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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-(Multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two dimensional Navier-Stokes equations, with periodic bound- ary conditions and for arbitrarily high Reynolds number. We propose a reformulation of statistical solutions in the vorticity-stream function form. The vorticity-stream function for- mulation is discretized with a finite difference scheme. We obtain a convergence ...Report -
Multilevel higher order Quasi-Monte Carlo Bayesian Estimation
(2016)Research reports / Seminar for Applied MathematicsReport -
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Multilevel Monte-Carlo 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 Multi-Level ...Report -
Adaptive anisotropic Petrov-Galerkin 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 Petrov-Galerkin 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 ReportSpace-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space H are developed. The well-posedness of these equations in the Hilbert space $L^2 (H, μ)$ of functions on $H$ , which are square-integrable with respect to a Gaussian measure $μ$ on $H$ is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time 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 hp-FEM on anisotropic meshes
(1997)SAM Research ReportMixed hp-FEM for incompressible fluid flow on anisotropic meshes are analyzed. A discrete inf-sup condition is proved with a constant independent of the meshwidth and the aspect ratio. For each polynomial degree $k\geq 2$, velocity-pressure subspace pairs are presented which are stable on quadrilateral mesh-patches, independently of the element aspect ratio implying in particular divergence stability on the so-called Shishkin-meshes. ...Report -
Analytic regularity and best N-term 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