Coarse Analysis of Microscopic Models using Equation-Free Methods

Abstract

Mathematical models of real-world problems from physics, biology and chemistry have become very complex over the last three decades. Although increasing computational power allows to solve even larger systems of differential equations, the number of differential equations is still a main limiting factor for the complexity of models, e.g., in real-time applications. With the increasing amount of data generated by computer simulations a challenge is to extract valuable information from the models in order to help scientists and managers in a decision-making process. Although the dynamics of these models might be high-dimensional, the properties of interest are usually macroscopic and lowdimensional in nature. Examples are numerous and not necessarily restricted to computer models. For instance, the power output, energy consumption and temperature of engines are interesting quantities for engineers, although the models they base their design on are described for the gas mixture (a system with many degrees-of-freedom) inside a combustion engine. Since good models are often not available on the macroscopic scale the necessary information has to be extracted from the microscopic, high-dimensional models. The goal of this thesis is to investigate such high-dimensional multiscale models and extract relevant low-dimensional information from them. Recently developed mathematical tools allow to reach this goal: a combination of so-called equation-free methods with numerical bifurcation analysis is used and further developed to gain insight into high-dimensional systems on a macroscopic level of interest. Based on a switching-procedure between a detailed microscopic and a coarse macroscopic level during simulations it is possible to obtain a closure-ondemand for the macroscopic dynamics by only using short simulation bursts of computationally-expensive complex models. Those information is subsequently used to construct bifurcation diagrams that show the parameter dependence of solutions of the system. The methods developed for this thesis have been applied to a wide range of relevant problems. Applications include the learning behavior in the barn owl’s auditory system, traffic jam formation in an optimal velocity model for circular car traffic and oscillating behavior of pedestrian groups in a counter-flow through a corridor with narrow door. The methods do not only quantify interesting properties in these models (learning outcome, traffic jam density, oscillation period), but also allow to investigate unstable solutions, which are important information to determine basins of attraction of stable solutions and thereby reveal information on the long-term behavior of an initial state.