A rank-adaptive robust integrator for dynamical low-rank approximation

Ceruti, Gianluca; Kusch, Jonas; Lubich, Christian

doi:10.48550/arxiv.2104.05247

“…An extension of this algorithm to rank adaptivity according to [2] is straight forward. We state the rank adaptive algorithm here, even though our numerical computations have all been done with the fixed rank integrator.…”

Section: Algorithmmentioning

confidence: 99%

“…Problems in which dynamical low-rank is successfully applied to reduce memory and computational costs are, e.g., kinetic theory [7,8,31,30,9,5,6,13,20] as well as uncertainty quantification [10,28,29,33,18]. Furthermore, DLRA allows for adaptive model refinement [4,2,32], where the main idea is to pick the rank of the solution representation adaptively. Recently, an approach to employ DLRA for computing rightmost eigenpairs has been proposed in [12].…”

Section: Introductionmentioning

confidence: 99%

A low-rank power iteration scheme for neutron transport criticality problems

Kusch¹,

Whewell²,

McClarren³

2022

Preprint

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Computing effective eigenvalues for neutron transport often requires a fine numerical resolution. The main challenge of such computations is the high memory effort of classical solvers, which limits the accuracy of chosen discretizations. In this work, we derive a method for the computation of effective eigenvalues when the underlying solution has a low-rank structure. This is accomplished by utilizing dynamical low-rank approximation (DLRA), which is an efficient strategy to derive time evolution equations for low-rank solution representations. The main idea is to interpret the iterates of the classical inverse power iteration as pseudotime steps and apply the DLRA concepts in this framework. In our numerical experiment, we demonstrate that our method significantly reduces memory requirements while achieving the desired accuracy. Analytic investigations show that the proposed iteration scheme inherits the convergence speed of the inverse power iteration, at least for a simplified setting.

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“…The unconventional integrator has recently been extended to allow for rank adaptivity [8]. I.e., given a tolerance parameter ϑ, the integrator adapts the rank in time.…”

Section: Rank Adaptive Unconventional Integratormentioning

confidence: 99%

“…More exact dose calculation can be achieved by an appropriate Monte Carlo (MC) algorithm, where individual interacting particles are directly simulated [4]. However, while recent performance-tuned MC The efficiency of dynamical low-rank approximation has been demonstrated in several applications, including kinetic theory [14,15,37,36,16,12,13,8,30,11]. Two main challenges of DLRA in the context of kinetic theory and radiation transport specifically are the preservation of mass as well as capturing the asymptotic limit.…”

Section: Introductionmentioning

confidence: 99%

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

Kusch¹,

Stammer²

2021

Preprint

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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.

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

Kusch

¹

,

Stammer

²

2023

View full text Add to dashboard Cite

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.

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A rank-adaptive robust integrator for dynamical low-rank approximation

Cited by 6 publications

References 18 publications

A low-rank power iteration scheme for neutron transport criticality problems

A low-rank power iteration scheme for neutron transport criticality problems

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

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

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