A Deterministic Dynamical Low-rank Approach for Charged Particle Transport

TL;DR

This work tackles the high computational cost of deterministic charged-particle transport at fine spatial and angular resolutions by introducing a collided-uncollided split and a dynamical low-rank approximation (DLRA) on the collided component, with energy treated as a pseudo-time. The uncollided flux is computed via a raytracer, while the collided flux is updated on a low-rank manifold using a rank-adaptive augmented BUG integrator, achieving substantial reductions in both runtime and memory without sacrificing accuracy relative to full-rank deterministic solutions and showing reasonable agreement with TOPAS Monte Carlo. The method is demonstrated on 90 MeV proton-therapy-like scenarios in both homogeneous and heterogeneous media, illustrating impressive gains in angular resolution feasibility and providing insight into the rank dynamics across energy. The results suggest significant practical impact for high-fidelity dose calculations, with potential extensions to multi-direction beams and non-water-equivalent tissues.

Abstract

Deterministically solving charged particle transport problems at a sufficient spatial and angular resolution is often prohibitively expensive, especially due to their highly forward peaked scattering. We propose a model order reduction approach which evolves the solution on a low-rank manifold in time, making computations feasible at much higher resolutions and reducing the overall run-time and memory footprint. For this, we use a hybrid dynamical low-rank approach based on a collided-uncollided split, i.e., the transport equation is split through a collision source method. Uncollided particles are described using a ray tracer, facilitating the inclusion of boundary conditions and straggling, whereas collided particles are represented using a moment method combined with the dynamical low-rank approximation. Here the energy is treated as a pseudo-time and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. We can reproduce the results of a full-rank reference code at a much lower rank and thus computational cost and memory usage. The solution further achieves comparable accuracy with respect to TOPAS MC as previous deterministic approaches.

Figures (5)

• Figure 1: Cuts through the deposited energy for Monte Carlo vs. DLRA with different angular resolutions and tolerances for homogeneous case.
• Figure 2: Deposited energy of MC vs. DLRA with P$_{95}$, $\vartheta=0.1$ at x=1cm in the homogeneous test case.
• Figure 3: (a-b) Ranks chosen by rank-adaptive integrator for different tolerances and angular resolutions. (c) Fraction of captured information in top k singular valuesof P$_{95}$ solution at different energies.
• Figure 4: Cuts through the deposited energy for Monte Carlo vs. DLRA with different angular resolutions and tolerances in heterogeneous case.
• Figure 5: Deposited energy of MC vs. DLRA with P$_{95}$, $\vartheta=0.1$ at x=1cm (top) and y=1cm (bottom) in heterogeneous case.