An angular multigrid method for computing mono-energetic particle beams in Flatland

Börgers, Christoph; MacLachlan, Scott

doi:10.1016/j.jcp.2009.12.023

Cited by 4 publications

(16 citation statements)

References 24 publications

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“…Since the transport equation (2.1a) is elliptic in the θ-direction but advective in the x-and y-directions, the angular multigrid method described in [3] combines relaxation sweeps ordered parallel to the (x, y) plane with coarse-grid correction over the θ-direction. The discretization of the transport equation (2.1a) that we wish to solve can briefly be represented as…”

Section: The Angular Multigrid Methodmentioning

confidence: 99%

“…Table 3 shows effective convergence factors for V(0, 1)-cycles applied to the transport equation where the Downloaded 12/08/14 to 130.88.90.140. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Table 3 Effective convergence factors, i. e., fourth roots of convergence factors, per iteration in first 24 V(0, 1)-cycles for the test problem, screened Rutherford scattering, discretized using second-order upstream differencing, the three-point discretization of the Fokker-Planck operator, and effective convergence factors per iteration in first 12 V(0, 1)-cycles with Henyey-Greenstein scattering, taken from [3,Tables 5,11,12]. Table 3, i. e., effective convergence factors that do not degrade as n s and n θ increase.…”

Section: The Angular Multigrid Methodmentioning

confidence: 99%

“…The relaxation scheme corresponds to the waveform relaxation discussed by Vandewalle and Horton in [24], and it is analogous to the relaxation used in the angular multigrid method of [3].…”

Section: The Two-grid Methodmentioning

confidence: 99%

“…In the following, we will review the model problem of [4] and the angular multigrid method of [3] for this model problem.…”

Section: Transport In Flatlandmentioning

confidence: 99%

“…Applied to electron-beam cancer therapy planning, the equation is assumed to be linear, 1 c ∂f ∂t (1.1c) dimensions and the angular multigrid method of [3] for this model problem, including experimentally measured convergence factors. In section 3, we introduce generalized diffusion problems as a simple subclass of the model problem and describe different space-time relaxation schemes and a two-grid method that corresponds to a two-grid version of the angular multigrid method considered in section 2.…”

Section: Introductionmentioning

confidence: 99%

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Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes

Friedhoff¹,

MacLachlan²,

Börgers³

2013

SIAM J. Sci. Comput.

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We consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning. While Monte Carlo methods are typically used for simulations of the forward-peaked scattering behavior of electron beams, rough calculations suggest that grid-based discretizations can provide more efficient simulations if the discretizations can be made sufficiently accurate, and optimal solvers can be found for the resulting linear systems. The multigrid method for model two-dimensional transport problems presented in [C. Börgers and S. MacLachlan, J. Comput. Phys., 229 (2010), pp. 2914-2931 shows the necessary optimal scaling with some dependence on the choice of scattering kernel. In order to understand this behavior, local Fourier analysis can be applied to the two-grid cycle. Using this approach, expressions for the error-propagation operators of the coarse-grid correction and relaxation steps, projected onto the fine-grid harmonic spaces, can be found. In this paper, we consider easier problems of the form of generalized diffusion problems in space-time that are analogous to model two-dimensional transport problems. We present local Fourier analysis results for these space-time model problems and compare with convergence factors of Börgers and MacLachlan. Since one of our model problems is the diffusion equation itself, we also compare to convergence factors for the diffusion equation of [S. Vandewalle and G. Horton, Computing, 54 (1995), pp. 317-330]. The results presented here show that local Fourier analysis does not offer its usual predictivity of the convergence behavior of the diffusion equation and the generalized diffusion equations until we consider unrealistically long time intervals.

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Section: The Angular Multigrid Methodmentioning

confidence: 99%

Section: The Angular Multigrid Methodmentioning

confidence: 99%

“…The relaxation scheme corresponds to the waveform relaxation discussed by Vandewalle and Horton in [24], and it is analogous to the relaxation used in the angular multigrid method of [3].…”

Section: The Two-grid Methodmentioning

confidence: 99%

“…In the following, we will review the model problem of [4] and the angular multigrid method of [3] for this model problem.…”

Section: Transport In Flatlandmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes

Friedhoff¹,

MacLachlan²,

Börgers³

2013

SIAM J. Sci. Comput.

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Composite‐grid multigrid for diffusion on the sphere

Adler

Lashuk

MacLachlan

2017

Numerical Linear Algebra App

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SummaryRecently, there has been much interest in the solution of differential equations on surfaces and manifolds, driven by many applications whose dynamics take place on such domains. Although increasingly powerful algorithms have been developed in this field, many straightforward questions remain, particularly in the area of coupling advanced discretizations with efficient linear solvers. In this paper, we develop a structured refinement algorithm for octahedral triangulations of the surface of the sphere. We explain the composite-grid finite-element discretization of the Laplace-Beltrami operator on such triangulations and extend the fast adaptive composite-grid scheme to provide an efficient solution of the resulting linear system. Supporting numerical examples are presented, including the recovery of second-order accuracy in the case of a nonsmooth solution. KEYWORDSfast adaptive composite-grid (FAC) algorithm, finite-element discretizations on surfaces, Laplace-Beltrami operator, multigrid

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Robust and adaptive multigrid methods: comparing structured and algebraic approaches

MacLachlan

Moulton

Chartier

2012

Numerical Linear Algebra App

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SUMMARY Although there have been significant advances in robust algebraic multigrid (AMG) methods in recent years, numerical studies and emerging hardware architectures continue to favor structured‐grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the black box multigrid (BoxMG) solver, have been shown to be 10 times faster than AMG for three‐dimensional heterogeneous diffusion problems on structured grids. BoxMG offers important features such as operator‐induced interpolation for robustness, while taking advantage of direct data access and bounded complexity in the Galerkin coarse‐grid operator. Moreover, because BoxMG uses a variational framework, it can be used to explore advances of modern adaptive AMG approaches in a structured setting. In this paper, we show how to extend the adaptive multigrid methodology to the BoxMG setting. This extension not only retains the favorable properties of the adaptive framework but also sheds light on the relationship between BoxMG and AMG. In particular, we show how classical BoxMG can be viewed as a special case of classical AMG and how this viewpoint leads to a richer family of adaptive BoxMG approaches. We present numerical results that explore this family of adaptive methods and compare its robustness and efficiency to the classical BoxMG solver.Copyright © 2012 John Wiley & Sons, Ltd.

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An angular multigrid method for computing mono-energetic particle beams in Flatland

Cited by 4 publications

References 24 publications

Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes

Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes

Composite‐grid multigrid for diffusion on the sphere

Robust and adaptive multigrid methods: comparing structured and algebraic approaches

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