@MASTERSTHESIS\{IMM2013-06659,
    author       = "E. B. K{\ae}rgaard",
    title        = "Spectral Methods for Uncertainty Quantification",
    year         = "2013",
    school       = "Technical University of Denmark, Department of Applied Mathematics and Computer Science / {DTU} Co",
    address      = "Matematiktorvet, Building 303B, {DK-}2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk",
    type         = "",
    note         = "{DTU} supervisor: Allan P. Engsig-Karup, apek@dtu.dk, {DTU} Compute",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "This thesis has investigated the field of Uncertainty Quantification with regard to differential equations. The focus is on spectral methods and especially the stochastic Collocation method. Numerical tests are conducted and the stochastic Collocation method for multivariate problems is investigated. The Smolyak sparse grid method is applied in combination with the stochastic Collocation method on two multivariate stochastic differential equations. The numerical tests showed that the sparse grids can reduce the computational
effort and at the same time produce very accurate results.
The first part of the thesis introduces the mathematical background for working with Uncertainty Quantification, the theoretical background for generalized Polynomial Chaos and the three methods for the Uncertainty Quantification. These methods are the Monte Carlo sampling, the stochastic Galerkin method and the stochastic Collocation method. The three methods have been tested numerically on the univariate stochastic Test equation to test accuracy and efficiency and further numerical tests of the Galerkin method and the Collocation method have been conducted with regard to the univariate stochastic Burgers' equation. The numerical tests have been carried out in Matlab.
The last part of the thesis involves an introduction of the multivariate stochastic Collocation method and numerical tests. Furthermore a few methods for reducing the computational effort in high dimensions is introduced. One of these methods - the Smolyak sparse grids - is applied together with the stochastic Collocation method on the multivariate Test equation and the multivariate Burgers' equation. This resulted in accurate and efficient estimates of the statistics"
}