A multilevel Monte Carlo finite element method for the stochastic Cahn–Hilliard–Cook equation

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...

Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by sw...

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

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. ...

A Spectral Monte Carlo Method for the Poisson Equation. a Spectral Monte Carlo Method for the Poisson Equation *

Using a sequential Monte Carlo algorithm, we compute a spectral approximation of the solution of the Poisson equation in dimension 1 and 2. The Feyman-Kac computation of the pointwise solution is achieved using either an integral representation or a modified walk on spheres method. The variances decrease geometrically with the number of steps. A global solution is obtained, accurate up to the i...