In this thesis, we develop multiscale models for particle simulations in population dynamics. These models are characterised by prescribing particle motion on two spatial scales: microscopic and macroscopic.At the microscopic level, each particle has its own mass, position and velocity, while at the macroscopic level the particles are interpolated to a continuum quantity whose evolution is governed by a system of transport equations.This way, one can prescribe various types of interactions on a global scale, whilst still maintaining high simulation speed for a large number of particles. In addition, the interplay between particle motion and interaction is well tuned in both regions of low and high densities. We analyse links between models on these two scales and prove that under certain conditions, a system of interacting particles converges to a nonlinear coupled system of transport equations.We use this as a motivation to derive a model defined on both modelling scales and prescribe the intercommunication between them. Simulation takes place in inhomogeneous domains with arbitrary conditions at inflow and outflow boundaries. We realise this by modelling obstacles, sources and sinks.Integrating these aspects into the simulation requires a route planning algorithm for the particles. Several algorithms are considered and evaluated on accuracy, robustness and efficiency. All aspects mentioned above are combined in a novel open source prototyping simulation framework called Mercurial. This computational framework allows the design of geometries and is built for high performance when large numbers of particles are involved. Mercurial supports various types of inhomogeneities and global systems of equations. We apply our framework to simulate scenarios in crowd dynamics.We compare our results with test cases from literature to assess the quality of the simulations. / <p>Master Thesis in Industrial and Applied Mathematics</p>

The emergence of collective patterns from repeated local interactions between individuals is a common feature to most living systems, spanning a variety of scales from cells to animals and humans. Subjects of this thesis are two aspects of emergent complexity in living systems: collective movement and transport network formation. For collective movement, this thesis studies the role of movement-mediated information transfer in fish decision-making. The second project on collective movement takes inspiration from granular media and soft mode analysis and develops a new approach to describe the emergence of collective phenomena from physical interactions in extremely dense crowds. As regards transport networks, this thesis proposes a model of network growth to extract simple, biologically plausible rules that reproduce topological properties of empirical ant trail networks.  In the second project on transport networks, this thesis starts from the simple rule of “connecting each new node to the closest one”, that describes ants building behavior, to study how balancing local building costs and global maintenance costs influences the growth and topological properties of transport networks. These projects are addressed through a modeling approach and with the aim of identifying minimal sets of basic mechanisms that are most likely responsible of large-scale complex patterns. Mathematical models are always based on empirical observations and are, when possible, compared to experimental data.

No description available.

<p>   </p> <p>Historically, roadway performance measures have focused almost exclusively on vehicular movement. In most urban settings, pedestrian movements typically outnumber vehicular movements significantly. However,  historically there has been no way to collect such data at scale in a systematic manner. With the widespread introduction of cameras for monitoring vehicular flow, there is an opportunity to leverage this infrastructure to acquire insights into the patterns and trends of pedestrian activities at signalized intersections in an automated and systematic manner. Such data and performance measures are critical inputs for detailed analysis of pedestrian movements. Overall, addressing this issue is a vital component of transportation agencies that seek to develop equitable treatment of all transportation system users including vulnerable road users. This dissertation addresses the gap in the literature regarding detailed characterization of pedestrian movement patterns and trends. The dissertation leverages data from signalized intersection cameras to (1) quantify the required duration for the pedestrian walk-interval based on pedestrian volume and geometric features of the intersection, (2) carry out time series analysis to acquire insights on pedestrian demand patterns and the influential variables, and (3) build machine learning algorithms to accurately predict pedestrian volumes and tie it to signal timing, to enhance service for all roadway users.</p> <p>The first study provides quantitative guidance for walk time interval selection. This part reports on 1,500 pedestrian movement observations from 12 signalized intersections with varying pedestrian demand, pedestrian storage areas, and pedestrian push-button locations. That data were used to develop a model predicting start-up time with an R2 of 0.89. The study concludes by presenting a quantitative table with four timing categories ranging from negligible volume to high volume and corresponding appropriate durations for the pedestrian walk interval time, based on the demand per cycle, storage area for pedestrians, and offset of the pedestrian push-button from the crosswalk.</p> <p>The second study describes several scalable techniques for measuring and analyzing the movement of pedestrians on a typical university campus. Approximately 35.6 million pedestrian movements over 19 months were tabulated in 15-minute counts of pedestrian volumes by intersection. Counts are used in evaluating pedestrian activity dependency on select explanatory variables at both the network and intersection levels at each time step for the entire analysis period. The study reports on time series correlation and cross-correlation and measures the time-dependency between pedestrian activities and influential factors such as the academic calendar, football games, basketball games, and graduation ceremonies. It provides a comprehensive understanding of the factors that are most influential of  pedestrian volumes at intersections.</p> <p>The third study presents a data-driven approach to predict pedestrian volume per intersection quadrant at 15-minute intervals, and to connect this information to signal timing. Machine learning random forest and XGBoost classification models were trained on a large dataset of pedestrian counts consisting of approximately 2.6 million observations collected through 19 months at 13 exclusive pedestrian service intersections. The predicted pedestrian volumes were then categorized per the pedestrian walk-interval categories to provide optimal signal timing for each intersection quadrant, thus enabling potential dynamic pedestrian signal timing at exclusive service intersections. The results of this study showed that the developed models accurately predict pedestrian volumes per 15-minute intervals for each quadrant of an intersection, with a high degree of precision and a prediction accuracy of 82.3%. Signal timing optimization based on predicted pedestrian volume can significantly improve pedestrian mobility and maximize traffic flow. </p> <p>The findings of this study provide valuable insights for traffic engineers and planners interested in developing and deploying dynamic pedestrian signal timing systems. It is a practical and effective solution for improving mobility for all roadway users at intersections with exclusive pedestrian service.</p> <p>  </p>

