Search
Results
-
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 -
Fast numerical solution of the linearized Molodensky problem
(1999)SAM Research ReportWhen standard boundary element methods (BEM) are used to solve the linearized vector Molodensky problem we are confronted with two problems: (i) the absence of $O(|x|^{-2})$ terms in the decay condition is not taken into account, since the single layer ansatz, which is commonly used as representation of the perturbation potential, is of the order $O(|x|^{-1})$ as $x \to \infty$. This implies that the standard theory of Galerkin BEM is not ...Report -
Generalized FEM for Homogenization Problems
(2001)SAM Research ReportWe introduce the concept of generalized Finite Element Method (gFEM) for the numerical treatment of homogenization problems. These problems are characterized by highly oscillatory periodic (or patchwise periodic) pattern in the coefficients of the differential equation and their solutions exhibit a multiple scale behavior: a macroscopic behavior superposed with local characteristics at micro length scales. The gFEM is based on two-scale ...Report -
On coupled problems for viscous flow in exterior domains
(1996)SAM Research ReportThe use of the complete Navier-Stokes system in an unbounded domain is not always convenient in computations and, therefore, the Navier-Stokes problem is often truncated to a bounded domain. In this paper we simulate the interaction between the flow in this domain and the exterior flow with the aid of a coupled problem. We propose in particular a linear approximation of the exterior flow (here the Stokes flow or potential flow) coupled ...Report -
High-dimensional finite elements for elliptic problems with multiple scales
(2003)SAM Research ReportElliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. They are discretized by a sparse tensor product finite element method (FEM) which resolves all scales of the solution throughout the physical domain. We prove that this FEM has accuracy, work and memory requirement comparable of FEM for single scale problems in the physical domain ...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 -
hp-DG-QTT solution of high-dimensional degenerate diffusion equations
(2012)SAM Research ReportWe consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous We consider the discretization of degenerate, time-inhomogeneous ...Report -
Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients
(2012)SAM Research ReportThis paper is a sequel to our previous work $({Kuo, Schwab, Sloan, SIAM J.\ Numer.\ Anal., 2013})$ where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented in a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as ...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