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Multilevel Monte Carlo approximations of statistical solutions to the Navier-Stokes equations
(2013)SAM Research ReportReport -
Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs
(2012)SAM Research ReportIn this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on $n \in \mathbb{N}$ 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 nest physical length scale, but the coarsest FE mesh does not, ...Report -
Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations
(2011)SAM Research ReportWe analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time ...Report -
Multilevel Monte Carlo method for parabolic stochastic partial differential equations
(2011)SAM Research ReportWe analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to ...Report -
Multi-Level Monte Carlo Finite Element Method for elliptic PDEs with stochastic coefficients
(2010)SAM Research ReportIt 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. With the Multi--level Monte Carlo method ...Report