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Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences

  • Giacomo Dimarco
  • Lorenzo Pareschi
  • Mattia Zanella
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 14)

Abstract

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker–Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.

Notes

Acknowledgements

The research that led to the present survey was partially supported by the research grant Numerical methods for uncertainty quantification in hyperbolic and kinetic equations of the group GNCS of INdAM. MZ acknowledges support from GNCS and “Compagnia di San Paolo” (Torino, Italy).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Giacomo Dimarco
    • 1
  • Lorenzo Pareschi
    • 1
  • Mattia Zanella
    • 2
  1. 1.University of FerraraFerraraItaly
  2. 2.Politecnico di TorinoTorinoItaly

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