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by Bertrand Maury (Author), Sylvain Faure (Author)
About the Author
Bertrand Maury is Professor of Mathematics at the Université Paris-Sud, Orsay (France). He is also adjunct Professor at the Ecole Polytechnique, Docent Professor at the University of Jyväskylä (Finland), and junior member of the Institut Universitaire de France. His domains of interest extend from Computational Fluid Dynamics and Numerical Analysis, to modeling in life sciences and Optimal Transportation.
About this book
In this book, intended to graduate students and researchers in mathematics, Sylvain Faure and Bertrand Maury invite us to discover the challenges and the first successes of mathematics applied to social sciences. As a preamble, they clearly explain the difficulties of the exercise, due, in particular, to the freedom of individuals and to the decision processes which are neither symmetric, nor interchangeable.
The book comes with the rigorous analysis of some models, essentially at the microscopic scale, which serve as mathematical prototypes and exhibit interesting phenomenologies. Starting from this picture, the authors propose eventually to extract the minimal elements which should be contained in a mathematical model in order to reproduce some typical and sometimes paradoxical properties of crowd motions: “Faster-is-Slower” effect, “Stop-and-Go” waves, and fluidizing effects of an obstacle. This is fascinating!
The book contains self-contained descriptions of existing models, accompanied by critical analyses of their properties both from a theoretical and practical standpoint. It aims to develop 'modeling skills' within the readers, giving them the ability to develop their own models and improve existing ones. Written in connection with a full, open source Python Library, this project also enables readers to run the simulations discussed within the text.
Brief contents
1. Introduction 1
1.1 From Passive to Active Entities 1
1.2 Basics on Crowd Motion Modeling 3
1.3 The Mathematical Standpoint 5
1.4 How to Use this Book? 12
2. One-Dimensional Microscopic Models 13
2.1 Follow-the-Leader Model 14
2.2 Accounting for Inertia/Delays 30
3. Social Force Model, Native and Overdamped Forms 37
3.1 Inertial Social Force Model 37
3.2 Overdamped Social Force Model 49
3.3 Alternative Approaches 56
4. Granular Models 59
4.1 One-Dimensional Model 59
4.2 Two-Dimensional Model 61
4.3 Numerical Scheme 64
4.4 Numerical Experiments 66
4.5 Mathematical Issues 68
4.6 Critical Discussion 76
5. Cellular Automata 83
5.1 Cellular Automata: General Principles 84
5.2 Algorithms 85
5.3 Variations, Extensions 92
5.4 Cellular Automata, Mathematical Issues 93
6. Compartment Models 97
6.1 Compartment Models: Toy Versions and General Setting 97
6.2 Numerical Solution 101
6.3 Extensions 102
6.4 Numerical Illustration 104
6.5 Mathematical Framework: A Cascade of Gradient Flows 104
7. Toward Macroscopic Models 111
7.1 One-Dimensional Macroscopic Traffic Model 112
7.2 Two-Dimensional Models 115
7.3 Granular Models: Hard Congestion 117
7.4 Micro–Macro Issues 123
7.5 Alternative Macroscopic Models 125
8. Computing Distances and Desired Velocities 127
8.1 Shortest Path Problemon a Graph 130
8.2 Shortest Path on a Domain: The Eikonal Equation 132
8.3 Non-homogenous Domains, Various Extensions 135
8.4 Shortest Paths in a Dynamic Environment 139
8.5 Alternative Way to Compute Desired Velocities 142
8.6 Illustrations 143
9. Data, Observable Phenomena 145
9.1 Diameters 145
9.2 Proxemics, Interpersonal Distances, Density 146
9.3 Cone of Vision 148
9.4 Pedestrian Speed, Fundamental Diagram 148
9.5 Door Capacity 151
9.6 Capacity Drop Phenomenon 151
9.7 Faster-is-Slower Effect 152
9.8 Influence of an Obstacle 154
9.9 Stop-and-Go Waves 157
9.10 Further Considerations on Human Behavior 158
10. A Wider Look on Characteristic Phenomena in Crowds 161
10.1 Faster-is-Slower Effect 161
10.2 Fluidizing Effect of an Obstacle 168
10.3 Damping, Propagation, and Stop-and-Go Waves 170
Appendix 177
A.1 Ordinary Differential Equations 177
A.2 Constrained Optimization 180
Bibliography 183
Index 189
Series: Advanced Textbooks in Mathematics
Pages: 200 pages
Publisher: World Scientific Publishing Europe Ltd (September 5, 2018)
Language: English
ISBN-10: 178634551X
ISBN-13: 978-1786345516
WS__Crowds In Equations.pdf
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