This dissertation concerns the self-organization behaviors in different types of multi-agent systems, and possible ways to apply interventions on top ofthat to achieve certain goals. A bounded confidence opinion dynamics modelis considered for the first two papers. Theoretical analysis of the model isperformed and modifications of the model are given so that it will have better properties in some aspect. Leader-follower based models are studied in the third to fifth papers where various optimal control problems are considered. Different methods such as Pontryagin minimum principle and dynamic programming are used to solve those optimal control problem. For complex problems, one may only get approximate solutions or suboptimal solutions.In Paper A and Paper B, we consider the continuous-time Hegselmann-Krause (H-K) model and its variations and target the problem of reaching consensus. A sufficient condition on the initial opinion distribution is givento guarantee consensus for the original continuous-time H-K model. A modified model is provided and proven to be able to lead a larger range of initial opinions to synchronization. An H-K model with an exo-system is also studied where sufficient conditions on the exo-system are given for the purpose of consensus.In Paper C and Paper D, optimal control problems with leader-followerbased multi-agent systems are discussed. Analytic solutions are derived if the dynamics is linear by applying Pontryagin minimum principle. For generalnon-linear leader-follower interactions, we provide a method that use sstatistic moments of the follower crowd to approximate the optimal control.The dynamic programming approach is used and certain approximation ofthe Hamilton-Jacobi-Bellman equations is needed. The computational burdenis so heavy that model predictive control method is required in practical applications.In Paper E, we apply a similar method to the approach used in PaperD to target a pollutant elimination problem. It implies that we can use themethod to attack optimal control problem with partial differential equation constraints by discretization in space. The dimension of the discretization is not related to the computational complexity since only the statistic moments are needed. / <p>QC 20161201</p>

In recent years, there has been an increasing interest in modeling crowd and evacuation dynamics. Pedestrian models are based on macroscopic or microscopic behavior. In this work, we are interested in developing models that can be used for evacuation control strategies. Hence, we use macroscopic modeling approach, where pedestrians are treated in an aggregate way and detailed interactions are overlooked. In this dissertation, we developed two-dimensional space crowd dynamic models to allow bi-directional low by modifying and enhancing various features of existing traffic and fluid dynamic models. In this work, four models based on continuum theory are developed, and conservation laws such as the continuity and momentum equations are used. The first model uses a single hyperbolic partial differential equation with a velocity-density relationship, while the other three models are systems of hyperbolic partial differential equations. For one of the system models presented, we show how it can be derived independently from a microscopic crowd model. The models are nonlinear, time-varying, hyperbolic partial differential equations, and the numerical simulation results given for the four macroscopic models were based on computational fluid dynamics schemes. We also started an initial control design that synthesizes the feedback linearization method for the one-dimensional traffic flow problem applied directly on the distributed parameter system. In addition, we suggest and discuss the information technology requirements for an evacuation system. This research was supported in part from the National Science Foundation through grant no. CMS-0428196 with Dr. S. C. Liu as the Program Director. This support is gratefully acknowledged. Any opinion, findings, and conclusions or recommendations expressed in this study are those of the writer and do not necessarily reflect the views of the National Science Foundation. / Ph. D.

