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Volume 25, Issue 2
Particle Based gPC Methods for Mean-Field Models of Swarming with Uncertainty

José Antonio Carrillo, Lorenzo Pareschi & Mattia Zanella

DOI: 10.4208/cicp.OA-2017-0244

Commun. Comput. Phys., 25 (2019), pp. 508-531.

Published online: 2018-10

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean-field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

  • Keywords

Uncertainty quantification, stochastic Galerkin, mean-field equations, swarming dynamics.

  • AMS Subject Headings

35Q83, 65C05, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-508, author = {José Antonio Carrillo, Lorenzo Pareschi and Mattia Zanella}, title = {Particle Based gPC Methods for Mean-Field Models of Swarming with Uncertainty}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {2}, pages = {508--531}, abstract = {

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean-field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0244}, url = {http://global-sci.org/intro/article_detail/cicp/12761.html} }
TY - JOUR T1 - Particle Based gPC Methods for Mean-Field Models of Swarming with Uncertainty AU - José Antonio Carrillo, Lorenzo Pareschi & Mattia Zanella JO - Communications in Computational Physics VL - 2 SP - 508 EP - 531 PY - 2018 DA - 2018/10 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0244 UR - https://global-sci.org/intro/article_detail/cicp/12761.html KW - Uncertainty quantification, stochastic Galerkin, mean-field equations, swarming dynamics. AB -

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean-field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

José Antonio Carrillo, Lorenzo Pareschi and Mattia Zanella. (2018). Particle Based gPC Methods for Mean-Field Models of Swarming with Uncertainty. Communications in Computational Physics. 25 (2). 508-531. doi:10.4208/cicp.OA-2017-0244
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