39 research outputs found
Harmonic mappings valued in the Wasserstein space
We propose a definition of the Dirichlet energy (which is roughly speaking
the integral of the square of the gradient) for mappings mu : Omega -> (P(D),
W\_2) defined over a subset Omega of R^p and valued in the space P(D) of
probability measures on a compact convex subset D of R^q endowed with the
quadratic Wasserstein distance. Our definition relies on a straightforward
generalization of the Benamou-Brenier formula (already introduced by Brenier)
but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit
of approximate Dirichlet energies, and to the definition of Reshetnyak of
Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e.
minimizers of the Dirichlet energy provided that the values on the boundary d
Omega are fixed. The notion of constant-speed geodesics in the Wasserstein
space is recovered by taking for Omega a segment of R. As the Wasserstein space
(P(D), W\_2) is positively curved in the sense of Alexandrov we cannot apply
the theory of Koorevaar, Schoen and Jost and we use instead arguments based on
optimal transport. We manage to get existence of harmonic mappings provided
that the boundary values are Lipschitz on d Omega, uniqueness is an open
question. If Omega is a segment of R, it is known that a curve valued in the
Wasserstein space P(D) can be seen as a superposition of curves valued in D. We
show that it is no longer the case in higher dimensions: a generic mapping
Omega -> P(D) cannot be represented as the superposition of mappings Omega ->
D. We are able to show the validity of a maximum principle: the composition
F(mu) of a function F : P(D) -> R convex along generalized geodesics and a
harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We
also study the special case where we restrict ourselves to a given family of
elliptically contoured distributions (a finite-dimensional and geodesically
convex submanifold of (P(D), W\_2) which generalizes the case of Gaussian
measures) and show that it boils down to harmonic mappings valued in the
Riemannian manifold of symmetric matrices endowed with the distance coming from
optimal transport
Unconditional convergence for discretizations of dynamical optimal transport
The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework
Lifting functionals defined on maps to measure-valued maps via optimal transport
How can one lift a functional defined on maps from a space X to a space Y
into a functional defined on maps from X into P(Y) the space of probability
distributions over Y? Looking at measure-valued maps can be interpreted as
knowing a classical map with uncertainty, and from an optimization point of
view the main gain is the convexification of Y into P(Y). We will explain why
trying to single out the largest convex lifting amounts to solve an optimal
transport problem with an infinity of marginals which can be interesting by
itself. Moreover we will show that, to recover previously proposed liftings for
functionals depending on the Jacobian of the map, one needs to add a
restriction of additivity to the lifted functional
Dynamical Optimal Transport on Discrete Surfaces
We propose a technique for interpolating between probability distributions on
discrete surfaces, based on the theory of optimal transport. Unlike previous
attempts that use linear programming, our method is based on a dynamical
formulation of quadratic optimal transport proposed for flat domains by Benamou
and Brenier [2000], adapted to discrete surfaces. Our structure-preserving
construction yields a Riemannian metric on the (finite-dimensional) space of
probability distributions on a discrete surface, which translates the so-called
Otto calculus to discrete language. From a practical perspective, our technique
provides a smooth interpolation between distributions on discrete surfaces with
less diffusion than state-of-the-art algorithms involving entropic
regularization. Beyond interpolation, we show how our discrete notion of
optimal transport extends to other tasks, such as distribution-valued Dirichlet
problems and time integration of gradient flows
Dynamical optimal transport on discrete surfaces
We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between
distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows
Mechanical magnetometry of Cobalt nanospheres deposited by focused electron beam at the tip of ultra-soft cantilevers
Using focused-electron-beam-induced deposition, Cobalt magnetic nanospheres
with diameter ranging between 100 nm and 300 nm are grown at the tip of
ultra-soft cantilevers. By monitoring the mechanical resonance frequency of the
cantilever as a function of the applied magnetic field, the hysteresis curve of
these individual nanospheres are measured. This enables to evaluate their
saturation magnetization, found to be around 430 emu/cm^3 independently of the
size of the particle, and to infer that the magnetic vortex state is the
equilibrium configuration of these nanospheres at remanence
Unconditional convergence for discretizations of dynamical optimal transport
The dynamical formulation of optimal transport, also known as Benamou–Brenier formulation or computational fluid dynamics formulation, amounts to writing the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this paper, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional problem for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer, and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien, and Solomon, fit in this framework