A low-rank method for two-dimensional time-dependent radiation transport calculations

Peng, Zhuogang; McClarren, Ryan G.; Frank, Martin

doi:10.1016/j.jcp.2020.109735

Cited by 60 publications

(53 citation statements)

References 25 publications

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“…A dynamical low-rank approximation for random wave equations has been studied in [34]. Examples of DLRA for kinetic equations, which include hyperbolic advection terms are [26,35,36,37,38,39]. Similar to our work, filters have been used in [38] to mitigate oscillations in the low-rank approximation.…”

Section: Introductionmentioning

confidence: 88%

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Dynamical Low-Rank Approximation for Burgers' Equation With Uncertainty

Kusch

Ceruti

Einkemmer

et al. 2022

Int. J. UncertaintyQuantification

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Quantifying uncertainties in hyperbolic equations is a source of several challenges. First, the solution forms shocks leading to oscillatory behaviour in the numerical approximation of the solution. Second, the number of unknowns required for an effective discretization of the solution grows exponentially with the dimension of the uncertainties, yielding high computational costs and large memory requirements. An efficient representation of the solution via adequate basis functions permits to tackle these difficulties. The generalized polynomial chaos (gPC) polynomials allow such an efficient representation when the distribution of the uncertainties is known. These distributions are usually only available for input uncertainties such as initial conditions, therefore the efficiency of this ansatz can get lost during runtime. In this paper, we make use of the dynamical low-rank approximation (DLRA) to obtain a memory-wise efficient solution approximation on a lower dimensional manifold. We investigate the use of the matrix projector-splitting integrator and the unconventional integrator for dynamical low-rank approximation, deriving separate time evolution equations for the spatial and uncertain basis functions, respectively. This guarantees an efficient approximation of the solution even if the underlying probability distributions change over time. Furthermore, filters to mitigate the appearance of spurious oscillations are implemented, and a strategy to enforce boundary conditions is introduced. The proposed methodology is analyzed for Burgers' equation equipped with uncertain initial values represented by a two-dimensional random vector. The numerical experiments validate that the results of a standard filtered Stochastic-Galerkin (SG) method are consistent with the numerical results obtained via the use of numerical integrators for dynamical low-rank approximation. Significant reduction of the memory requirements is obtained, and the important characteristics of the original system are well captured.

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Section: Introductionmentioning

confidence: 88%

“…Examples of DLRA for kinetic equations, which include hyperbolic advection terms are [26,35,36,37,38,39]. Similar to our work, filters have been used in [38] to mitigate oscillations in the low-rank approximation. Furthermore, [36] uses diffusion terms to dampen oscillatory artifacts.…”

Section: Introductionmentioning

confidence: 89%

Dynamical Low-Rank Approximation for Burgers' Equation With Uncertainty

Kusch

Ceruti

Einkemmer

et al. 2022

Int. J. UncertaintyQuantification

Self Cite

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“…The reference solution to this problem is given by the standard de facto Ganapol's benchmark test [10] and this problem has been investigated for dynamical low-rank approximations in [25,24]. As time increases, the scalar flux Φ(t, x) = 1 −1 f (t, x, µ) dµ moves to the left and right side of the spatial domain, showing a discontinuous (or shock) profile at the front.…”

Section: Radiation Transport Equationmentioning

confidence: 99%

“…In Section 5 we present numerical experiments with examples from the fields of kinetic equations and uncertainty quantification, where dynamical low-rank approximation has recently found much interest, e.g. in [6][7][8]25] and [9,22,23,27], beyond the original application area of quantum dynamics, e.g. [21,20] and [11,12].…”

Section: Introductionmentioning

confidence: 99%

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

2022

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A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator. Keywordsdynamical low-rank approximation • rank adaptivity • structurepreserving integrator • matrix and tensor differential equations Mathematics Subject Classification (2000) 65L05 • 65L20 • 65L70 • 15A69

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“…This approach does not only provide the ability to derive stable discretizations [19], but in the radiation transfer context allows an efficient implementation of scattering [19,20]. 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.…”

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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A low-rank method for two-dimensional time-dependent radiation transport calculations

Cited by 60 publications

References 25 publications

Dynamical Low-Rank Approximation for Burgers' Equation With Uncertainty

Dynamical Low-Rank Approximation for Burgers' Equation With Uncertainty

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

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

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