A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Abstract

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L 2 -stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

Adaptation of the urban climate model PALM-4U for Wuerzburg, Germany

“…to the presented energy stable mass conservative DLRA solution from (19). The total mass at any time t n shall be defined as m n = ∆x j u n j0 + B n j .…”

“…At a given time point t end = 3.16 this waves can be seen in Figure 3 where we display numerical results for the full solution f (x, µ), the scalar flux Φ = ⟨f ⟩ µ and the temperature T . We compare the solution of the full coupled-implicit system differing from (28) by an additional source term to the presented energy stable mass conservative DLRA solution from (19). Further, the evolution of the rank in time is presented for a tolerance parameter of ϑ = 10 −2 ∥Σ∥ 2 .…”

“…We compare the solution of the two-dimensional full system corresponding to (28) to the two-dimensional DLRA solution corresponding to (19). In Figure 4 we show numerical results for the scalar flux Φ = S 2 f (t, x, •) dΩ and the temperature T at the time t = 0.5.…”

“…DLRA's core idea is to represent and evolve the solution on the low-rank manifold of rank r functions. Past work in the area of radiative transfer has focused on asymptotic-preserving schemes [9,8], mass conservation [29], stable discretizations [18], imposing boundary conditions [19,15] and implicit time discretizations [30]. A discontinuous Galerkin discretization of the DLRA evolution equations for thermal radiative transfer has been proposed in [4].…”

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation

“…Here, we use the BUG integrator which only evolves the solution forward in time, thereby facilitating the construction of stable spatial discretizations [35]. Moreover, the BUG integrator enables a straightforward basis augmentation step [6] which simplifies the construction of rank adaptive methods [6,36,20] and allows for conservation properties [6,14].…”