A multilevel Monte Carlo finite difference method for random scalar degenerate convection-diffusion equations

作者：Koley Ujjwal^*; Risebro Nils Henrik; Schwab Christoph; Weber Franziska

来源：Journal of Hyperbolic Differential Equations, 2017, 14(3): 415-454.

DOI：10.1142/S021989161750014X

• Degenerate convection-diffusion equation
• rate of convergence
• random entropy solution
• finite difference schemes
• Monte Carlo method

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate 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 equations. 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 deterministic 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 obtained for p = 2, for finite volume convergence rate of 1/3. We conclude with numerical experiments.

• 出版日期2017-9