scholarly journals A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE

2010 ◽  
Vol 20 (10) ◽  
pp. 1787-1821 ◽  
Author(s):  
BERTRAND MAURY ◽  
AUDE ROUDNEFF-CHUPIN ◽  
FILIPPO SANTAMBROGIO
Keyword(s):  
Ad Hoc ◽  
Gradient Flow ◽  
Emergency Evacuation ◽  
Probability Measures ◽  
Motion Model ◽  
Exit Door ◽  
Incompressibility Constraint ◽  
Macroscopic Approach ◽  
Crowd Motion ◽  
Wasserstein Space

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U (x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknown is a density ρ(t,x). If a gradient structure is given, say U = -∇D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad hoc study of the convergence of a discrete scheme.

Thermomechanical Equations Governing a Material With Prescribed Temperature-Dependent Density With Application to Nonisothermal Plane Poiseuille Flow

1996 ◽  
Vol 63 (4) ◽  
pp. 1011-1018 ◽  
Author(s):  
D. Cao ◽  
S. E. Bechtel ◽  
M. G. Forest
Keyword(s):  
Poiseuille Flow ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Incompressible Material ◽  
Momentum Equation ◽  
Plane Poiseuille Flow ◽  
Temperature Dependent ◽  
Incompressibility Constraint ◽  
Expansion Cooling ◽  
Dependent Density

The standard practice in the literature for modeling materials processing in which changes in temperature induce significant volume changes is based on the a posteriori substitution of a temperature-dependent expression for density into the governing equations for an incompressible material. In this paper we show this ad hoc approach misses important terms in the equations, and by example show the ad hoc equations fail to capture important physical effects. First we derive the three-dimensional equations which govern the deformation and heat transfer of materials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanical constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (“pressure”) enters into the energy equation as well as the momentum equation. Then we demonstrate the effect of the correct constraint response by comparing solutions of our thermomechanical theory with solutions of the ad hoc theory in plane Poiseuille flow. The differences are significant, both quantitatively and qualitatively. In particular, the observed phenomenon of expansion cooling is captured by the thermomechanically constrained theory, but not by the ad hoc theory.

scholarly journals Modeling Pedestrian Motion in Crowded Scenes Based on the Shortest Path Principle

2021 ◽  
Vol 12 (1) ◽  
pp. 381
Author(s):  
Yi Zou ◽  
Yuncai Liu
Keyword(s):  
Shortest Path ◽  
State Of The Art ◽  
Meaningful Work ◽  
Movement Behavior ◽  
Motion Model ◽  
Human Dynamics ◽  
Motion Trajectories ◽  
Crowd Motion ◽  
Crowded Scenes ◽  
Part Motion

In the computer vision field, understanding human dynamics is not only a great challenge but also very meaningful work, which plays an indispensable role in public safety. Despite the complexity of human dynamics, physicists have found that pedestrian motion in a crowd is governed by some internal rules, which can be formulated as a motion model, and an effective model is of great importance for understanding and reconstructing human dynamics in various scenes. In this paper, we revisit the related research in social psychology and propose a two-part motion model based on the shortest path principle. One part of the model seeks the origin and destination of a pedestrian, and the other part generates the movement path of the pedestrian. With the proposed motion model, we simulated the movement behavior of pedestrians and classified them into various patterns. We next reconstructed the crowd motions in a real-world scene. In addition, to evaluate the effectiveness of the model in crowd motion simulations, we created a new indicator to quantitatively measure the correlation between two groups of crowd motion trajectories. The experimental results show that our motion model outperformed the state-of-the-art model in the above applications.

A mathematical framework for a crowd motion model

2008 ◽  
Vol 346 (23-24) ◽  
pp. 1245-1250 ◽  
Author(s):  
Bertrand Maury ◽  
Juliette Venel
Keyword(s):  
Mathematical Framework ◽  
Motion Model ◽  
Crowd Motion

First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces

2012 ◽  
Vol 55 (4) ◽  
pp. 723-735 ◽  
Author(s):  
Nicola Gigli ◽  
Shin-Ichi Ohta
Keyword(s):  
Variational Inequality ◽  
Gradient Flow ◽  
Wasserstein Distance ◽  
Convex Functional ◽  
Alexandrov Spaces ◽  
Variation Formula ◽  
First Variation ◽  
Evolution Variational Inequality ◽  
Wasserstein Space ◽  
Bounded Below

AbstractWe extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space ((X),W2) satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

scholarly journals A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

2012 ◽  
Vol 04 (04) ◽  
pp. 515-542 ◽  
Author(s):  
JÉRÔME BERTRAND ◽  
BENOÎT KLOECKNER
Keyword(s):  
Large Scale ◽  
Euclidean Plane ◽  
Optimal Transport ◽  
Probability Measures ◽  
Hadamard Space ◽  
Geodesic Boundary ◽  
Wasserstein Space ◽  
Scale Properties ◽  
Positively Curved ◽  
Geometric Study

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].

scholarly journals Prototipe Pengendali Pintu Daurat Menggunakan Mikrokontroler ATMega 16

2018 ◽  
Vol 5 (2) ◽  
pp. 122
Author(s):  
Abdul Zain ◽  
Arief Muliawan
Keyword(s):  
Emergency Evacuation ◽  
Ultrasonic Sensor ◽  
Security System ◽  
Lcd Display ◽  
Exit Door ◽  
User Facility ◽  
Emergency Exit

In a building should have a good security system,especially Emergency Exit (emergency evacuation path).Emergency Exit is important because when an emergency occursin the building, the first thing to do is get out of the lab building.One thing to note is the emergency exit door. With the emergencydoor that can be opened quickly so all workers can get outquickly as well from the building when an emergency occurs.So,we need a model (prototype) controlling an emergency door thatcan function properly in an emergency. Prototype newemergency door controller is expected to be implemented intoactual emergency door. Prototype can be accessed with a pushbutton by the user facility. The prototype controller usemicrocontroller ATMega16. And for each emergency exitmovement is monitored and its status is expressed in LEDindicator lights and LCD display on 16x2. Modeling controllingemergency exit of a series of tests using a prototype have workedwell except on the 3rd test caused an error when calibrating thedistance between Ultrasonic sensor with Emergency Door.

scholarly journals An Optimal Transport View of Schrödinger's Equation

2012 ◽  
Vol 55 (4) ◽  
pp. 858-869 ◽  
Author(s):  
Max-K. von Renesse
Keyword(s):  
Symplectic Structure ◽  
Optimal Transport ◽  
Riemannian Metric ◽  
Symplectic Space ◽  
Probability Measures ◽  
Extended System ◽  
Third Law ◽  
Classical Potential ◽  
Wasserstein Space ◽  
Complex Valued

AbstractWe show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential μ → F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.

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