arXiv subjects

Young-Heon Kim

Publications and source records attributed to Young-Heon Kim.

At least 37 records · Page 2Linked to original sources

arXiv2020

Hidden convexity in a problem of nonlinear elasticity

We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.

math.AP↗

arXiv2020

A solution to the Monge transport problem for Brownian martingales

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable, as well as a stochastic version of the standard "twist condition" frequently used in deterministic Monge transport theory. This setting includes in particular the case of the distance cost $c(x,y)=|x-y|$. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

math.AP↗

arXiv2019

Optimal transport and barycenters for dendritic measures

We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our motivation is the comparison of and interpolation between plants' root systems. We characterize barycenters with respect to this metric, and establish that the interpolations of root-like measures, using this new metric, are also root like, in a certain sense; this property fails for conventional Wasserstein barycenters. We also establish geodesic convexity with respect to this metric for a variety of functionals, some of which we expect to have biological importance.

math.OC↗

arXiv2019

Stochastic optimal transport with free end time

We consider a stochastic transportation problem between two prescribed probability distributions (a source and a target) over processes with general drift dependence and with free end times. First, and in order to establish a dual principle, we associate two equivalent formulations of the primal problem in order to guarantee its convexity and lower semi-continuity with respect to the source and target distributions. We exhibit an equivalent Eulerian formulation, whose dual variational principle is given by Hamilton-Jacobi-Bellman type variational inequalities. In the case where the dependence on the drift is bounded, regularity results on the minimizers of the Eulerian problem then enable us to prove attainment in the corresponding dual problem. We also address attainment when the drift component of the cost defining Lagrangian $L$ is superlinear $L \approx |u|^p$ with $1<p<2$, in which case the setting is reminiscent of our approach -- in a previous work -- on deterministic controlled transport problems with free end time. We finally address criteria under which the optimal drift and stopping time are unique, namely strict convexity in the drift component and monotonicity in time of the Lagrangian.

math.OC↗

arXiv2019

Optimal Brownian stopping when the source and target are radially symmetric distributions

Given two probability measures $μ, ν$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $τ$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $μ$ and with final distribution --once stopped at $τ$-- equal to $ν$. Under the assumption of radial symmetry on $μ$ and $ν$, we show that in dimension $d \geq 3$ and $α\neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales, and establish a duality result. This paper is an expanded version of a previously posted but not published work by the authors.

math.AP↗

arXiv2019

PDE Methods For Optimal Skorokhod Embeddings

We consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. PDEs and a free boundary problem approach are used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths. We introduce an Eulerian---mass flow---formulation of the problem, whose dual is given by Hamilton-Jacobi-Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglböck, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

math.AP↗

arXiv2018

Optimal Transport with Controlled Dynamics and Free End Times

We consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal "stopping plans," and the corresponding Monge-Kantorovich duality involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality. We discuss both Lagrangian and Eulerian formulations of the problem, and its natural connection to Pontryagin's maximum principle. We also exhibit a purely dynamic situation, where the optimal stopping plan is a hitting time of a barrier given by the free boundary problem associated to the dual variational inequality. This problem was motivated by its stochastic counterpart, which will be studied in a companion paper.

math.OC↗

arXiv2017

Optimal Brownian Stopping between radially symmetric marginals in general dimensions

Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stopped at $τ$-- equal to $B_τ\sim ν$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in "subharmoic order" into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely the optimal subharmonic martingale transport. Under the assumption of radial symmetry on $μ$ and $ν$, we show that the optimal stopping time is a hitting time of a suitable barrier, hence is non-randomized and is unique.

math.PR↗

arXiv2017

A canonical barycenter via Wasserstein regularization

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $μ$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given measure $μ$ forward to a measure on $P(X)$. We then regularize the measure by the Wasserstein distance to the reference measure ${\bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $μ$. We investigate various properties of $B(μ)$

