A MIXED VARIATIONAL FRAMEWORK FOR THE RADIATIVE TRANSFER EQUATION

@article{Egger2012AMV,
  title={A MIXED VARIATIONAL FRAMEWORK FOR THE RADIATIVE TRANSFER EQUATION},
  author={H. Egger and Matthias Schlottbom},
  journal={Mathematical Models and Methods in Applied Sciences},
  year={2012},
  volume={22},
  pages={1150014}
}
We present a rigorous variational framework for the analysis and discretization of the radiative transfer equation. Existence and uniqueness of weak solutions are established under rather general assumptions on the coefficients. Moreover, weak solutions are shown to be regular and hence also strong solutions of the radiative transfer equation. The relation of the proposed variational method to other approaches, including least-squares and even-parity formulations, is discussed. Moreover, the… Expand

Figures from this paper

Stationary radiative transfer with vanishing absorption
We investigate the unique solvability of radiative transfer problems without strictly positive lower bounds on the absorption and scattering parameters. The analysis is based on a reformulation ofExpand
Energy dependent radiative transfer equation and energy discretization
TLDR
This first paper of the series focuses on the well-posedness analysis and energy discretization of the energy dependent RTE and uses a mixed formulation so that the analysis covers both cases of non-vanishing absorption and vanishing absorption. Expand
Numerical methods for parameter identification in stationary radiative transfer
TLDR
By establishing the weak-continuity of the parameter-to-solution map, this work is able to ensure the existence of minimizers and thus the well-posedness of the regularization method. Expand
An Lp theory for stationary radiative transfer
Abstract We present a self-contained analysis of the stationary radiative transfer equation in weighted spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for underExpand
On robustly convergent and efficient iterative methods for anisotropic radiative transfer
TLDR
This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere and develops preconditioned Richardson iterations in Hilbert spaces. Expand
A class of Galerkin Schemes for Time-Dependent Radiative Transfer
TLDR
The starting point for the considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure to first order hyperbolic systems in acoustics or electrodynamics. Expand
On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer
TLDR
It is shown numerically that the preconditioned iteration is in practice robust in the diffusion limit, and computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. Expand
A numerical method for generalized Fokker-Planck equations
Generalized Fokker-Planck (GFP) equations have been employed to approximate the radiative transfer equation in applications of highly forward peaked biological media. In this paper, we discuss aExpand
Time reversal for radiative transport with applications to inverse and control problems
In this paper we develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, andExpand
A perfectly matched layer approach for radiative transfer in highly scattering regimes
We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by anExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 32 REFERENCES
Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes
A systematic solution approach for the neutron transport equation, based on a least-squares finite-element discretization, is presented. This approach includes the theory for the existence andExpand
Existence and Uniqueness Theorems for the Neutron Transport Equation
In an attempt to understand the conditions under which the neutron transport equation has solutions, and the properties of those solutions, a number of existence and uniqueness theorems are proved.Expand
Solution of Radiative Transfer Problems with Finite Elements
Mathematical modeling for monochromous radiative transfer problems is reviewed. Suitable boundary conditions for well-posed problems are introduced. Finite element discretizations for the integralExpand
A Boundary Functional for the Least-Squares Finite- Element Solution of Neutron Transport Problems
TLDR
One of the key features of the least-squares approach is that it produces a posteriori error bounds that is demonstrated in numerical examples for a spatial discretization using trilinear finite elements on a uniform tessellation into cubes. Expand
CONVERGENCE OF A FULLY DISCRETE SCHEME FOR TWO-DIMENSIONAL NEUTRON TRANSPORT*
We prove an error estimate for a fully discrete method for the numerical solution of a two-dimensional model problem in neutron transport theory based on using the discrete ordinates method for theExpand
Sparse adaptive finite elements for radiative transfer
TLDR
An a priori error analysis shows that the sparse tensor product method is clearly superior to a discrete ordinates method, as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in R^n. Expand
Analysis of a fully discrete scheme for neutron transport in two-dimensional geometry
We derive error estimates for a fully discrete scheme for the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry obtained by using a special quadrature ruleExpand
Reconstruction in optical tomography using the P-N approximations
In this paper, we consider the inverse problem of reconstructing the absorption and scattering coefficients of the radiative transfer equation (RTE) from measurements of photon current transmittedExpand
Fast iterative methods for discrete-ordinates particle transport calculations
TLDR
This Review discusses the theoretical foundations of the development of acceleration methods for iterative convergence of discrete-ordinates simulations, the important results that have been accomplished, and remaining open questions. Expand
A finite element-spherical harmonics radiation transport model for photon migration in turbid media
Abstract In this paper, we solve the steady-state form of the Boltzmann transport equation in homogeneous and heterogeneous tissue-like media with a finite element-spherical harmonics (FE-PN)Expand
...
1
2
3
4
...