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Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs
(2011)SAM Research ReportWe investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a-priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor-, Legendre- and ...Report -
Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions
(2011)SAM Research ReportSpace-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space $H$ are developed. The well-posedness of these equations in the Hilbert space $L^2(H,μ)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $μ$ on $H$, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin ...Report -
Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficient
(2013)SAM Research ReportIn this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in $R^d$ (d=1,2,3), with diffusion coefficient a(x,ω) given as a lognormal random field, i.e., a(x,ω)=exp(Z(x,ω)) where x is the spatial variable and Z(x,⋅) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of ...Report -
Sparse tensor spherical harmonics approximation in radiative transfer
(2010)SAM Research ReportThe stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, inflow boundary conditions have to be prescribed. The sparse tensor product discretization has been successfully applied to finite element methods in radiative transfer with wavelet discretization of the angular domain (Widmer2009a). In this report we show that the ...Report -
Covariance structure of parabolic stochastic partial differential equations
(2012)SAM Research ReportIn this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity
(2010)SAM Research ReportFor a linear second order elliptic partial differential operator $A: V → V'$, we consider the boundary value problems $Au=f$ with stationary Gaussian random data $f$ over the dual $V'$ of the separable Hilbert space $V$ in which the solution u is sought. The operator $A$ is assumed to be deterministic and bijective. The unique solution $u= A^-$$^1f $ is a Gaussian random field over $V$. It is characterized by its mean field $E_u$ and ...Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 2: Exponential convergence
(2010)SAM Research ReportWe prove exponential rates of convergence of a class of $hp$ Galerkin Finite Element approximations of solutions to a model tensor non-hypoelliptic equation in the unit square □ = (0,1)$^2$ which exhibit singularities on ∂□ and on the diagonal ∆ = {($x,y$) ∈ □ : $x$ = $y$}, but are otherwise analytic in □. As we explained in the first part [6] of this work, such problems arise as deterministic second moment equations of linear, second ...Report -
Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs
(2011)SAM Research ReportA class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and $n$ known, separated microscopic length scales $\epsilon_i$, $i=1,...,n$ in a bounded domain $D\subset R^d$ is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge $P$-a.s, as $\epsilon_i\rightarrow 0$, to a stochastic, ...Report -
Sparse Approximation Algorithms for High Dimensional Parametric Initial Value Problems
(2013)SAM Research ReportWe consider the efficient numerical approximation on nonlinear systems of initial value Ordinary Differential Equations (ODEs) on Banach state spaces $\mathcal{S}$ over $\mathbb{R}$ or $\mathbb{C}$. We assume the right hand side depends $in$ $affine$ $fashion$ on a vector $y =(y_j)_{j \geq 1}$ of possibly countably many parameters, normalized such that $|y_j| \leq 1$. Such affine parameter dependence of the ODE arises, among others, in ...Report -
Sparse Tensor Approximation of Parametric Eigenvalue Problems
(2010)SAM Research ReportWe design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method ...Report