A Multimodes Monte Carlo Finite Element Method for Elliptic Partial Differential Equations with Random Coefficients

Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity, and lead to var...

Multi-level Higher Order Qmc Galerkin Discretization for Affine Parametric Operator Equations

We 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 qMC-spectral method for elliptic PDEs with random coefficients on the unit sphere

We present a quasi-Monte Carlo spectral method for a class of elliptic partial differential equations (PDEs) with random coefficients defined on the unit sphere. The random coefficients are parametrised by the Karhunen-Loève expansion, while the exact solution is approximated by the spherical harmonics. The expectation of the solution is approximated by a quasi-Monte Carlo integration rule. A m...

Multi-Level Monte Carlo Finite Element Method for Elliptic Partial Differential Equations with Stochastic Data

Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identi...