Des ensembles complexes, tels que les foules de piétons peuvent être soumis à de très fortes fluctuations de vitesses et de densités. Les individus formant ces ensembles sont les propres vecteurs de leur mouvement. Ils répondent à des règles locales microscopiques de déplacement qui, sous certaines conditions de densité, peuvent avoir un impact macroscopique sur la dynamique de l'ensemble. A la différence des animaux purement collectifs, comme ceux composant les bancs de poissons, les nuées d'oiseaux ou bien les troupeaux de gnous, les piétons partagent des intentions à la fois collectives et individuelles. De cette nature particulière apparaissent des comportements d'auto-organisation singuliers, dépendant de la nature cognitive des interactions entre les piétons, de la géométrie de l'espace, de la pluralité des objectifs, ainsi que la densité et la vitesse. L'objectif de cette thèse a été de reproduire via la simulation numérique les principaux phénomènes observés à l'aide d'hypothèses physiques et comportementales. Nous avons adopté une approche microscopique continue prenant en compte leur capacité d'anticipation via la recherche des différents temps de collision et l'utilisation d'une fonction d'estimation des angles de déviation possibles. Nous avons alors comparé notre modèle avec ceux dont il s'inspire face à différentes géométries. Nous avons aussi étudié des situations types comme l'évacuation à travers une porte avec différents jeux de paramètres : la taille de la porte, la vitesse et l'injection d'un bruit. Enfin, nous avons reproduit le phénomène d'oscillation de passages de deux groupes autour d'une porte en introduisant quelques règles de priorité. / Complex bodies, as crowds of pedestrian can present Strong Velocity and Density fluctuations. The persons shaping theses bodies are their own Drive motors. They follow local microscopic motion rules which, under certains Density conditions, can play a major role on the whole dynamics at the macroscopic scale. Unlike the purely collective animals, as those composing schools of fish, flocks of birds or herds of wildebeest, the pedestrians share at the same time collective and individual intentions. From this particular nature appear singular self-organisation behaviors, depending on the cognitive aspect of the pedestrian interactions, the geometry of the space, the plurality of targets, as well as the density and the velocity. The goal of this thesis has been to reproduce via numerical simulation tools the principal observed phenomena using physical and behavioral hypothesis. For that, we have adopted a continuous microscopic approach. The dynamics for each pedestrian is done locally trough a Newton-like equation which takes in account their anticipation ability via the search of the different collision time and the use of an estimation function of the possible deviation angles. Then, we have compared the behaviour differences between our model and those that inspired it in differents Spatial geometries. We have also studied generic situations as the evacuation trough a door with different sets of parameters : the size of the door, the velocity and the injection of a noise in the equation of forces. Finally, we have reproduced the oscillation phenomena of passings of two groups around a door by introducing some priority rules.

Within the design of any building, there is a requirement for designers to understand the intended purposes of the building and the elements that influence performance. These elements can be as tangible as providing a lecture hall within a university or relatively intangible such as the environmental temperatures of the rooms. The elements involved are generally recognised within the design industry and a combined force of engineers, architects, and specialist advisors work together to ensure all of the elements are in place for each new design. However, one element affecting performance that has not yet been comprehensively covered (at least for many building types) is that relating to occupant movement and the influence this has on experience and hence performance. For example, the number of times people have to negotiate cross-flow environments in a train station before becoming agitated is unknown. Also, the average distance people will walk through a shopping centre before becoming tired and ending the activity is unknown. Even so, they will both be impacted upon by the design and they will both reflect back on the performance of the design. Before starting this research, it was realised by the research engineer that there was only a limited understanding and application of people flow analyses within industry and, where it existed, it was solely related to transport terminals, pedestrian walkways/crossings, sports stadia arrivals/egress, and evacuation analyses.