Emiliano Cristiani

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows one to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This way it is possible to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.

Multiscale Modeling of Pedestrian Dynamics

1 An Introduction to the Modeling of Crowd Dynamics.- 2 Problems and Simulations.- 3 Psychological Insights.- 4 An Overview of the Modeling of Crowd Dynamics.- 5 Multiscale Modeling by Time-Evolving Measures.- 6 Basic Theory of Measure-Based Models.- 7 Evolution in Measure Spaces with Wasserstein Distance.- 8 Generalizations of the Multiscale Approach.- 9 Appendices: A Basics of Measure and Probability Theory B Pseudo-Code for the Multiscale Algorithm.

Fast Semi-Lagrangian Schemes for the Eikonal Equation and Applications

We introduce and analyze a fast version of the semi-Lagrangian algorithm for front propagation originally proposed in [M. Falcone, “The minimum time problem and its applications to front propagation,” in Motion by Mean Curvature and Related Topics, A. Visintin and G. Buttazzo, eds., de Gruyter, Berlin, 1994, pp. 70-88]. The new algorithm is obtained using the local definition of the approximate solution typical of semi-Lagrangian schemes and redefining the set of “neighboring nodes” necessary for fast marching schemes. A new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences. We show that the new algorithm converges to the viscosity solution of the problem and that its complexity is

Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints

O(N \log N_{nb})

Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations

, as it is for the fast marching method based on finite difference (

Effects of anisotropic interactions on the structure of animal groups

N

A Patchy Dynamic Programming Scheme for a Class of Hamilton--Jacobi--Bellman Equations

and

Modeling rationality to control self-organization of crowds: an environmental approach

N_{nb}

An adaptive domain-decomposition technique for parallelization of the fast marching method

being, respectively, the total number of nodes and the number of nodes in the narrow band). A new sufficient condition for the convergence of the standard finite difference fast marching method is also given. We present several tests comparing the two algorithms and other fast methods (e.g., fast sweeping) on a series of benchmarks which include the minimum time problem and the shape-from-shading problem.

Numerical algorithms for perspective shape from shading

This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different self-organized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration.