Spectral Methods for Uncertainty Quantification | Christian Brams
| | Abstract | The goal of the thesis is to apply uncertainty quantification by generalized polynomial chaos with spectral methods on systems of partial differential equations, and implement the methods in the Python programming language.
We start off by introducing the mathematical basis for spectral methods for numerical computations. Deriving standard forms of differential operators enable us to implement the spectral collocation method on most partial differential equations. We implement the spectral methods in Python, creating a standardized method for solving a numerical problem spectrally using the spectral collocation method.
Afterwards we derive the stochastic collocation and Galerkin methods, allowing us to combine the spectral methods with the generalized polynomial chaos methods in order to achieve exponential convergence in both the numerical solution as well as the quantification of uncertainty.
Using Python as the medium, we implement these combined methods on a lid driven cavity problem as well as a two dimensional tank with nonlinear free surface movement, in order to examine the impact uncertainty on the input can have on a system of partial differential equations, and how to efficiently quantify the impact. It is discovered that using uncertainty quantification, we can describe the actual effect of the variables, by looking at what changes when the variable is subject to uncertainty, even for nonlinear systems.
The methods derived in this thesis combine excellently, and are easy to implement on most partial differential equations, allowing great versatility in implementing these methods of uncertainty quantification on different differential systems. | | Type | Bachelor thesis [Academic thesis] | | Year | 2013 | | Publisher | Technical University of Denmark, DTU Compute, E-mail: compute@compute.dtu.dk | | Address | Matematiktorvet, Building 303-B, DK-2800 Kgs. Lyngby, Denmark | | Series | IMM-B.Sc.-2013-03 | | Note | DTU supervisor: Allan P. Engsig Karup, apek@imm.dtu.dk, DTU Compute | | Electronic version(s) | [pdf] | | Publication link | http://www.compute.dtu.dk/English.aspx | | BibTeX data | [bibtex] | | IMM Group(s) | Scientific Computing |
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