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Multilevel Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems
(2012)SAM Research ReportWe consider stochastic multidimensional linear hyperbolic systems of conservation laws. We prove existence and uniqueness of a random weak solution, provide estimates for the regularity of the solution in terms of regularities of input data, and show existence of statistical moments. Bounds for mean square error vs. expected work are proved for the MultiLevel Monte Carlo Finite Volume algorithm which is used to approximate the moments ...Report 
Multilevel Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws
(2012)Research ReportReport 
QMC Galerkin discretization of parametric operator equations
(2012)SAM Research ReportWe extend recent results of QMC quadrature and Finite Element discretization for parametric, scalar second order elliptic partial differential equations to general QMCGalerkin discretizations of parametric operator equations, which depend on possibly countably many parameters. Such problems typically arise in the numerical solution of differential and integral equations with random field inputs. The present setting covers general second ...Report 
Sparse tensor edge elements
(2012)SAM Research ReportWe consider the tensorized operator for the Maxwell cavity source problem in frequency domain. We establish a discrete infsup condition for its RitzGalerkin discretization on sparse tensor product edge element spaces built on nested sequences of meshes. Our main tool is a generalization of the edge element Fortin projector to a tensor product setting. The techniques extend to the surface boundary edge element discretization of tensorized ...Report 
Exponential convergence of the hp version of isogeometric analysis in 1D
(2012)Research ReportWe review the recent results of [21, 22], and establish the exponential convergence of hpversion discontinuous Galerkin finite element methods for the numerical approximation of linear secondorder elliptic boundaryvalue problems with homogeneous Dirichlet boundary conditions and constant coefficients in threedimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.Report 
Sparse, adaptive Smolyak algorithms for Bayesian inverse problems
(2012)SAM Research ReportBased on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite dimensional, separable Banach spaces proposed in [28], we develop a practical computational algorithm whose convergence rates are provably higher than those of MonteCarlo (MC) and MarkovChain MonteCarlo methods, in terms of the number of solutions of the forward problem. In the formulation of [28], the forward ...Report 
Intrinsic Fault Tolerance of Multi Level Monte Carlo Methods
(2012)SAM Research ReportMonte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to solvers for Partial Differential Equations with random input data are shown to exhibit intrinsic failure resilience. Sufficient conditions are provided for nonrecoverable loss of a random fraction of samples not to fatally damage the asymptotic accuracy vs. work of an MC simulation. Specifically, the convergence behavior of MLMC methods on massively parallel hardware ...Report 
Sparse MCMC gpc Finite Element Methods for Bayesian Inverse Problems
(2012)SAM Research ReportSeveral classes of MCMC methods for the numerical solution of Bayesian Inverse Problems for partial differential equations (PDEs) with unknown random field coefficients are considered. A general framework for their numerical analysis is presented. The complexity of MCMC sampling for the unknown fields from the posterior density, as well as the convergence of the discretization error of the PDE of interest in the forward response map, is ...Report 
Exponential convergence of hpDGFEM for elliptic problems in polyhedral domains
(2012)SAM Research ReportWe review the recent results of [21, 22], and establish the exponential convergence of hpversion discontinuous Galerkin finite element methods for the numerical approximation of linear secondorder elliptic boundaryvalue problems with homogeneous Dirichlet boundary conditions and constant coefficients in threedimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.Report 
High order Galerkin appoximations for parametric second order elliptic partial differential equations
(2012)SAM Research ReportLet $D \subset \mathbb{R}^d, d=2,3$, be a bounded domain with piecewise smooth boundary $\partial D$ and let $U$ be an open subset of Banach space $Y$. We consider a parametric family $P_y$ of uniformly strongly elliptic, parametric second order partial differential operators $P_y$ on $D$ in divergence form, where the parameter $y$ ranges in the parameter domain $U$ so that, for a given set of data $f_y$, the solution $u$ and the ...Report