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A Multilevel Monte Carlo Finite Difference Method for Random Scalar Degenerate Convection Diffusion Equations

by U. Koley and N.H. Risebro and Ch. Schwab and F. Weber

(Report number 2013-32)

Abstract
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. 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, 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.

Keywords:

@Techreport{KRSW13_529,
  author = {U. Koley and N.H. Risebro and Ch. Schwab and F. Weber},
  title = {A Multilevel Monte Carlo Finite Difference Method for Random Scalar Degenerate Convection Diffusion Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-32.pdf },
  year = {2013}
}

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