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Uncertainty quantification for kinetic equations of emergent phenomena

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FKT - Frontiers in kinetic theory: connecting microscopic to macroscopic scales - KineCon 2022

Kinetic equations play a leading role in the modelling of large systems of interacting particles/agents with a recognized effectiveness in describing real world phenomena ranging from plasma physics to multi-agent dynamics. The derivation of these models has often to deal with physical, or even social, forces that are deduced empirically and of which we have limited information. Hence, to produce realistic descriptions of the underlying systems it is of paramount importance to consider the effects of uncertain quantities as a structural feature in the modelling process. In this talk, we focus on a class of numerical methods that guarantee the preservation of main physical properties of kinetic models with uncertainties. In contrast to a direct application of classical uncertainty quantification methods, typically leading to the loss of positivity of the numerical solution of the problem, we discuss the construction of novel schemes that are capable of achieving high accuracy in the random space without losing nonnegativity of the solution [1,3]. Applications of the developed methods are presented in the classical RGD framework and in related models in life sciences. In particular, we concentrate on the interplay of this class of models with mathematical epidemiology where the assessment of uncertainties in data assimilation is crucial to design efficient interventions, see [2].   Bibliography: [1] J. A. Carrillo, L. Pareschi, M. Zanella. Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys., 25(2): 508-531, 2019.  [2] G. Dimarco, B. Perthame, G. Toscani, M. Zanella. Kinetic models for epidemic dynamics with social heterogeneity. J. Math. Biol., 83, 4, 2021.   [3] L. Pareschi, M. Zanella. Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case. J. Comput. Phys. 423:109822, 2020.

This talk is part of the Isaac Newton Institute Seminar Series series.

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