High Dimensional Uncertainty Quantification And A Perfectly Matched Layer For The Boltzmann Equation

In the first part of this thesis, we work on the high-dimensional problems related touncertainty quantification and the ANOVA expansion. First of all, we discuss adaptivesparse grid algorithms for stochastic differential equations with a particular focus on ap-plications to electromagnetic scattering by structures with holes of uncertain size, loca-tion, and quantity. Secondly, we discuss how the combination of ANOVA expansions,different sparse grid techniques, and the total sensitivity index (TSI) as a pre-selectivemechanism enables the modeling of problems with hundred of parameters. Thirdly, weemploy the relationship between sparse grid and Dirac ANOVA expansion to argue thatthe optimal choice of this anchor point is the center point of a sparse grid quadraturein the Dirac ANOVA expansion case. We demonstrate the accuracy and efficiency ofproposed methods on a number of standard benchmark problems and challenging testcases drawn from engineering and science in this part.In the second part, we develop a long time stable perfectly matched layer (PML)for the Boltzmann equation by properly choosing a set of auxiliary parameters. Theresults are compared with reference solutions computed from a larger domain in severalstandard test examples, confirming the accuracy and efficiency of the proposed PML.