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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 -
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 -
Higher order QMC Galerkin discretization for parametric operator equations
(2013)SAM Research ReportWe construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of ...Report