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@PHDTHESIS{Gruber:750850,
author = {Gruber, Felix Josef},
othercontributors = {Dahmen, Wolfgang and Torrilhon, Manuel},
title = {{A}daptive source term iteration : a stable formulation for
radiative transfer},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
reportid = {RWTH-2018-230893},
pages = {1 Online-Ressource (107 Seiten) : Illustrationen},
year = {2018},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University 2019; Dissertation, RWTH Aachen University, 2018},
abstract = {The radiative transfer problem is a model used to describe
particles moving in a medium with which the particles might
interact. It is used in a broad variety of fields including
nuclear physics, medical imaging and astrophysics. From a
numerical perspective, it is a challenging problem, due to
its transport character and relatively high dimensionality
with a 2d−1 dimensional solution (d spatial and d−1
directional dimensions). An integral operator over the
directional domain introduces a global coupling of all
directions that further complicates the high dimensionality.
Solving the radiative transfer problem is traditionally done
either using the non-deterministic Monte Carlo method or
with deterministic solvers like the method of moments and
the discrete ordinates method. Those deterministic methods
usually use rather strong assumptions to obtain a priori
estimates on the discretization error that might not hold in
realistic physical settings. In this thesis, we propose a
new deterministic method for solving the radiative transfer
problem that gives rigorous a posteriori error estimates on
the discrete solution. This method is based on an ideal
fixed-point iteration in an infinite-dimensional setting
that is solved approximately with dynamically updated
accuracy. Thus, we call this new method Adaptive Source Term
Iteration or ASTI for short. The use of a posteriori error
estimates allows us to solve problems with less regular
solutions and also reduces the computational costs by using
adaptively chosen grids. The main difference with regard to
existing Source Term Iteration methods, which iterate in
fixed discrete spaces, is that ASTI adapts the spaces, in
both the spatial and directional domain, during the
iteration. This way, we can control the error of our
iteration to guarantee convergence towards the exact
solution. For the transport solver, we use a Discontinuous
Petrov-Galerkin (DPG) method from Broersen, Dahmen and
Stevenson. It is well suited for the kind of linear
trans-port problems we obtain from the Adaptive Source Term
Iteration and gives reliable a posteriori error estimates.
This is based on Banach-Nečas-Babuška stability theory
which centers around the existence of inf-sup estimates. All
this adaptivity theory based upon a posteriori error
estimators is new in the context of radiative transfer
problems. As the analysis gets more involved, we also have
to solve new implementational challenges. This can
especially be seen in the grid management which involves
combining transport solutions living on different adaptively
refined grids. Our implementation of the Adaptive Source
Term Iteration is built upon the general-purpose Dune-DPG
library which was extended by code for an adaptive
scattering approximation and for combining solutions living
on differently adapted grids. Finally, we give two example
problems computed with our ASTI implementation which
illustrate how the adaptivity keeps the size of the
discretized formulation in a feasible range while
guaranteeing certified error bounds.},
cin = {111410 / 110000},
ddc = {510},
cid = {$I:(DE-82)111410_20170801$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2018-230893},
url = {https://publications.rwth-aachen.de/record/750850},
}
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