arXiv subjects

Yuanlong Ruan

Publications and source records attributed to Yuanlong Ruan.

6 recordsLinked to original sources

arXiv2026

Central limit theorem for Wasserstein projection - the case of convex order

The main focuses of the article are limit theorems of Wasserstein projection in the convex order which are useful for inference tasks. The main results rely on the establishment of dual attainment, stability and several useful observations. The first is a clean criterion to invoke the Wasserstein projection dualities on the classical cost $c=h(x-y)$. A new strategy was introduced to establish the dual attainment. Backward and forward dual transport are proved to enjoy a conjugate relationship. Sufficient conditions are given that ensure uniqueness of the optimal dual potential. These ingredients laid the groundwork for the proof of optimal dual potential stability, thereby permitting us to prove the central limit theorems under mild moment assumptions. Additional discussions show that the moment assumptions are sharp. These results also answered several open questions raised by Professor Benjamin Jourdain.

math.PR↗

arXiv2025

Limit of quasilinear equations and related extremal problems

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies. However the solutions converge uniformly to solution of a limiting equation irrelevant to the Lipschitz constant of g. The limiting equation has no coercivity in u. We prove that the limiting equation admits a weak comparison principle and has a unique viscosity solution. We also obtain a Poincare inequality in the Sobolev-Orlicz space for discontinuous operator, which paves the way for our study of an extremal problem where its operator becomes unbounded in a subdomain. Upon giving proper meaning to its solution, we show that the extremal problem has a unique solution. It turns out the solution has sufficient continuity, although operator is discontinuous. In the appendix we provide some technical inequalities which play crucial roles in the proof of uniqueness and we believe will be of independent interest.

math.AP↗

arXiv2025

Statistical inference of convex order by Wasserstein projection

Ranking distributions according to a stochastic order has wide applications in diverse areas. Although stochastic dominance has received much attention, convex order, particularly in general dimensions, has yet to be investigated from a statistical point of view. This article addresses this gap by introducing a simple statistical test for convex order based on the Wasserstein projection distance. This projection distance not only encodes whether two distributions are indeed in convex order, but also quantifies the deviation from the desired convex order and produces an optimal convex order approximation. Lipschitz stability of the backward and forward Wasserstein projection distance is proved, which leads to elegant consistency and concentration results of the estimator we employ as our test statistic. Combining these with state of the art results regarding the convergence rate of empirical distributions, we also derive upper bounds for the $p$-value and type I error of our test statistic, as well as upper bounds on the type II error for an appropriate class of strict alternatives. With proper choices of families of distributions, we further attain that the power of the proposed test increases to one as the number of samples grows to infinity. Lastly, we provide an efficient numerical scheme for our test statistic, by way of an entropic Frank-Wolfe algorithm. Experiments based on synthetic data sets illuminate the success of our approach.

stat.ME↗

arXiv2017

A partial differential equation for the rank one convex envelope

In this article we introduce a Partial Differential Equation (PDE) for the rank one convex envelope. Rank one convex envelopes arise in non-convex vector valued variational problems \cite{BallElasticity, kohn1986optimal1, BallJames87, chipot1988equilibrium}. More generally, we study a PDE for directional convex envelopes, which includes the usual convex envelope \cite{ObermanConvexEnvelope} and the rank one convex envelope as special cases. Existence and uniqueness of viscosity solutions to the PDE is established. Wide stencil elliptic finite difference schemes are built. Convergence of finite difference solutions to the viscosity solution of the PDE is proven. Numerical examples of rank one and other directional convex envelopes are presented. Additionally, laminates are computed from the rank one convex envelope.

math.AP↗

arXiv2016

Nontrivial Periodic Minimizer for Landau-Brazovskii Model with Constraint

Block copolymer, a synthesized polymer material, has found many applications in industry. It is consisting of multiple sequences of monomer alternating in series with different monomer blocks. The combination of different polymers endows the polymer material with rich properties, which are the key to their important applications. In this paper, we model the copolymers with Landau-Brazovskii model with additional constraints reflecting physical structures, which is in the form of a second order variational problem. Critical points of the functional are interpreted as states of polymers. By reducing to handy situations, we find a nontrivial periodic minimal solution. Moreover, the proof is kept as simple and self-contained as possible in our specific case.

math-ph↗

arXiv2015

An efficient linear programming method for Optimal Transportation

An efficient method for computing solutions to the Optimal Transportation (OT) problem with a wide class of cost functions is presented. The standard linear programming (LP) discretization of the continuous problem becomes intractible for moderate grid sizes. A grid refinement method results in a linear cost algorithm. Weak convergence of solutions is stablished. Barycentric projection of transference plans is used to improve the accuracy of solutions. The method is applied to more general problems, including partial optimal transportation, and barycenter problems. Computational examples validate the accuracy and efficiency of the method. Optimal maps between nonconvex domains, partial OT free boundaries, and high accuracy barycenters are presented.

math.NA↗