Vol. 16, No. 1, 2021

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From interacting agents to density-based modeling with stochastic PDEs

Luzie Helfmann, Nataša Djurdjevac Conrad, Ana Djurdjevac, Stefanie Winkelmann and Christof Schütte

Vol. 16 (2021), No. 1, 1–32
DOI: 10.2140/camcos.2021.16.1
Abstract

Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation of such agent-based models is often intractable when the system becomes too large. In this paper, starting from a stochastic spatiotemporal agent-based model (ABM), we present a reduced model in terms of stochastic PDEs that describes the evolution of agent number densities for large populations while retaining the inherent model stochasticity. We discuss the algorithmic details of both approaches; regarding the SPDE model, we apply finite element discretization in space, which not only ensures efficient simulation but also serves as a regularization of the SPDE. Illustrative examples for the spreading of an innovation among agents are given and used for comparing ABM and SPDE models.

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Keywords
agent-based modeling, model reduction, Dean–Kawasaki model, SPDEs, finite element method
Mathematical Subject Classification 2010
Primary: 60H15, 60H35, 91B69, 91B74
Milestones
Received: 10 May 2019
Revised: 30 October 2020
Accepted: 1 November 2020
Published: 19 January 2021
Authors
Luzie Helfmann
Institut für Mathematik
Freie Universität Berlin
Berlin
Germany
Zuse-Institut Berlin
Berlin
Germany
Nataša Djurdjevac Conrad
Zuse-Institut Berlin
Berlin
Germany
Ana Djurdjevac
Institut für Mathematik
Freie Universität Berlin
Berlin
Germany
Institut für Mathematik
Technische Universität Berlin
Berlin
Germany
Stefanie Winkelmann
Zuse-Institut Berlin
Berlin
Germany
Christof Schütte
Institut für Mathematik
Freie Universität Berlin
Berlin
Germany
Zuse-Institut Berlin
Berlin
Germany