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Exponential convergence of simplicial hp-FEM for H^1-functions with isotropic singularities
(2015)Lecture Notes in Computational Science and Engineering ~ Spectral and High Order Methods for Partial Differential Equations (ICOSAHOM 2014)For functions u ∈ H 1(Ω) in an open, bounded polyhedron Ω⊂Rd of dimension d = 1, 2, 3, which are analytic in Ω¯¯¯¯∖S with point singularities concentrated at the set S⊂Ω¯¯¯¯ consisting of a finite number of points in Ω¯¯¯¯, the exponential rate exp(−bN−−√d+1) of convergence of h p-version continuous Galerkin finite element methods on families of regular, simplicial meshes in Ω can be achieved. The simplicial meshes are assumed to be ...Report -
Scaling Limits in Computational Bayesian Inversion
(2014)Research ReportComputational Bayesian inversion of operator equations with distributed uncertain input parameters is based on an infinite-dimensional version of Bayes’ formula established in [31] and its numerical realization in [27, 28]. Based on the sparsity of the posterior density shown in [29], dimensionadaptive Smolyak quadratures afford higher convergence rates than MCMC in terms of the number M of solutions of the forward (parametric operator) ...Report -
Multi-Level Monte Carlo Finite Volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium
(2014)Research ReportWe consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated ...Report -
Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations
(2014)Research ReportWe analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space X. Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of X render the random inputs and the solutions of the forward problem countably parametric. ...Report -
Covariance regularity and H-matrix approximation for rough random fields
(2014)Research ReportReport -
Multi-level higher order QMC Galerkin discretization for affine parametric operator equations
(2014)SAM Research ReportWe develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-Level first order analysis in [F.Y. Kuo, Ch. Schwab, and I.H. Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential ...Report -
Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
(2014)Research ReportReport -
A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
(2014)Research ReportWe analyze a-posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countablyparametric, elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this ...Report -
Computational Higher Order Quasi-Monte Carlo Integration
(2014)Research ReportThe efficient construction of higher-order interlaced polynomial lattice rules introduced recently in [4] is considered. After briefly reviewing the principles of their construction by the “fast component-by-component” (CBC) algorithm due to [1, 10] as well as recent theoretical results on their convergence rates, we indicate algorithmic details of their construction. Instances of such rules are applied to highdimensional test integrands ...Report -
Multilevel Monte Carlo approximations of statistical solutions to the Navier-Stokes equations
(2013)SAM Research ReportReport