48 research outputs found
A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction
We propose a Semi-Lagrangian scheme coupled with Radial Basis Function
interpolation for approximating a curvature-related level set model, which has
been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces
from sparse, possibly noisy data sets. The main advantages of the proposed
scheme are the possibility to solve the level set method on unstructured grids,
as well as to concentrate the reconstruction points in the neighbourhood of the
data set, with a consequent reduction of the computational effort. Moreover,
the scheme is explicit. Numerical tests show the accuracy and robustness of our
approach to reconstruct curves and surfaces from relatively sparse data sets.Comment: 14 pages, 26 figure
A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system
In this paper we study a fully discrete Semi-Lagrangian approximation of a
second order Mean Field Game system, which can be degenerate. We prove that the
resulting scheme is well posed and, if the state dimension is equals to one, we
prove a convergence result. Some numerical simulations are provided, evidencing
the convergence of the approximation and also the difference between the
numerical results for the degenerate and non-degenerate cases.Comment: 21 pages, 8 figure
A model problem for Mean Field Games on networks
In [14], Gueant, Lasry and Lions considered the model problem ``What time
does meeting start?'' as a prototype for a general class of optimization
problems with a continuum of players, called Mean Field Games problems. In this
paper we consider a similar model, but with the dynamics of the agents defined
on a network. We discuss appropriate transition conditions at the vertices
which give a well posed problem and we present some numerical results
The Hughes model for pedestrian dynamics and congestion modelling
In this paper we present a numerical study of some variations of the Hughes
model for pedestrian flow under different types of congestion effects. The
general model consists of a coupled non-linear PDE system involving an eikonal
equation and a first order conservation law, and it intends to approximate the
flow of a large pedestrian group aiming to reach a target as fast as possible,
while taking into account the congestion of the crowd.
We propose an efficient semi-Lagrangian scheme (SL) to approximate the
solution of the PDE system and we investigate the macroscopic effects of
different penalization functions modelling the congestion phenomena.Comment: 6 page
Numerical Analysis of time-dependent Hamilton-Jacobi Equations on Networks
A new algorithm for time dependent Hamilton Jacobi equations on networks,
based on semi Lagrangian scheme, is proposed. It is based on the definition of
viscosity solution for this kind of problems recently given in. A thorough
convergence analysis, not requiring weak semilimits, is provided. In
particular, the check of the supersolution property at the vertices is
performed through a dynamical technique which seems new. The scheme is
efficient, explicit, allows long time steps, and is suitable to be implemented
in a parallel algorithm. We present some numerical tests, showing the advantage
in terms of computational cost over the one proposed in [7
Semi-Lagrangian schemes for mean field game models
In this work we consider first and second order Mean Field Games (MFGs) systems, introduced in \cite{LasryLions06i,LasryLions06ii,LasryLions07}. For the first order case, we recall a fully-discrete Semi-Lagrangian (SL) scheme introduced in \cite{CS12} and its main properties. We propose the natural extension of this scheme for the second order case and we present some numerical simulations
A Generalized Fast Marching Method for dislocation dynamics
International audienceIn this paper, we consider a Generalized Fast Marching Method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hyper-surface in (with for physical applications) is given by its normal velocity which is a non-local function of the whole shape of the hyper-surface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension
A high-order scheme for mean field games
In this paper we propose a high-order numerical scheme for time-dependent
mean field games systems. The scheme, which is built by combining
Lagrange-Galerkin and semi-Lagrangian techniques, is consistent and stable for
large time steps compared with the space steps. We provide a convergence
analysis for the exactly integrated Lagrange-Galerkin scheme applied to the
Fokker-Planck equation, and we propose an implementable version with inexact
integration. Finally, we validate the convergence rate of the proposed scheme
through the numerical approximation of two mean field games systems
Convergence of a Generalized Fast Marching Method for a non-convex eikonal equation
International audienceWe present a new Fast Marching algorithm for a non-convex eikonal equation modeling front evolutions in the normal direction. The algorithm is an extension of the Fast Marching Method since the new scheme can deal with a \emph{time-dependent} velocity without \emph{any restriction on its sign}. We analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in