Fast, low-memory numerical methods for radiative transfer via hp-adaptive mesh refinement

Du, Shukai; Stechmann, Samuel N.

doi:10.1016/j.jcp.2023.112021

Cited by 4 publications

(6 citation statements)

References 92 publications

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“…Proof. We simply choose v * ∈ V 0 h in (15) and then take the difference between ( 14) and ( 15). This completes the proof.…”

Section: The Error Estimatorsmentioning

confidence: 99%

“…, where the Galerkin orthogonality identity ( 16) is used for the second equal sign. Then, by the definition of the bilinear form (8a) and its consistency property (15), we can proceed as follows:…”

Section: The Error Estimatorsmentioning

confidence: 99%

“…Specifically, the (smoothed) point measurement is put at (L x , 15 21 L y ). In the first test, we use the error estimator η K defined in (18) to guide mesh refinement.…”

Section: Goal-oriented Amr For the Forward Problemmentioning

confidence: 99%

“…This suggests the great potential of using hp-AMR to reduce the DOFs for radiative transfer. Our recent work [15] demonstrates this potential for the forward problem, while here we consider solving the inverse radiative transfer problem.…”

Section: Introductionmentioning

confidence: 99%

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Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

Du,

Stechmann

2023

Inverse Problems

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The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp-adaptive mesh refinement (AMR) method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element (DGSE) methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp-adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method.

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“…Proof. We simply choose v * ∈ V 0 h in (15) and then take the difference between ( 14) and ( 15). This completes the proof.…”

Section: The Error Estimatorsmentioning

confidence: 99%

Section: The Error Estimatorsmentioning

confidence: 99%

“…Specifically, the (smoothed) point measurement is put at (L x , 15 21 L y ). In the first test, we use the error estimator η K defined in (18) to guide mesh refinement.…”

Section: Goal-oriented Amr For the Forward Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

Du,

Stechmann

2023

Inverse Problems

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“…(1989), W. C. Skamarock and Klemp (1993), Staniforth (1997), Jablonowski et al. (2006, 2009), Harris and Durran (2010), and Marsico and Stechmann (2020) for weather and climate prediction; Blayo and Debreu (1999) and Debreu and Blayo (2008) for ocean modeling; Evans (1998) and Du and Stechmann (2023a, 2023b, 2023c) for radiative transfer; Berger and Oliger (1984), Berger and Colella (1989), and Bell et al. (1994) for hyperbolic systems and shock waves; and Pelties et al.…”

Section: Introductionmentioning

confidence: 99%

A Universal Predictor‐Corrector Approach for Minimizing Artifacts Due To Mesh Refinement

Du,

Stechmann

2023

J Adv Model Earth Syst

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With nested grids or related approaches, it is known that numerical artifacts can be generated at the interface of mesh refinement. Most of the existing methods of minimizing these artifacts are either problem‐dependent or numerical methods‐dependent. In this paper, we propose a universal predictor‐corrector approach to minimize these artifacts. By its construction, the approach can be applied to a wide class of models and numerical methods without modifying the existing methods but instead incorporating an additional step. The idea is to use an additional grid setup with a refinement interface at a different location, and then to correct the predicted state near the refinement interface by using information from the other grid setup. We give some analysis for our method in the setting of a one‐dimensional advection equation, showing that the key to the success of the method depends on an optimized way of choosing the weight functions, which determine the strength of the corrector at a certain location. Furthermore, the method is also tested in more general settings by numerical experiments, including shallow water equations, multi‐dimensional problems, and a variety of underlying numerical methods including finite difference/finite volume and spectral element. Numerical tests suggest the effectiveness of the method on reducing numerical artifacts due to mesh refinement.

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A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

Shao,

Chen

2025

Applied Numerical Mathematics

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Fast, low-memory numerical methods for radiative transfer via hp-adaptive mesh refinement

Cited by 4 publications

References 92 publications

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

A Universal Predictor‐Corrector Approach for Minimizing Artifacts Due To Mesh Refinement

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

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