Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities

Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities

While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with uncertain coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC finite element approach based on deterministic adaptive mesh refinement for the arising ”pathwise” problems and outline a convergence theory in terms of ...

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...

Multilevel Monte Carlo Methods

We study Monte Carlo approximations to high dimensional parameter dependent integrals. We survey the multilevel variance reduction technique introduced by the author in [4] and present extensions and new developments of it. The tools needed for the convergence analysis of vector-valued Monte Carlo methods are discussed, as well. Applications to stochastic solution of integral equations are give...

Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs

In this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on n ∈ N a-priori known, separated length scales are considered. The convergence of multilevel MC FE (MLMC FE) discretizations is analyzed. In particular, it is considered that the multilevel FE discretization resolves the finest physical length scale, bu...

Stochastic Methods: Monte Carlo Approach