35 research outputs found
Sweeping process by prox-regular sets in Riemannian Hilbert manifolds
In this paper, we deal with sweeping processes on (possibly
infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful
notions (proximal normal cone, prox-regularity) already defined in the setting
of a Hilbert space to the framework of such manifolds. Especially we introduce
the concept of local prox-regularity of a closed subset in accordance with the
geometrical features of the ambient manifold and we check that this regularity
implies a property of hypomonotonicity for the proximal normal cone. Moreover
we show that the metric projection onto a locally prox-regular set is
single-valued in its neighborhood. Then under some assumptions, we prove the
well-posedness of perturbed sweeping processes by locally prox-regular sets.Comment: 27 page
A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients
We perform the a posteriori error analysis of residual type of a transmission
problem with sign changing coefficients. According to [6] if the contrast is
large enough, the continuous problem can be transformed into a coercive one. We
further show that a similar property holds for the discrete problem for any
regular meshes, extending the framework from [6]. The reliability and
efficiency of the proposed estimator is confirmed by some numerical tests.Comment: 15 page
Existence of solutions for second-order differential inclusions involving proximal normal cones
In this work, we prove global existence of solutions for second order
differential problems in a general framework. More precisely, we consider
second order differential inclusions involving proximal normal cone to a
set-valued map. This set-valued map is supposed to take admissible values (so
in particular uniformly prox-regular values, which may be non-smooth and
non-convex). Moreover we require the solution to satisfy an impact law,
appearing in the description of mechanical systems with inelastic shocks.Comment: 37 page
A discrete contact model for crowd motion
The aim of this paper is to develop a crowd motion model designed to handle
highly packed situations. The model we propose rests on two principles: We
first define a spontaneous velocity which corresponds to the velocity each
individual would like to have in the absence of other people; The actual
velocity is then computed as the projection of the spontaneous velocity onto
the set of admissible velocities (i.e. velocities which do not violate the
non-overlapping constraint). We describe here the underlying mathematical
framework, and we explain how recent results by J.F. Edmond and L. Thibault on
the sweeping process by uniformly prox-regular sets can be adapted to handle
this situation in terms of well-posedness. We propose a numerical scheme for
this contact dynamics model, based on a prediction-correction algorithm.
Numerical illustrations are finally presented and discussed.Comment: 22 page
Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme
Here we present well-posedness results for first order stochastic
differential inclusions, more precisely for sweeping process with a stochastic
perturbation. These results are provided in combining both deterministic
sweeping process theory and methods concerning the reflection of a Brownian
motion. In addition, we prove convergence results for a Euler scheme,
discretizing theses stochastic differential inclusions.Comment: 30 page