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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 -
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 -
Multi-level Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems
(2012)SAM Research ReportWe consider stochastic multi-dimensional 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 Multi-Level Monte Carlo Finite Volume algorithm which is used to approximate the moments ...Report -
Isotropic Gaussian random fields on the sphere: regularity, fast simulation, and stochastic partial differential equations
(2013)SAM Research ReportIsotropic Gaussian random fields on the sphere are characterized by Karhunen-Loève expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Loève ...Report -
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 -
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 -
Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs. I: Analytic regularity and gpc-approximation
(2010)SAM Research ReportFor initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an apriori error analysis for sparse tensor, space-time discretizations. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameterspace by Galerkin projection onto finitely supported ...Report -
Regularity and generalized polynomial chaos approximation of parametric and random 2nd order hyperbolic partial differential equations
(2010)SAM Research ReportInitial boundary value problems of linear second order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform ...Report -
Sparsity in Bayesian Inversion of Parametric Operator Equations
(2013)SAM Research ReportWe establish posterior sparsity in Bayesian inversion for systems with distributed parameter uncertainty subject to noisy data. We generalize the particular case of scalar diffusion problems with random coefficients in [29] to broad classes of operator equations. For countably parametric, deterministic representations of uncertainty in the forward problem which belongs to a certain sparsity class, we quantify analytic regularity of the ...Report -
Multilevel Monte Carlo for random degenerate scalar convection diffusion equation
(2013)SAM Research ReportThis paper proposes a Finite Difference Multilevel Monte Carlo algorithm for degenerate parabolic convection diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end we design a multilevel Monte Carlo method based on ...Report