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A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit

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Monte Carlo and Quasi-Monte Carlo Methods (MCQMC 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 324))

Abstract

We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments.

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Acknowledgements

We thank Pieterjan Robbe for many helpful discussions on the multilevel Monte Carlo method. We also thank the anonymous reviewers for their helpful suggestions for improving the quality of this work. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation—Flanders (FWO) and the Flemish Government—department EWI.

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Authors and Affiliations

  1. Department of Computer Science, NUMA Section, Celestijnenlaan 200A box 2402, 3001, Leuven, Belgium

    Emil Løvbak, Giovanni Samaey & Stefan Vandewalle

Authors
  1. Emil Løvbak
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  2. Giovanni Samaey
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  3. Stefan Vandewalle
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Correspondence to Emil Løvbak .

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Editors and Affiliations

  1. INRIA Rennes Bretagne-Atlantique, Campus Universtaire de Beaulieu, Rennes, France

    Bruno Tuffin

  2. Département d’informatique et de Recherche Opérationnelle (DIRO), Université de Montréal, Montreal, QC, Canada

    Pierre L'Ecuyer

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Løvbak, E., Samaey, G., Vandewalle, S. (2020). A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit. In: Tuffin, B., L'Ecuyer, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2018. Springer Proceedings in Mathematics & Statistics, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-030-43465-6_19

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