math.OC↗

arXiv2016

Emergence of phase concentration for the Kuramoto-Sakaguchi equation

We study the asymptotic phase concentration phenomena for the Kuramoto-Sakaguchi(K-S) equation in a large coupling strength regime. For this, we analyze the detailed dynamics of the order parameters such as the amplitude and the average phase. For the infinite ensemble of oscillators with the identical natural frequency, we show that the total mass distribution concentrates on the average phase asymptotically, whereas the mass around the antipodal point of the average phase decays to zero exponentially fast in any positive coupling strength regime. Thus, generic initial kinetic densities evolve toward the Dirac measure concentrated on the average phase. In contrast, for the infinite ensemble with distributed natural frequencies, we find a certain time-dependent interval whose length can be explicitly quantified in terms of the coupling strength. Provided that the coupling strength is sufficiently large, the mass on such an interval is eventually non-decreasing over the time. We also show that the amplitude order parameter has a positive lower bound that depends on the size of support of the distribution function for the natural frequencies and the coupling strength. The proposed asymptotic lower bound on the order parameter tends to unity, as the coupling strength increases to infinity. This is reminiscent of practical synchronization for the Kuramoto model, in which the diameter for the phase configuration is inversely proportional to the coupling strength. Our results for the K-S equation generalize the results in [19] on the emergence of phase-locked states for the Kuramoto model in a large coupling strength regime.

math.AP↗

arXiv2016

Structure of optimal martingale transport plans in general dimensions

Given two probability measures $μ$ and $ν$ in "convex order" on $\R^d$, we study the profile of one-step martingale plans $π$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all martingales having $μ$ and $ν$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure $μ$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $μ$-almost every $x$ in $\R^d$ is transported by the optimal martingale plan into a probability measure $π_x$ concentrated on the extreme points of the closed convex hull of its support. This will be established in full generality in the 2-dimensional case, and also for any $d\geq 3$ as long as the marginals are in "subharmonic order". In some cases, $π_x$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of $\R^d\times \R^d$ into a collection of mutually disjoint components by means of a "convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to natural and intriguing constructions of higher dimensional Nikodym sets.

math.AP↗

arXiv2015

Wasserstein Barycenters over Riemannian manifolds

We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures $Ω\in P(P(M))$ on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier \cite{ac} to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger \cite{c-ems} for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where $Ω$ is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

math.AP↗

arXiv2015

Dynamics of Optimal Partial Transport

This paper considers the evolution dynamics of the free boundaries in terms of the change of $m$, the allowed amount of transported mass or the change of $λ$, the transportation cost cap, i.e. the allowed maximum cost for a unit mass to be transported. Focusing on the quadratic cost function, we show Hölder and Lipschitz estimates on the speed of the free boundary motion in terms of $m$ and $λ$, respectively. It is also shown that the parameter $m$ is a Lipschitz function of $λ$, which previously was known only to be a continuous increasing function \cite{Ca-Mc}.

math.AP↗

arXiv2015

Nonpositive curvature, the variance functional, and the Wasserstein barycenter

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space $P(M)$ of probability measures over $M$. We show that $M$ has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if and only if the variance functional on $P(M)$ is displacement convex. This is followed by a Jensen type inequality for the variance functional with respect to Wasserstein barycenters, as well as by a result comparing the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$ (that is, an element of $P(P(M))$). These results are applied to invariant measures under isometry group actions, giving a comparison for the variance functional between the Wasserstein projection and the $L^2$ projection to the set of invariant measures.

math.DG↗

arXiv2014

Prohibiting isolated singularities in optimal transport

We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular points.

math.AP↗

arXiv2013

A general condition for Monge solutions in the multi-marginal optimal transport problem

We develop a general condition on the cost function which is sufficient to imply Monge solution and uniqueness results in the multi-marginal optimal transport problem. This result unifies and generalizes several results in the rather fragmented literature on multi-marginal problems. We also provide a systematic way to generate new examples from old ones.

math.AP↗

arXiv2013

Multi-marginal optimal transport on Riemannian manifolds

We study a multi-marginal optimal transportation problem on a Riemannian manifold, with cost function given by the average distance squared from multiple points to their barycenter. Under a standard regularity condition on the first marginal, we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable, thus inducing a Monge solution. This result generalizes McCann's polar factorization theorem on manifolds from 2 to several marginals, in the same sense that a well known result of Gangbo and Swiech generalizes Brenier's polar factorization theorem on $\mathbb{R}^n$.

math.AP↗

arXiv2012

On the degeneracy of optimal transportation

We extend a dimensional upper bound on how much an optimal transport map can degenerate for the quadratic transportation cost, originally due to Caffarelli, to cost functions that satisfy the curvature condition of Ma, Trudinger, and Wang.

math.AP↗