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Abstract
Mathematical models of realworld 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 realtime 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 decisionmaking process. Although the dynamics of these models might be highdimensional, 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 degreesoffreedom) 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, highdimensional models.
The goal of this thesis is to investigate such highdimensional multiscale models and extract relevant lowdimensional information from them. Recently developed mathematical tools allow to reach this goal: a combination of socalled equationfree methods with numerical bifurcation analysis is used and further developed to gain insight into highdimensional systems on a macroscopic level of interest. Based on a switchingprocedure between a detailed microscopic and a coarse macroscopic level during simulations it is possible to obtain a closureondemand for the macroscopic dynamics by only using short simulation bursts of computationallyexpensive 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 counterflow 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 longterm behavior of an initial state.
The goal of this thesis is to investigate such highdimensional multiscale models and extract relevant lowdimensional information from them. Recently developed mathematical tools allow to reach this goal: a combination of socalled equationfree methods with numerical bifurcation analysis is used and further developed to gain insight into highdimensional systems on a macroscopic level of interest. Based on a switchingprocedure between a detailed microscopic and a coarse macroscopic level during simulations it is possible to obtain a closureondemand for the macroscopic dynamics by only using short simulation bursts of computationallyexpensive 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 counterflow 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 longterm behavior of an initial state.
Original language  English 

Place of Publication  Kgs. Lyngby 

Publisher  Technical University of Denmark 
Number of pages  189 
Publication status  Published  2014 
Series  DTU Compute PHD2014 

Number  342 
ISSN  09093192 
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Dive into the research topics of 'Coarse Analysis of Microscopic Models using EquationFree Methods'. Together they form a unique fingerprint.Projects
 1 Finished

Analysis of Pattern Formation on Networks
Marschler, C. (PhD Student), Starke, J. (Main Supervisor), Christiansen, L. E. (Examiner), Sugiyama, Y. (Examiner) & Barkley, D. (Examiner)
Technical University of Denmark
01/08/2011 → 21/11/2014
Project: PhD