Uncertainty quantification for kinetic models in socio-economic and life sciences

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.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic Equations

Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

A realizable filtered intrusive polynomial moment method

Intrusive uncertainty quantification methods for hyperbolic problems exhibit spurious oscillations at shocks, which leads to a significant reduction of the overall approximation quality. Furthermore, a challenging task is to preserve hyperbolicity of the gPC moment system. An intrusive method which guarantees hyperbolicity is the intrusive polynomial moment (IPM) method, which performs the gPC expansion on the entropy variables. The method, while still being subject to oscillations, requires solving a convex optimization problem in every spatial cell and every time step. The aim of this work is to mitigate oscillations in the IPM solution by applying filters. Filters reduce oscillations by damping high order gPC coefficients. Naive filtering, however, may lead to unrealizable moments, which means that the IPM optimization problem does not have a solution and the method breaks down. In this paper, we propose and analyze two separate strategies to guarantee the existence of a solution to the IPM problem. First, we propose a filter which maintains realizability by being constructed from an underlying Fokker-Planck equation. Second, we regularize the IPM optimization problem to be able to cope with non-realizable gPC coefficients. Consequently, standard filters can be applied to the regularized IPM method. We demonstrate numerical results for the two strategies by investigating the Euler equations with uncertain shock structures in one- and two-dimensional spatial settings. We are able to show a significant reduction of spurious oscillations by the proposed filters

Quantifying multiple uncertainties in modelling shallow water-sediment flows: A stochastic Galerkin framework with Haar wavelet expansion and an operator-splitting approach

The interactive processes of shallow water flow, sediment transport, and morphological evolution constitute a hierarchy of multi-physical problems of significant interests in a spectrum of engineering and science areas. To date, modelling shallow water hydro-sediment-morphodynamic (SHSM) processes is subject to multiple sources of uncertainty arising from input data and incomplete understanding of the underlying physics. A stochastic SHSM model with multiple uncertainties has yet to be developed as most SHSM models still concern deterministic problems and only one has been recently extended to a stochastic setting, but is restricted to a single source of uncertainty. Here we first present a new probabilistic SHSM model incorporating multiple uncertainties within the stochastic Galerkin framework using a multidimensional tensor product of Haar wavelet expansion to capture local, nonlinear variations in joint probability distributions and an operator-splitting-based method to ensure that the modelling system remains hyperbolic. Then, we verify the proposed model via benchmark probabilistic numerical tests with joint uncertainties introduced in initial and boundary conditions, matching established experiments of flow-sediment-bed evolutions driven by a sudden dam break and by a landslide dam failure and large-scale rapid flow-sediment-bed evolution in response to flash flood. The present work facilitates a promising modelling framework for quantifying multiple uncertainties in practical shallow water hydro-sediment-morphodynamic modelling applications