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Sparse wavelet methods for option pricing under stochastic volatility
(2004)Research ReportReport 
Numerical approximation of statistical solutions of incompressible flow
(2015)Research ReportWe present a finite difference(Multilevel) Monte Carlo algorithm to efficiently compute statistical solutions of the two dimensional NavierStokes equations, with periodic bound ary conditions and for arbitrarily high Reynolds number. We propose a reformulation of statistical solutions in the vorticitystream function form. The vorticitystream function for mulation is discretized with a finite difference scheme. We obtain a convergence ...Report 
Multilevel higher order QuasiMonte Carlo Bayesian Estimation
(2016)Research reports / Seminar for Applied MathematicsReport 

Multilevel MonteCarlo front tracking for random scalar conservation laws
(2012)SAM Research ReportWe consider random scalar hyperbolic conservation laws (RSCLs) in spatial dimension $d\ge 1$ with bounded random flux functions which are $\mathbb{P}$a.s. Lipschitz continuous with respect to the state variable, for which there exists a unique random entropy solution (i.e., a measurable mapping from the probability space into $C(0,T;L^1(\mathbb{R}^d))$ with finite second moments). We present a convergence analysis of a MultiLevel ...Report 
Adaptive anisotropic PetrovGalerkin methods for first order transport equations
(2016)SAM Research ReportThis paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of PetrovGalerkin type. The key issues can be summarized as follows. The variational formulations allow us to employ meshes with cells of arbitrary aspect ratios. We develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems. We establish approximation ...Report 
Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions
(2011)SAM Research ReportSpacetime variational formulations of infinitedimensional FokkerPlanck (FP) and OrnsteinUhlenbeck (OU) equations for functions on a separable Hilbert space H are developed. The wellposedness of these equations in the Hilbert space $L^2 (H, μ)$ of functions on $H$ , which are squareintegrable with respect to a Gaussian measure $μ$ on $H$ is proved. Specifically, for the infinitedimensional FP equation, adaptive spacetime Galerkin ...Report 
Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs
(2011)SAM Research ReportA class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and $n$ known, separated microscopic length scales $\epsilon_i$, $i=1,...,n$ in a bounded domain $D\subset R^d$ is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge $P$a.s, as $\epsilon_i\rightarrow 0$, to a stochastic, ...Report 
Mixed hpFEM on anisotropic meshes
(1997)SAM Research ReportMixed hpFEM for incompressible fluid flow on anisotropic meshes are analyzed. A discrete infsup condition is proved with a constant independent of the meshwidth and the aspect ratio. For each polynomial degree $k\geq 2$, velocitypressure subspace pairs are presented which are stable on quadrilateral meshpatches, independently of the element aspect ratio implying in particular divergence stability on the socalled Shishkinmeshes. ...Report 
Analytic regularity and best Nterm approximation of high dimensional parametric initial value problems
(2011)SAM Research ReportWe consider nonlinear systems of ordinary differential equations (ODEs) on a Banach state space S over R or C, where the right hand side depends affinely linear on a parameter vector y = (yj)j_1, normalized such that yj  _ 1. Under suitable analyticity assumptions on the ODEs, we prove that the solution {X(t; y) : 0 _ t _ T} of the corresponding IVP depends holomorphically on the parameter vector y, as a mapping from the infinite ...Report