A conjugate gradient method for solving the non-LTE line
 radiation transfer problem

Paletou, F.; Anterrieu, Éric

doi:10.1051/0004-6361/200912491

Cited by 12 publications

(6 citation statements)

References 13 publications

(13 reference statements)

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“…More recently, iterative approaches based on Krylov methods, which could yield better performances than the aforementioned operator-splitting methods, have been proposed (e.g., Paletou & Anterrieu 2009;Anusha et al 2009;Benedusi et al 2021).…”

Section: Continuum Contribution To Radiative Transfer Coefficientsmentioning

confidence: 99%

The transfer of polarized radiation in resonance lines with partial frequency redistribution, J-state interference, and arbitrary magnetic fields

Ballester

Belluzzi

Bueno

2022

A&A

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Aims. We present the theoretical framework and numerical methods we have implemented to solve the problem of the generation and transfer of polarized radiation in spectral lines without assuming local thermodynamical equilibrium, while accounting for scattering polarization, partial frequency redistribution (due to both the Doppler effect and elastic collisions), J-state interference, and hyperfine structure. The resulting radiative transfer code allows one to model the impact of magnetic fields of an arbitrary strength and orientation through the Hanle, incomplete Paschen-Back, and magneto-optical effects. We also evaluate the suitability of a series of approximations for modeling the scattering polarization in the wings of strong resonance lines at a much lower computational cost, which is particularly valuable for the numerically intensive case of three-dimensional radiative transfer. Methods. We examine the suitability of the considered approximations by using our radiative transfer code to model the Stokes profiles of the Mg ii h & k lines and of the H i Lyman-α line in magnetized one-dimensional models of the solar atmosphere. Results. Neglecting Doppler redistribution in the scattering processes that are unperturbed by elastic collisions (i.e., treating them as coherent in the observer's frame) produces a negligible error in the scattering polarization wings of the Mg ii resonance lines and a minor one in the Lyman-α wings, although it is unsuitable to model the cores of these lines. For both lines, the scattering processes that are perturbed by elastic collisions only give a significant contribution to the intensity component of the emissivity. Neglecting collisional as well as Doppler redistribution (so that all scattering processes are coherent) represents a rough but suitable approximation for the wings of the Mg ii resonance lines, but a very poor one for the Lyman-α wings. The magnetic sensitivity in the scattering polarization wings of the considered lines can be modeled by accounting for the magnetic field in only the η I and ρ V coefficients of the Stokes-vector transfer equation (i.e., using the zero-field expression for the emissivity).

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Section: Continuum Contribution To Radiative Transfer Coefficientsmentioning

confidence: 99%

The transfer of polarized radiation in resonance lines with partial frequency redistribution, J-state interference, and arbitrary magnetic fields

Ballester

Belluzzi

Bueno

2022

A&A

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“…In particular, Krylov methods have already been employed to linear transfer problems of unpolarized radiation, showing promising results. We would like to mention the first convergence studies on the BICGSTAB method by Paletou & Anterrieu (2009) and Anusha et al (2009). Nagendra et al (2009) first applied BICGSTAB to the transfer of polarized radiation in 1D atmospheric models.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions to linear transfer problems of polarized radiation II. Krylov methods and matrix-free implementation

Benedusi,

Janett,

Belluzzi

et al. 2021

Preprint

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Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods and direct solution strategies. Methods. After a brief introduction to the concept of Krylov methods, we report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, (S)SOR, or an incomplete LU factorization. Furthermore, specific numerical tests were performed to study the robustness of the various methods as the problem size grew. Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness (with respect to the problem size) of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use. Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods.

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“…Other methods exists: Gauss-Seidel (GS) iteration (Trujillo Bueno & Fabiani Bendicho 1995;Trujillo Bueno & Manso Sainz 1999) converges four times as fast as the diagonal operator, but cannot be parallelized as efficiently as Jacobi iteration (Tsitsiklis et al 1988), a serious disadvantage in parallel solvers that are used for 3D applications. Conjugate gradient methods are under development, but have not yet been implemented in a general 3D non-LTE RT code (Paletou & Anterrieu 2009).…”

Section: Introductionmentioning

confidence: 99%

Numerical non-LTE 3D radiative transfer using a multigrid method

Bjørgen

Leenaarts

2017

A&A

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Context. 3D non-LTE radiative transfer problems are computationally demanding, and this sets limits on the size of the problems that can be solved. So far, multilevel accelerated lambda iteration (MALI) has been the method of choice to perform high-resolution computations in multidimensional problems. The disadvantage of MALI is that its computing time scales as O(n 2 ), with n the number of grid points. When the grid becomes finer, the computational cost increases quadratically. Aims. We aim to develop a 3D non-LTE radiative transfer code that is more efficient than MALI. Methods. We implement a non-linear multigrid, fast approximation storage scheme, into the existing Multi3D radiative transfer code. We verify our multigrid implementation by comparing with MALI computations. We show that multigrid can be employed in realistic problems with snapshots from 3D radiative magnetohydrodynamics (MHD) simulations as input atmospheres.Results. With multigrid, we obtain a factor 3.3-4.5 speed-up compared to MALI. With full-multigrid, the speed-up increases to a factor 6. The speed-up is expected to increase for input atmospheres with more grid points and finer grid spacing. Conclusions. Solving 3D non-LTE radiative transfer problems using non-linear multigrid methods can be applied to realistic atmospheres with a substantial increase in speed.

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A conjugate gradient method for solving the non-LTE line radiation transfer problem

Cited by 12 publications

References 13 publications

The transfer of polarized radiation in resonance lines with partial frequency redistribution, J-state interference, and arbitrary magnetic fields

The transfer of polarized radiation in resonance lines with partial frequency redistribution, J-state interference, and arbitrary magnetic fields

Numerical solutions to linear transfer problems of polarized radiation II. Krylov methods and matrix-free implementation

Numerical non-LTE 3D radiative transfer using a multigrid method

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