We study how uncertainty in the input data of the Radiative Transport equation (RTE), aﬀects the distribution of (functionals of) its solution (the output data). The RTE is an integro-diﬀerential equation, in up to seven independent variables, that models the behaviour of rareﬁed particles (such as photons and neutrons) in a domain. Its applications include nuclear reactor design, radiation shielding, medical imaging, optical tomography and astrophysics. We focus on the RTE in the context of nuclear reactor physics where, to design and maintain safe reactors, understanding the eﬀects of uncertainty is of great importance. There are many potential sources of uncertainty within a nuclear reactor. These include the geometry of the reactor, the material composition and reactor wear. Here we consider uncertainty in the macroscopic cross-sections (‘the coeﬃcients’), representing them as correlated spatial random ﬁelds. We wish to estimate the statistics of a problem-speciﬁc quantity of interest (under the inﬂuence of the given uncertainty in the cross-sections), which is deﬁned as a functional of the scalar ﬂux. This is the forward problem of Uncertainty Quantiﬁcation. We seek accurate and eﬃcient methods for estimating these statistics. Thus far, the research community studying Uncertainty Quantiﬁcation in radiative transport has focused on the Polynomial Chaos expansion. However, it is known that the number of terms in the expansion grows exponentially with respect to the number of stochastic dimensions and the order of the expansion, i.e. polynomial chaos suﬀers from the curse of dimensionality. Instead, we focus our attention on variants of Monte Carlo sampling - studying standard and quasi-Monte Carlo methods, and their multilevel and multi-index variants. We show numerically that the quasiMonte Carlo rules, and the multilevel variance reduction techniques, give substantial gains over the standard Monte Carlo method for a variety of radiative transport problems. Moreover, we report problems in up to 3600 stochastic dimensions, far beyond the capability of polynomial chaos. A large part of this thesis is focused towards a rigorous proof that the multilevel Monte Carlo method is superior to the standard Monte Carlo method, for the RTE in one spatial and one angular dimension with random cross-sections. This is the ﬁrst rigorous theory of Uncertainty Quantiﬁcation for transport problems and the ﬁrst rigorous theory for Uncertainty Quantiﬁcation for any PDE problem which accounts for a path-dependent stability condition. To achieve this result, we ﬁrst present an error analysis (including a stability bound on the discretisation parameters) for the combined spatial and angular discretisation of the spatially heterogeneous RTE, which is explicit in the heterogeneous coeﬃcients. We can then extend this result to prove probabilistic bounds on the error, under assumptions on the statistics of the cross-sections and provided the discretisation satisﬁes the stability condition pathwise. The multilevel Monte Carlo complexity result follows. Amongst other novel contributions, we: introduce a method which combines a direct and iterative solver to accelerate the computation of the scalar ﬂux, by adaptively choosing the fastest solver based on the given coeﬃcients; numerically test an iterative eigensolver, which uses a single source iteration within each loop of a shifted inverse power iteration; and propose a novel model for (random) heterogeneity in concrete which generates (piecewise) discontinuous coeﬃcients according to the material type, but where the composition of materials are spatially correlated.
|Date of Award||13 Feb 2019|
|Supervisor||Ivan Graham (Supervisor) & Rob Scheichl (Supervisor)|