A MULTIMODES MONTE CARLO FINITE ELEMENT METHOD FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS

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...

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC ...

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

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R (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 ...

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. ...