A hierarchical finite element Monte Carlo method for stochastic two-scale elliptic equations

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

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

This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon a multimodes representation of the solution as a power series of the perturbation parameter, and the Monte Carlo technique for sampling the probability space...

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

It is a well–known property of Monte Carlo methods that quadrupling the sample size halves the error. In the case of simulations of a stochastic partial differential equations, this implies that the total work is the sample size times the discretization costs of the equation. This leads to a convergence rate which is impractical for many simulations, namely in finance, physics and geosciences. ...

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Multi-Level Monte-Carlo Finite Element (MLMC–FE) methods for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “mean-square” and “pathwise” formulations. S...

A Finite Element Method for Elliptic Equations on Surfaces

Abstract. In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a fl...