Sparse Discrete Ordinates Method in Radiative Transfer

Grella, Konstantin; Schwab, Christoph

doi:10.2478/cmam-2011-0017

Cited by 16 publications

(22 citation statements)

References 19 publications

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“…In Section 6 we describe a numerical setup for treating parametric problems analogous to sparse tensorization of discrete ordinate methods as in [22,23,21], but with the directionally adaptive discretizations in physical space.…”

Section: Introductionmentioning

confidence: 99%

Adaptive anisotropic Petrov–Galerkin methods for first order transport equations

Dahmen

Kutyniok

Lim

et al. 2018

Journal of Computational and Applied Mathematics

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This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov-Galerkin type. The key issues can be summarized as follows. The variational formulations allow us to employ meshes with cells of arbitrary aspect ratios. We develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems. We establish approximation rates for N -term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions. In contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here. The approximation rate results serve then as a benchmark for subsequent applications to adaptive Galerkin solvers for transport equations. We outline a new class of directionally adaptive, Petrov-Galerkin discretizations for such equations. In numerical experiments, the new algorithms track C 2 -curved shear layers and discontinuities stably and accurately, and realize essentially optimal rates. Finally, we treat parameter dependent transport problems, which arise in kinetic models as well as in radiative transfer. In heterogeneous media these problems feature propagation of singularities along curved characteristics precluding, in particular, fast marching methods based on ray-tracing. Since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality. We show computationally, for a model parametric transport problem in heterogeneous media in 2 + 1 dimension, that sparse tensorization of the presently proposed spatial directionally adaptive scheme with hierarchic collocation in ordinate space based on a stable variational formulation high-dimensional phase space, the curse of dimensionality can be removed when approximating averaged bulk quantities.

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Section: Introductionmentioning

confidence: 99%

Adaptive anisotropic Petrov–Galerkin methods for first order transport equations

Dahmen

Kutyniok

Lim

et al. 2018

Journal of Computational and Applied Mathematics

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“…Specifically, taking η n from (3.9) we get, on account of (3.5), 16) and hence the same type of bound as in (3.10) for the transport dominated case. Clearly, according to (3.15), the work horse in this scheme is Algorithm 1 which acts as a preconditioner in each step of the outer iteration.…”

Section: Perturbed Iterationmentioning

confidence: 76%

“…Modern strategies to face the complexity issues posed by 1) are sparse tensor methods based on sparse grid or hyperbolic cross approximations. Answering 2) is then based on suitable a priori regularity assumptions such as the validity of a certain order of mixed smoothness, see e. g. [20,1,2,16].…”

Section: Problem Formulationmentioning

confidence: 99%

An adaptive nested source term iteration for radiative transfer equations

2020

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We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov-Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov-Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.

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“…As indicated in Subsection 1.5, this can be utilised for solving more involved transport equations (1.6), based on the fact that the ridgelet frame Φ covers all direction s simultaneously, while the multiscale structure makes it possible to alleviate the curse of dimensionality, for example by the "sparse discrete ordinates method" (see e.g. [GS11]).…”

Section: Resultsmentioning

confidence: 99%

“…There are several ways to utilise a solver for (1.2) to solve (1.6). If we neglect the scattering term for the moment, the ridgelet-based solver we develop could be used to solve (1.6) by either tensor product or collocation methods in s (similar to techniques used in [GS11], where one can additionally make use of the multiscale structure of Φ to alleviate the curse of dimensionality by balancing resolution in angle with resolution in space).…”

Section: Impactmentioning

confidence: 99%

On the approximation of functions with line singularities by ridgelets

Obermeier

2019

Journal of Approximation Theory

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In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal -by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate.Structurally, the proof resembles [Can01], where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes [CDD01] we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

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Sparse Discrete Ordinates Method in Radiative Transfer

Cited by 16 publications

References 19 publications

Adaptive anisotropic Petrov–Galerkin methods for first order transport equations

Adaptive anisotropic Petrov–Galerkin methods for first order transport equations

An adaptive nested source term iteration for radiative transfer equations

On the approximation of functions with line singularities by ridgelets

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