A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Peng, Zhuogang; McClarren, Ryan G.

doi:10.1016/j.jcp.2021.110672

Cited by 25 publications

(9 citation statements)

References 65 publications

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“…Another open question in low-rank methods is conserving the intrinsic properties of the underlying problem such as conservation of particles or the appropriate asymtotic limits. As in our previous work [15], the high-order/loworder (HOLO) algorithm is one way to perform the fix. The implementation is straightforward since we can also calculate the Eddington tensor with the lowrank S N solution.…”

Section: Discussionmentioning

confidence: 99%

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A sweep-based low-rank method for the discrete ordinate transport equation

Peng,

McClarren

2022

Preprint

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The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via an "unconventional" basis update & Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

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Section: Discussionmentioning

confidence: 99%

“…In previous work, the authors applied the DLR method to transport calculations with a spherical harmonics expansion and an explicit time scheme [14]. Later, a high-order/low-order algorithm was developed in [15] to overcome the conservation loss in the low-rank evolution.…”

Section: Introductionmentioning

confidence: 99%

A sweep-based low-rank method for the discrete ordinate transport equation

Peng,

McClarren

2022

Preprint

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“…Such an approach has been shown to be effective for both kinetic equations that appear in plasma physics (such as the Vlasov-Poisson [19,28] or Vlasov-Maxwell [22] equations) and in a number of transport problems (e.g. radiation transport problems [49,12,48,37] or the Boltzmann-BGK equation [16,24,8]). In all of these cases ξ 1 contains the spatial variables and ξ 2 contains all the velocity variables.…”

Section: Dynamical Low-rank Algorithmmentioning

confidence: 99%

“…While such methods can be applied in a rather generic way to ordinary or partial differential equation, an efficient algorithm is only obtained if a suitable decomposition of variables is chosen that allows us to run the simulation with a small to moderate rank. For kinetic problems primarily a decomposition between spatial and velocity variables has been performed (see [16,22,23,49,48,12,8,38]; some work on tensor decomposition also exists [36,19,1,28]). It turns out that for kinetic problems this has a number of advantages.…”

Section: Introductionmentioning

confidence: 99%

A robust and conservative dynamical low-rank algorithm

Einkemmer

Ostermann

Scalone

2023

Journal of Computational Physics

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“…To leverage the opportunity presented by the inherent structure of the equation in the diffusive regime and address the challenge especially of high dimensionality, projection based ROMs and tensor decomposition based low rank algorithms have been designed for the stationary and time-dependent RTE. Along the line of low rank algorithms based on tensor decomposition, dynamical low rank algorithm (DLRA) [35,14,34]and the proper generalized decomposition (PGD) [1,37,13] have been designed. Projection based ROMs have also been actively developed in the recent few years, for example the proper orthogonal decomposition (POD) and its variations [5,11,12,40,3,10,19], the dynamical mode decomposition (DMD) [26,27].…”

Section: Introductionmentioning

confidence: 99%

A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation

Peng¹,

Chen²,

Cheng³

et al. 2022

Preprint

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Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there is a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work towards that direction.Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [30] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed reduced basis method (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro-and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of: reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: 1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and 2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux.

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A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Cited by 25 publications

References 65 publications

A sweep-based low-rank method for the discrete ordinate transport equation

A sweep-based low-rank method for the discrete ordinate transport equation

A robust and conservative dynamical low-rank algorithm

A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation

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