Sammendrag
This paper proposes a Finite Difference Multilevel Monte Carlo algorithm for degener- ate parabolic convection diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo Finite Volume method, allowing in particular for low p -integrability of the random solution with 1 < p ≤ 2, and low deter- ministic convergence rates (here, the theoretical rate is 1 / 3). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error), is ob- tained for p = 2, for finite volume convergence rate of 1 / 3. We conclude with numerical experiments.
The final version of this research has been published in the Journal of Hyperbolic Differential Equations. © 2017 World Scientific Publishing