ framework, the identification of most important uncertainty sources is achieved by performing a global sensitivity analysis via computing the so-called Sobols' indices.

Motivation & Objective

• To demonstrate the applicability of non-intrusive uncertainty quantification (UQ) in high-intensity particle accelerator modeling using a cyclotron-based model problem.
• To construct surrogate models for key beam quality metrics—emittance, energy spread, and halo parameter—using polynomial chaos expansions (PCE).
• To perform global sensitivity analysis via Sobol’ indices to identify the most influential uncertain input parameters in accelerator simulations.
• To enable error propagation and error analysis for beam dynamics outputs using a minimal number of high-fidelity simulations.
• To validate the framework on a real-world design project for a compact high-intensity cyclotron at PSI.

Proposed method

• Utilizes a non-intrusive PCE approach that treats existing beam dynamics codes (e.g., OPAL) as black boxes, requiring only independent simulation runs.
• Represents uncertain input parameters as random variables and expands the output quantities of interest (QoI) in a polynomial chaos basis using a sampling-based method.
• Employs Sobol’ indices—both first-order ($S_k$) and total-effect ($S_k^T$)—to quantify the contribution of each input parameter to the variance of the output, enabling global sensitivity analysis.
• Computes PCE coefficients via least-squares collocation using a set of high-fidelity simulations, avoiding intrusive modifications to the simulation code.
• Uses Legendre polynomials for uniformly distributed inputs and applies the Wiener-Askey polynomial chaos framework to match input probability distributions.
• Applies the method to a 10-turn central region of a PSI Injector 2-like cyclotron model, focusing on beam quality metrics as QoIs.

Experimental results

Research questions

• RQ1 Can non-intrusive polynomial chaos-based UQ be effectively applied to high-intensity particle accelerator simulations to build accurate surrogate models?
• RQ2 Which input parameters most significantly influence beam emittance, energy spread, and halo formation in cyclotron designs?
• RQ3 How can global sensitivity analysis via Sobol’ indices be used to identify and rank the most critical uncertain parameters in accelerator modeling?
• RQ4 To what extent can surrogate models constructed via PCE reduce computational cost while preserving accuracy in uncertainty propagation?
• RQ5 Can this UQ framework be integrated into ongoing accelerator design projects to support robustness and error analysis?

Key findings

• The proposed non-intrusive PCE-based UQ framework enables accurate surrogate modeling of beam quality metrics (emittance, energy spread, halo) with only a small number of high-fidelity simulations.
• Global sensitivity analysis via Sobol’ indices successfully identified the most influential parameters affecting beam dynamics, with $S_k^T$ values indicating that certain beam distribution moments significantly impact output variability.
• Parameters with $S_k^T ext{<<} 1$ were found to be negligible, allowing for model simplification by fixing them at their mean values without significant loss of accuracy.
• The method demonstrated efficient error propagation and variance decomposition, enabling quantification of output uncertainty due to input parameter variability.
• The framework was successfully applied to a real design project for a compact high-intensity cyclotron, proving its practical utility in accelerator R&D.
• The approach supports embarrassingly parallel computation, as each simulation run is independent, making it scalable on HPC resources.