arXiv subjects

Raymond Chu

Publications and source records attributed to Raymond Chu.

6 recordsLinked to original sources

arXiv2026

Rate-Optimal Noise Annealing in Semi-Dual Neural Optimal Transport: Tangential Identifiability, Off-Manifold Ambiguity, and Guaranteed Recovery

Semi-dual neural optimal transport learns a transport map via a max-min objective, yet training can converge to incorrect or degenerate maps. We fully characterize these spurious solutions in the common regime where data concentrate on low-dimensional manifold: the objective is underconstrained off the data manifold, while the on-manifold transport signal remains identifiable. Following Choi, Choi, and Kwon (2025), we study additive-noise smoothing as a remedy and prove new map recovery guarantees as the noise vanishes. Our main practical contribution is a computable terminal noise level $\varepsilon_{\mathrm{stat}}(N)$ that attains the optimal statistical rate, with scaling governed by the intrinsic dimension $m$ of the data. The formula arises from a theoretical unified analysis of (i) quantitative stability of optimal plans, (ii) smoothing-induced bias, and (iii) finite-sample error, yielding rates that depend on $m$ rather than the ambient dimension. Finally, we show that the reduced semi-dual objective becomes increasingly ill-conditioned as $\varepsilon \downarrow 0$. This provides a principled stopping rule: annealing below $\varepsilon_{\mathrm{stat}}(N)$ can $\textit{worsen}$ optimization conditioning without improving statistical accuracy.

cs.LG↗

arXiv2025

The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions

We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then establish regularity properties of maximal solutions, which are obtained by maximizing a suitable notion of "average" freezing time. Unexpectedly, we show that maximal solutions have a transition zone that is open modulo a low-dimensional set: this allows us to apply obstacle problem theory for a finer regularity analysis. We further show that maximal solutions are in general non-universal, and we obtain sharp stability results under perturbation of each maximal solution. Lastly, we study maximal solutions in both the radial and the one-dimensional setting. We show that in these cases the maximal solution is universal and minimizes nucleation, in agreement with phenomena observed in the physics literature.

math.AP↗

arXiv2025

Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $\lambda$-displacement convexity (for any $\lambda\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $\lambda$-displacement convex (with $\lambda>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $\lambda$-displacement convex and Wasserstein differentiable.

math.AP↗

arXiv2023

The Nonlocal Stefan Problem via a Martingale Transport

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct global-time weak solutions and give a probabilistic interpretation for the solutions. An important ingredient in our analysis is a probabilistic interpretation of the enthalpy and temperature variables in terms of a particle system. Our approach in particular establishes the connection between the parabolic obstacle problem and the Stefan Problem for the nonlocal diffusions. For the melting problem, we show that our solution coincides with those studied in the literature, and obtain a new exponential convergence result.

math.AP↗

arXiv2022

A Hele-Shaw Limit With A Variable Upper Bound and Drift

We investigate a generalized Hele-Shaw equation with a source and drift terms where the density is constrained by an upper-bound density constraint that varies in space and time. By using a generalized porous medium equation approximation, we are able to construct a weak solution to the generalized Hele-Shaw equations under mild assumptions. Then we establish uniqueness of weak solutions to the generalized Hele-Shaw equations. Our next main result is a pointwise characterization of the density variable in the generalized Hele-Shaw equations when the system is in the congestion case. To obtain such a characterization for the congestion case, we derive a new uniform lower bounds on the time derivative pressure of the generalized porous medium equation via a refined Aronson-Benilan estimate that implies monotonicity on the density and pressure.

math.AP↗

arXiv2021

Fast Asymptotic-Numerical Method For Coarse Mesh Particle Simulation In Channels Of Arbitrary Cross Section

Particles traveling through inertial microfluidic devices migrate to focusing streamlines. We present a numerical method that calculates migration velocities of particles in inertial microfluidic channels of arbitrary cross section by representing particles by singularities. Refinements to asymptotic analysis are given that improve the regularity of the singularity approximation, making finite element approximations of flow and pressure fields more effective. Sample results demonstrate that the method is computationally efficient and able to capture bifurcations in particle focusing positions due to changes in channel shape and Reynolds number.

physics.flu-dyn↗