Nonlinear Radiative Transfer Equation Research Articles

Manifold Learning and Nonlinear Homogenization

We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on detailed analytical understanding of the multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.

An Asymptotic-Preserving IMEX Method for Nonlinear Radiative Transfer Equation

We present an asymptotic preserving method for the radiative transfer equations in the framework of \(P_N\) method. An implicit and explicit numerical scheme is proposed to solve the \(P_N\) system based on the order analysis of the expansion coefficients of the specific intensity, where the order of each expansion coefficient is derived by the Chapman-Enskog method. The coefficients at higher-order are treated explicitly while those at lower-order are treated implicitly in each equation of the \(P_N\) system. Energy inequality is proved for this numerical scheme. Several numerical examples validate the efficiency of this scheme in both optically thick and thin regions.

Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.

Reconstruction of the Emission Coefficient in the Nonlinear Radiative Transfer Equation

In this paper, we investigate an inverse problem for the radiative transfer equation that is coupled with a heat equation in a nonscattering medium in $\mathbb{R}^n, n\geq 2$. The two equations are...

Nonlinear radiative transfer in high temperature plasmas

An approximation to the nonlinear radiative transfer equation is considered in the context of magnetohydrodynamics. This approximation retains nonlinear terms which are responsible for the Compton plasma heating in addition to the radiation cooling. The effect is studied numerically. In particular, the Kelvin–Helmholtz instability and the Dai-Woodward case are modeled. It is shown that radiation-plasma coupling results in damping of small scale instabilities and alters the shock wave structure.

Qualitative analysis of the S N approximations of the transport equation and combined conduction-radiation heat transfer problem in a slab

A review is presented on the recent work carried out by our research group on mathematical aspects of the existence and uniqueness of solutions of the one-dimensional steady-state transport equation as well as an analytical study of the nonlinear radiative transfer equation in a slab. Dependence on control parameters about the solutions of the one-dimensional S N, approximations of the transport equation is also addressed.

A Nonlinear Physical Retrieval Algorithm—Its Application to theGOES-8/9Sounder

Abstract A nonlinear physical retrieval algorithm is developed and applied to the GOES-8/9 sounder radiance observations. The algorithm utilizes Newtonian iteration in which the maximum probability solution for temperature and water vapor profiles is achieved through the inverse solution of the nonlinear radiative transfer equation. The nonlinear physical retrieval algorithm has been tested for one year. It has also been implemented operationally by the National Oceanic and Atmospheric Administration National Environmental Satellite, Data and Information Service during February 1997. Results show that the GOES retrievals of temperature and moisture obtained with the nonlinear algorithm more closely agree with collocated radiosondes than the National Centers for Environmental Prediction (NCEP) forecast temperature and moisture profile used as the initial profile for the solution. The root-mean-square error of the total water vapor from the solution first guess, which is the NCEP 12-h forecast (referred to ...

A Linear-Discontinuous Spatial Differencing Scheme forSnRadiative Transfer Calculations

Various types of linear-discontinuous spatial differencing schemes have been developed for theSn(discrete-ordinates) equations approximating the linear Boltzmann transport equation. It has been shown through an asymptotic analysis that the 1D slab-geometry lumped linear-discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit, but it also yields the correct interior solution, even when boundary layers are left unresolved by the spatial mesh. In the present work we generalize this scheme to obtain a 1D slab-geometry lumped linear-discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation. We present a full nonlinear energy-dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium-diffusion limit. We find that this scheme goes over to a convergent and robust differencing of the equilibrium-diffusion equation on the interior of the mesh, but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh. Nevertheless, the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate. Computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique.

Iterative algorithms for nonlinear problems in remote sensing

First a brief review of the Backus and Gilbert method for the linear problems is given. Then a comprehensive presentation of iterative algorithms for constructing approximate solutions of nonlinear problems from erroneous inadequate data is given. Proofs of convergence and error estimates of these iterative algorithms are obtained. It is found that locally the limit of the iterates does not satisfy the original nonlinear operator equation, but a somewhat different nonlinear equation which depends on the initial iterate. A nonlinear radiative transfer equation in remote sensing of the atmospheric temperature profiles is used as an example to demonstrate the applicability of the iterative algorithms. Finally, a discussion of the present status of research in this domain and some personal views on what should be the useful lines for future research is given.

Numerical solutions of a nonlinear radiative transfer equation with inadequate data

Iterative algorithms for solving numerically a nonlinear radiative transfer equation with inadequate data are presented and tested by numerical simulations. It is found that these iterative algorithms do give excellent results.