592 research outputs found
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions
We address the discretization of optimization problems posed on the cone of
convex functions, motivated in particular by the principal agent problem in
economics, which models the impact of monopoly on product quality. Consider a
two dimensional domain, sampled on a grid of N points. We show that the cone of
restrictions to the grid of convex functions is in general characterized by N^2
linear inequalities; a direct computational use of this description therefore
has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of
discrete convex functions, associated to stencils which can be adaptively,
locally, and anisotropically refined. Numerical experiments optimize the
accuracy/complexity tradeoff through the use of a-posteriori stencil refinement
strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2.
Modifications are anecdotic.
Two geometrically frustrated magnets studied by neutron diffraction
In the pyrochlore compounds, TbTiO and TbSnO, only
the Tb ions are magnetic. They exhibit quite abnormal -- and, in view of
their chemical similarity, strikingly different -- magnetic behaviour, as
probed by neutron diffraction at ambient and applied pressure.
TbTiO is a cooperative paramagnet (`spin liquid'), without long
range order at ambient pressure; however, it does become ordered under
pressure. By contrast, TbSnO enters an "ordered spin ice" state
already at ambient pressure. We analyse a simple model which already clearly
exhibits some of the qualitative features observed experimentally. Overall,
comparing these two compounds emphasizes the power of small perturbations in
selecting low-temperature states in geometrically frustrated systems.Comment: 6 pages, 4 figures, International Conference on Neutron Scattering
Sydney(2005
Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction
We introduce a modification of the Fast Marching Algorithm, which solves the
generalized eikonal equation associated to an arbitrary continuous riemannian
metric, on a two or three dimensional domain. The algorithm has a logarithmic
complexity in the maximum anisotropy ratio of the riemannian metric, which
allows to handle extreme anisotropies for a reduced numerical cost. We prove
the consistence of the algorithm, and illustrate its efficiency by numerical
experiments. The algorithm relies on the computation at each grid point of a
special system of coordinates: a reduced basis of the cartesian grid, with
respect to the symmetric positive definite matrix encoding the desired
anisotropy at this point.Comment: 28 pages, 12 figure
Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance
We consider a two player game, where a first player has to install a
surveillance system within an admissible region. The second player needs to
enter the the monitored area, visit a target region, and then leave the area,
while minimizing his overall probability of detection. Both players know the
target region, and the second player knows the surveillance installation
details.Optimal trajectories for the second player are computed using a
recently developed variant of the fast marching algorithm, which takes into
account curvature constraints modeling the second player vehicle
maneuverability. The surveillance system optimization leverages a reverse-mode
semi-automatic differentiation procedure, estimating the gradient of the value
function related to the sensor location in time N log N
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
We introduce a numerical method for extracting minimal geodesics along the
group of volume preserving maps, equipped with the L2 metric, which as observed
by Arnold solve Euler's equations of inviscid incompressible fluids. The method
relies on the generalized polar decomposition of Brenier, numerically
implemented through semi-discrete optimal transport. It is robust enough to
extract non-classical, multi-valued solutions of Euler's equations, for which
the flow dimension is higher than the domain dimension, a striking and
unavoidable consequence of this model. Our convergence results encompass this
generalized model, and our numerical experiments illustrate it for the first
time in two space dimensions.Comment: 21 pages, 9 figure
A simple preconditioned domain decomposition method for electromagnetic scattering problems
We present a domain decomposition method (DDM) devoted to the iterative
solution of time-harmonic electromagnetic scattering problems, involving large
and resonant cavities. This DDM uses the electric field integral equation
(EFIE) for the solution of Maxwell problems in both interior and exterior
subdomains, and we propose a simple preconditioner for the global method, based
on the single layer operator restricted to the fictitious interface between the
two subdomains.Comment: 23 page
Monotone and Consistent discretization of the Monge-Ampere operator
We introduce a novel discretization of the Monge-Ampere operator,
simultaneously consistent and degenerate elliptic, hence accurate and robust in
applications. These properties are achieved by exploiting the arithmetic
structure of the discrete domain, assumed to be a two dimensional cartesian
grid. The construction of our scheme is simple, but its analysis relies on
original tools seldom encountered in numerical analysis, such as the geometry
of two dimensional lattices, and an arithmetic structure called the
Stern-Brocot tree. Numerical experiments illustrate the method's efficiency
Spin-Charge Separation in Two-dimensional Frustrated Quantum Magnets
The dynamics of a mobile hole in two-dimensional frustrated quantum magnets
is investigated by exact diagonalization techniques. Our results provide
evidence for spin-charge separation upon doping the kagome lattice, a prototype
of a spin liquid. In contrast, in the checkerboard lattice, a symmetry broken
Valence Bond Crystal, a small quasi-particle peak is seen for some crystal
momenta, a finding interpreted as a restoration of weak holon-spinon
confinement.Comment: 4 pages, 6 figure
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