140 research outputs found
A geometric optics method for high-frequency electromagnetic fields computations near fold caustics—Part II. The energy
AbstractWe present the computation of the amplitudes needed to evaluate the energy deposited by the laser wave in a plasma when a fold caustic forms. We first recall the Eulerian method designed in Benamou et al. (J. Comput. Appl. Math. 156 (2003) 93) to compute the caustic location and the phases associated to the two ray branches on its illuminated side. We then turn to the computation of the amplitudes needed to evaluate the energy. We use the classical geometrical form of the amplitudes to avoid the blow up problem at the caustic. As our proposed method is Eulerian we have to consider transport equations for these geometrical quantities where the advection field depends on the ray flow. The associated vector field structurally vanishes like the square root of the distance to the caustic when approaching the caustic. This introduces an additional difficulty as traditional finite difference scheme do not retain their accuracy for such advection fields. We propose a new scheme which remains of order 1 at the caustic and present a partial theoretical analysis as well as a numerical validation. We also test the capability of our Eulerian geometrical algorithm to produce numerical solution of the Helmholtz equation and attempt to check their frequency asymptotic accuracy
Limiting absorption principle for the dissipative Helmholtz equation
Adapting Mourre's commutator method to the dissipative setting, we prove a
limiting absorption principle for a class of abstract dissipative operators. A
consequence is the resolvent estimates for the high frequency Helmholtz
equation when trapped trajectories meet the set where the imaginary part of the
potential is non-zero. We also give the resolvent estimates in Besov spaces
A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations
We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the
dependence of the coefficients is nonlinear and nonlocal in time with respect
to the unknowns. We extend the numerical scheme proposed and studied recently
by the authors for a single FPK equation of this type. We analyse the
convergence of the scheme and we study its applicability in two examples. The
first one concerns a population model involving two interacting species and the
second one concerns two populations Mean Field Games
Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints
In this paper we consider a class of optimization problems with a strongly
convex objective function and the feasible set given by an intersection of a
simple convex set with a set given by a number of linear equality and
inequality constraints. A number of optimization problems in applications can
be stated in this form, examples being the entropy-linear programming, the
ridge regression, the elastic net, the regularized optimal transport, etc. We
extend the Fast Gradient Method applied to the dual problem in order to make it
primal-dual so that it allows not only to solve the dual problem, but also to
construct nearly optimal and nearly feasible solution of the primal problem. We
also prove a theorem about the convergence rate for the proposed algorithm in
terms of the objective function and the linear constraints infeasibility.Comment: Submitted for DOOR 201
Geodesics in the space of measure-preserving maps and plans
We study Brenier's variational models for incompressible Euler equations.
These models give rise to a relaxation of the Arnold distance in the space of
measure-preserving maps and, more generally, measure-preserving plans. We
analyze the properties of the relaxed distance, we show a close link between
the Lagrangian and the Eulerian model, and we derive necessary and sufficient
optimality conditions for minimizers. These conditions take into account a
modified Lagrangian induced by the pressure field. Moreover, adapting some
ideas of Shnirelman, we show that, even for non-deterministic final conditions,
generalized flows can be approximated in energy by flows associated to
measure-preserving maps
Fast Optimal Transport Averaging of Neuroimaging Data
Knowing how the Human brain is anatomically and functionally organized at the
level of a group of healthy individuals or patients is the primary goal of
neuroimaging research. Yet computing an average of brain imaging data defined
over a voxel grid or a triangulation remains a challenge. Data are large, the
geometry of the brain is complex and the between subjects variability leads to
spatially or temporally non-overlapping effects of interest. To address the
problem of variability, data are commonly smoothed before group linear
averaging. In this work we build on ideas originally introduced by Kantorovich
to propose a new algorithm that can average efficiently non-normalized data
defined over arbitrary discrete domains using transportation metrics. We show
how Kantorovich means can be linked to Wasserstein barycenters in order to take
advantage of an entropic smoothing approach. It leads to a smooth convex
optimization problem and an algorithm with strong convergence guarantees. We
illustrate the versatility of this tool and its empirical behavior on
functional neuroimaging data, functional MRI and magnetoencephalography (MEG)
source estimates, defined on voxel grids and triangulations of the folded
cortical surface.Comment: Information Processing in Medical Imaging (IPMI), Jun 2015, Isle of
Skye, United Kingdom. Springer, 201
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations
We establish a connection between Optimal Transport Theory and classical
Convection Theory for geophysical flows. Our starting point is the model
designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal
Transport problems. This model can be seen as a generalization of the
Darcy-Boussinesq equations, which is a degenerate version of the
Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate
different variants of the NSB equations (in particular what we call the
generalized Hydrostatic-Boussinesq equations) to various models involving
Optimal Transport (and the related Monge-Ampere equation. This includes the 2D
semi-geostrophic equations and some fully non-linear versions of the so-called
high-field limit of the Vlasov-Poisson system and of the Keller-Segel for
Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model
can be related to the magnetic relaxation model studied by Arnold and Moffatt
to obtain stationary solutions of the Euler equations with prescribed topology
Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
We study a singular-limit problem arising in the modelling of chemical
reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck
convection-diffusion equation with a double-well convection potential. This
potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the
solution concentrates onto the two wells, resulting into a limiting system that
is a pair of ordinary differential equations for the density at the two wells.
This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM
Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear
structure of the equation. In this paper we re-prove the result by using solely
the Wasserstein gradient-flow structure of the system. In particular we make no
use of the linearity, nor of the fact that it is a second-order system. The
first key step in this approach is a reformulation of the equation as the
minimization of an action functional that captures the property of being a
curve of maximal slope in an integrated form. The second important step is a
rescaling of space. Using only the Wasserstein gradient-flow structure, we
prove that the sequence of rescaled solutions is pre-compact in an appropriate
topology. We then prove a Gamma-convergence result for the functional in this
topology, and we identify the limiting functional and the differential equation
that it represents. A consequence of these results is that solutions of the
{\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference
A glimpse into the differential topology and geometry of optimal transport
This note exposes the differential topology and geometry underlying some of
the basic phenomena of optimal transportation. It surveys basic questions
concerning Monge maps and Kantorovich measures: existence and regularity of the
former, uniqueness of the latter, and estimates for the dimension of its
support, as well as the associated linear programming duality. It shows the
answers to these questions concern the differential geometry and topology of
the chosen transportation cost. It also establishes new connections --- some
heuristic and others rigorous --- based on the properties of the
cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page
Ricci curvature of finite Markov chains via convexity of the entropy
We study a new notion of Ricci curvature that applies to Markov chains on
discrete spaces. This notion relies on geodesic convexity of the entropy and is
analogous to the one introduced by Lott, Sturm, and Villani for geodesic
measure spaces. In order to apply to the discrete setting, the role of the
Wasserstein metric is taken over by a different metric, having the property
that continuous time Markov chains are gradient flows of the entropy.
Using this notion of Ricci curvature we prove discrete analogues of
fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that
Ricci curvature bounds are preserved under tensorisation. As a special case we
obtain the sharp Ricci curvature lower bound for the discrete hypercube.Comment: 39 pages, to appear in Arch. Ration. Mech. Ana
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