280 research outputs found
Finding Transition Pathways on Manifolds
We consider noise-induced transition paths in randomly perturbed dynami- cal
systems on a smooth manifold. The classical Freidlin-Wentzell large devia- tion
theory in Euclidean spaces is generalized and new forms of action functionals
are derived in the spaces of functions and the space of curves to accommodate
the intrinsic constraints associated with the manifold. Numerical meth- ods are
proposed to compute the minimum action paths for the systems with constraints.
The examples of conformational transition paths for a single and double rod
molecules arising in polymer science are numerically investigated
Time Discrete Geodesic Paths in the Space of Images
In this paper the space of images is considered as a Riemannian manifold
using the metamorphosis approach, where the underlying Riemannian metric
simultaneously measures the cost of image transport and intensity variation. A
robust and effective variational time discretization of geodesics paths is
proposed. This requires to minimize a discrete path energy consisting of a sum
of consecutive image matching functionals over a set of image intensity maps
and pairwise matching deformations. For square-integrable input images the
existence of discrete, connecting geodesic paths defined as minimizers of this
variational problem is shown. Furthermore, -convergence of the
underlying discrete path energy to the continuous path energy is proved. This
includes a diffeomorphism property for the induced transport and the existence
of a square-integrable weak material derivative in space and time. A spatial
discretization via finite elements combined with an alternating descent scheme
in the set of image intensity maps and the set of matching deformations is
presented to approximate discrete geodesic paths numerically. Computational
results underline the efficiency of the proposed approach and demonstrate
important qualitative properties.Comment: 27 pages, 7 figure
Ether theory of gravitation: why and how?
Gravitation might make a preferred frame appear, and with it a clear
space/time separation--the latter being, a priori, needed by quantum mechanics
(QM) in curved space-time. Several models of gravitation with an ether are
discussed: they assume metrical effects in an heterogeneous ether and/or a
Lorentz-symmetry breaking. One scalar model, starting from a semi-heuristic
view of gravity as a pressure force, is detailed. It has been developed to a
complete theory including continuum dynamics, cosmology, and links with
electromagnetism and QM. To test the theory, an asymptotic scheme of
post-Newtonian approximation has been built. That version of the theory which
is discussed here predicts an internal-structure effect, even at the
point-particle limit. The same might happen also in general relativity (GR) in
some gauges, if one would use a similar scheme. Adjusting the equations of
planetary motion on an ephemeris leaves a residual difference with it; one
should adjust the equations using primary observations. The same effects on
light rays are predicted as with GR, and a similar energy loss applies to
binary pulsars.Comment: Standard LaTeX, 60 pages. Invited contribution to the book ``Ether,
Spacetime and Cosmology" (M. C. Duffy, ed.), to appear at Hadronic Press. v2:
minor improvements, new refs., post-scriptum summarizing later wor
Handling congestion in crowd motion modeling
We address here the issue of congestion in the modeling of crowd motion, in
the non-smooth framework: contacts between people are not anticipated and
avoided, they actually occur, and they are explicitly taken into account in the
model. We limit our approach to very basic principles in terms of behavior, to
focus on the particular problems raised by the non-smooth character of the
models. We consider that individuals tend to move according to a desired, or
spontanous, velocity. We account for congestion by assuming that the evolution
realizes at each time an instantaneous balance between individual tendencies
and global constraints (overlapping is forbidden): the actual velocity is
defined as the closest to the desired velocity among all admissible ones, in a
least square sense. We develop those principles in the microscopic and
macroscopic settings, and we present how the framework of Wasserstein distance
between measures allows to recover the sweeping process nature of the problem
on the macroscopic level, which makes it possible to obtain existence results
in spite of the non-smooth character of the evolution process. Micro and macro
approaches are compared, and we investigate the similarities together with deep
differences of those two levels of description
An alternative approach to the galactic dark matter problem
We discuss scenarios in which the galactic dark matter in spiral galaxies is
described by a long range coherent field which settles in a stationary
configuration that might account for the features of the galactic rotation
curves. The simplest possibility is to consider scalar fields, so we discuss in
particular, two mechanisms that would account for the settlement of the scalar
field in a non-trivial configuration in the absence of a direct coupling of the
field with ordinary matter: topological defects, and spontaneous scalarization.Comment: 36 pages, 12 figures, Revtex, a brief discussion added, accepted for
publication in PR
Recommended from our members
Variational Methods for Evolution (hybrid meeting)
Variational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also time-incremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations)
thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multi-agent systems, and in data science.
This workshop brought together a broad spectrum of researchers from
calculus of variations, partial differential equations, metric
geometry, and stochastics, as well as applied and computational
scientists to discuss and exchange ideas. It focused on variational
tools such as minimizing movement schemes,
optimal transport, gradient flows, and large-deviation principles for
time-continuous Markov processes, -convergence and homogenization
Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics
In this PhD thesis, we develop a new approach to classical gravity starting
from Mach's principles and the idea that the local shape of spatial
configurations is fundamental. This new theory, "shape dynamics", is equivalent
to general relativity but differs in an important respect: shape dynamics is a
theory of dynamic conformal 3-geometry, not a theory of spacetime. Equivalence
is achieved by trading foliation invariance for local conformal invariance (up
to a global scale). After the trading, what is left is a gauge theory invariant
under 3d diffeomorphisms and conformal transformations that preserve the volume
of space. The local canonical constraints are linear and the constraint algebra
closes with structure constants. Shape dynamics, thus, provides a novel new
starting point for quantum gravity.
The procedure for the trading of symmetries was inspired by a technique
called "best matching". We explain best matching and its relation to Mach's
principles. The key features of best matching are illustrated through finite
dimensional toy models. A general picture is then established where relational
theories are treated as gauge theories on configuration space. Shape dynamics
is then constructed by applying best matching to conformal geometry. We then
study shape dynamics in more detail by computing its Hamiltonian and
Hamilton-Jacobi functional perturbatively.
This thesis is intended as a pedagogical but complete introduction to shape
dynamics and the Machian ideas that led to its discovery. The reader is
encouraged to start with the introduction, which gives a conceptual outline and
links to the relevant sections in the text for a more rigorous exposition. When
full rigor is lacking, references to the literature are given. It is hoped that
this thesis may provide a starting point for anyone interested in learning
about shape dynamics.Comment: 117 pages, 2 tables, 10 figures, PhD thesi
POD-based reduced order model for flows induced by rigid solids in forced rotation
This paper deals with the construction of reduced order models (ROMs) for the simulation of the interaction between a fluid and a rigid solid with imposed rotation velocity. The approach is a follows. First, we derive a monolithic description of the fluid/structure interaction by extending the Navier-Stokes equations from the fluid domain to the solid (rotor) domain similarly to the fictitious-domain approach. Second, we build a ROM by a proper orthogonal decomposition (POD) of the resulting multi-phases flow. This method consists in (i) constructing an optimal albeit empirical spatial basis for a very small sub-space of the solution space, and (ii) projecting the governing equations on this reduced basis. Third, we cope with the reconstruction of the high-dimensional velocity field needed to evaluate the imposed velocity constraint by a POD of the solid membership function. Fourth, we use state of the art method to interpolate between available POD bases to build the proposed POD-ROM for a range of parameters values. The proposed method is applied to an academic configuration and proves efficient in the reconstruction of the velocity in both the fluid and solid domains while substantially reducing the computational cost
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