Numerical radiative transfer modelled by a Markov process

A model based on random processes for radiative transfer is introduced and investigated. Based on the ideas of SimpleX, for a given point distribution of an astronomical object, the points are being connected with a number of nearest neighbours. On the edges of the resulting graph, probabilities are assigned that reflect the way how radiation is being transferred throughout the object. Here only diffuse transfer is considered, i.e. all radiation is divided equally among the neighbours. The idea of adding sources and sinks is introduced and used. All of this eventually results in a large, sparse matrix, and replaces the question of solving the radiation equation to finding the stationary distribution vector, the eigenvector of the matrix corresponding to eigenvalue one. Various results from Markov theory, the Perron Frobenius theorem and numerical linear algebra are used to find this solution. As it turns out, our matrix is irreducible and aperiodic, so that the dominant eigenvalue...Show moreA model based on random processes for radiative transfer is introduced and investigated. Based on the ideas of SimpleX, for a given point distribution of an astronomical object, the points are being connected with a number of nearest neighbours. On the edges of the resulting graph, probabilities are assigned that reflect the way how radiation is being transferred throughout the object. Here only diffuse transfer is considered, i.e. all radiation is divided equally among the neighbours. The idea of adding sources and sinks is introduced and used. All of this eventually results in a large, sparse matrix, and replaces the question of solving the radiation equation to finding the stationary distribution vector, the eigenvector of the matrix corresponding to eigenvalue one. Various results from Markov theory, the Perron Frobenius theorem and numerical linear algebra are used to find this solution. As it turns out, our matrix is irreducible and aperiodic, so that the dominant eigenvalue equals one and the corresponding eigenvector is the only eigenvector that has positive entries. Hence we know that a stationary distribution vector exists and is unique. By means of the Arnoldi algorithm one can compute this eigenvector, along with other eigenvectors and eigenvalues. As an example, a given point distribution describing the collision of two small galaxies is taken. The model is investigated by experimenting with the parameters of the model, i.e., including the number of nearest neighbours, the luminosity of the sources and the location of the sinks, to see if we get any physically true results. As it turns out, taking 20 nearest neighbours and a certain allocation of the sinks gives a physcially satisfying result. Finally, one scheme for absorption by the points in the distribution is considered. The conclusion of this thesis is that this model can be used to model radiative transfer, although one needs to work more on the absorption aspect, and there is still work to be done on various other types of radiative transfer.Show less

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Numerical radiative transfer modelled by a Markov process