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978-3-8439-0762-0, Reihe Mathematik
Unsteady Adaptive Stochastic Collocation Methods on Sparse Grids
153 Seiten, Dissertation Technische Universität Darmstadt (2012), Softcover, A5
This work incorporates uncertain quantities arising in nature or processes into numerical simulations. By doing so, computational results become more realistic and meaningful. Underlying mathematical models often consist of Partial Differential Equations (PDEs) with input data, that specify the describing system. If these input parameters are not explicitly known or subject to natural fluctuations, we arrive at PDEs with random parameters.
We focus on random parameters that can be described by correlated random fields. A parametrization into finitely many random variables yields problems with possibly high dimensional parameter space, that has to be discretized beside the deterministic dimensions. To this end, we use adaptive, anisotropic stochastic collocation on sparse grids. Similar to a Monte Carlo simulation, this approach decouples and hence parallelizes the stochastic problem into a set of deterministic problems. By means of a fluid flow example, we show impressively that the method is able to resolve a stochastic parameter space of 20 dimensions.
One aim of this work is to derive effective error estimates. Hierarchical error indicators usually provide acceptable solutions, but can also highly over- or underestimate the true approximation error. We extend the approach of adjoint error estimation to the stochastic framework. The difficulty consists in the accurate detection of both deterministic and stochastic parts of the overall error.
In order to control the arising extra costs, we suggest to make use of reduced models. Our numerical examples prove that this novel approach can indeed be applied successfully. A theoretical digression concerning analytic formulas of statistical moments for a class of parabolic problems completes this work. We show that certain problems and parameters allow for exact formulations of such moments without the need of stochastic discretization tools at all.
Our results gain theoretical and practical insight into the behaviour of PDEs with random parameters. Uncertainty quantification allows to better understand many technical or scientific problems and to represent them more accurately.