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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 inf-sup condition for its Ritz-Galerkin 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 hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value 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 Monte-Carlo (MC) and Markov-Chain Monte-Carlo 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 non-recoverable 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 hp-DGFEM for elliptic problems in polyhedral domains
(2012)SAM Research ReportWe review the recent results of [21, 22], and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value 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 -
Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond
(2012)SAM Research ReportThis paper is a contemporary review of QMC ("quasi-Monte Carlo") methods, i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube $[0; 1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original ...Report -
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 -
Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM
(2012)SAM Research ReportWe study the approximation of harmonic functions by means of harmonic polynomials in twodimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a $\delta$-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the ...Report