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hp-DGFEM for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes
(2011)SAM Research ReportWe analyze the discretization of non-local degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes, in particular in option pricing problems when dealing with Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes. The discontinuous Galerkin (DG) Finite Element Method is analyzed. In the DG-FEM, ...Report -
Adaptive Petrov-Galerkin methods for first order transport equations
(2011)SAM Research ReportWe propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport \cs{and evolution} equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations ...Report -
High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data
(2011)SAM Research ReportWe analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the ...Report -
Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients
(2011)SAM Research ReportIn this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably innite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear ...Report -
The multi-level Monte Carlo Finite Element Method for a stochastic Brinkman problem
(2011)SAM Research ReportWe present the formulation and the numerical analysis of the Brinkman problem derived rigorously in [2, 3] with a random permeability tensor. The random permeability tensor is assumed to be a lognormal random field taking values in the symmetric matrices of size d×d , where d denotes the spatial dimension of the physical domain D . We prove that the solutions admit bounded moments of any finite order with respect to the random input's ...Report -
Sparse deterministic approximation of Bayesian inverse problems
(2011)SAM Research ReportWe present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a ...Report -
N-term Wiener Chaos Approximation Rates for elliptic PDEs with lognormal Gaussian random inputs
(2011)SAM Research ReportWe consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner-Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on $\mathbb{R}^\mathbb{N}$. It is shown that the weak solution can ...Report -
Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs
(2011)SAM Research ReportThis paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the number of parameters may be countable infinite, i.e., $\sigma_j$ with $j\in N$, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in ...Report -
Sparse Twisted Tensor Frame Discretization of Parametric Transport Operators
(2011)SAM Research ReportWe propose a novel family of frame discretizations for linear, high-dimensional parametric transport operators. Our approach is based on a least squares formulation in the phase space associated with the transport equation and by subsequent Galerkin discretization with a novel, sparse tensor product frame construction in the possibly high-dimensional phase space. The proposed twisted tensor product frame construction exploits invariance ...Report -
First order k-th moment finite element analysis of nonlinear operator equations with stochastic data
(2011)SAM Research ReportWe develop and analyze a class of efficient algorithms for uncertainty quantification of nonlinear operator equations. The algorithm are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, for a class of abstract nonlinear, parametric operator equations J(α,u) = 0 for random parameters α ith realizations in a neighborhood of a nominal parameter α0. Under some structural assumptions ...Report