arXiv subjects

Yuxi Han

Publications and source records attributed to Yuxi Han.

11 recordsLinked to original sources

arXiv2026

Selection principle for Weak KAM solutions on $\mathbb{R}^n$ via large deviation principle for diffusions

Stationary Hamilton-Jacobi equations generally admit many weak KAM solutions. The question is how to select one weak KAM solution that reflects a meaningful physical principle. In this paper, we propose a selection principle for Hamilton-Jacobi equations arising from the large deviation principle of general drift-diffusion processes on $\mathbb{R}^n$ based on the exponential asymptotics of invariant measures. We first characterize the Aubry set as the finitely many equilibria of the drift, using an action estimate from the orthogonal decomposition of the drift. For any boundary data on the Aubry set compatible with the Peierls barrier, we construct the unique coercive weak KAM solution with these boundary data, which is also the maximal locally Lipschitz viscosity solution. The Freidlin-Wentzell graph formula then provides the boundary data on the Aubry set that select the unique weak KAM solution. The selected solution is exactly the rate function in the large deviation principle for invariant measures in the zero noise limit. We also show that this rate function is the vanishing viscosity limit, to which the viscous approximations converge locally uniformly. Finally, under additional regularity assumptions, we show that the long-time and zero-noise limits commute for suitably prepared initial densities.

math.AP↗

arXiv2026

Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability

We study the dependence of the effective Hamiltonian $\overline H_Ω(p)$ on the geometry of periodic perforated domains for convex Hamilton--Jacobi equations with state-constraint boundary conditions. We establish an inf--max representation over smooth test functions for domains with $C^1$ boundary. Using this representation, we derive a quantitative stability estimate in terms of the Hausdorff distance between domains and the reach of one boundary. This estimate yields convergence for non-nested domain approximations whose Hausdorff error is negligible relative to their boundary reach. Examples demonstrate the sharpness of the continuity modulus and show that Hausdorff convergence alone can fail to preserve the effective Hamiltonian, even for smooth domains and the quadratic Hamiltonian.

math.AP↗

arXiv2026

Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space

We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution $U^\varepsilon$, which is the value function of a mean field control problem, converges uniformly as $\varepsilon \to 0$ to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions with multiscale dependence, we prove that the rate of convergence is $O(\sqrt{\varepsilon})$. When the Hamiltonian depends only on the fast variable and the momentum, we establish the sharp convergence rate $O(\varepsilon)$. To the best of our knowledge, this is the first quantitative convergence result extending the optimal rate for first-order Hamilton-Jacobi equations in finite dimensions to the Wasserstein space. Finally, we show that our analysis extends to dynamic optimal transport problems, where the terminal condition imposes a constraint on the final distribution.

math.AP↗

arXiv2025

Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions

We study the periodic homogenization of convex Hamilton-Jacobi equations on perforated domains with Dirichlet boundary conditions. By analyzing the optimal control representation of the solutions and the properties of the metric function associated with the running cost, we establish the optimal convergence rate $\mathcal{O}(\varepsilon)$ for homogenization. A key aspect of our approach is the treatment of the singularity that arises when the optimal path does not fully utilize the available time.

math.AP↗

arXiv2025

State-Constrained Chemical Reactions: Discrete-to-Continuous Hamilton--Jacobi Equations and Large Deviations

We study the macroscopic behavior of chemical reactions modeled as random time-changed Poisson processes on discrete state spaces. Using the WKB reformulation, the backward equation of the rescaled process leads to a discrete Hamilton--Jacobi equation with state constraints. As the grid size tends to zero, the limiting solution and its associated variational representation are closely connected to the good rate function of the large deviation principle for state-constrained chemical reactions in the thermodynamic limit. In this work, we focus on the limiting behavior of discrete Hamilton--Jacobi equations defined on bounded domains with state-constraint boundary conditions. For a single chemical reaction, we show that, under a suitable reparametrization, the solution of the discrete Hamilton--Jacobi equation converges to the solution of a continuous Hamilton--Jacobi equation with a Neumann boundary condition. Building on this convergence result and the associated variational representation, we establish the large deviation principle for the rescaled chemical reaction process in bounded domains.

math.AP↗

arXiv2025

Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations

We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $\mathcal{O}(\sqrt{\varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $\mathcal{O}(\varepsilon)$ for nonnegative compactly supported data and $\mathcal{O}(\varepsilon^{1/(p-\frac{1}{2})})$ (where $1<p\leq 2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overlineΩ)$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.

math.AP↗

arXiv2024

CiliaGraph: Enabling Expression-enhanced Hyper-Dimensional Computation in Ultra-Lightweight and One-Shot Graph Classification on Edge

Graph Neural Networks (GNNs) are computationally demanding and inefficient when applied to graph classification tasks in resource-constrained edge scenarios due to their inherent process, involving multiple rounds of forward and backward propagation. As a lightweight alternative, Hyper-Dimensional Computing (HDC), which leverages high-dimensional vectors for data encoding and processing, offers a more efficient solution by addressing computational bottleneck. However, current HDC methods primarily focus on static graphs and neglect to effectively capture node attributes and structural information, which leads to poor accuracy. In this work, we propose CiliaGraph, an enhanced expressive yet ultra-lightweight HDC model for graph classification. This model introduces a novel node encoding strategy that preserves relative distance isomorphism for accurate node connection representation. In addition, node distances are utilized as edge weights for information aggregation, and the encoded node attributes and structural information are concatenated to obtain a comprehensive graph representation. Furthermore, we explore the relationship between orthogonality and dimensionality to reduce the dimensions, thereby further enhancing computational efficiency. Compared to the SOTA GNNs, extensive experiments show that CiliaGraph reduces memory usage and accelerates training speed by an average of 292 times(up to 2341 times) and 103 times(up to 313 times) respectively while maintaining comparable accuracy.

cs.LG↗

arXiv2024

Quantitative homogenization of state-constraint Hamilton--Jacobi equations on perforated domains and applications

We study the periodic homogenization problem of state-constraint Hamilton--Jacobi equations on perforated domains in the convex setting and obtain the optimal convergence rate. We then consider a dilute situation in which the holes' diameter is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed.

math.AP↗

arXiv2023

Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with Multiscales

We study the rate of convergence in periodic homogenization for convex Hamilton--Jacobi equations with multiscales, where the Hamiltonian $H=H(x, y, p): \mathbb{R}^n \times \mathbb{T}^n \times \mathbb{R}^n \to \mathbb{R }$ depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is $O(\sqrtε)$ by optimal control formulas, scale separations and curve cutting techniques. We also show the rate $O(\sqrtε)$ of homogenization for the static problem based on the same idea. Additionally, we provide examples that illustrate the rate of convergence for the Cauchy problem is optimal.

math.AP↗

arXiv2022

Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints

We focus on the global semiconcavity of solutions to first-order Hamilton--Jacobi equations with state constraints, especially for the Hamiltonian $H(x, β):=|β|^p-f(x)$ with $p \in (1, 2]$. We first show that the solution is locally semiconcave, and the semiconcavity constant at each point depends on the first time a corresponding minimizing curve emanating from this point hits the boundary. Then, with appropriate conditions on $Df$, we prove that for any such minimizing curve, the time it takes to hit the boundary of the domain is $+\infty$, and as a consequence, the solution is globally semiconcave. Moreover, the condition on $Df$ is essentially optimal with examples in one-dimensional space. The proofs employ the Euler-Lagrange equations and techniques in weak KAM theory.

math.AP↗

arXiv2020

Risk-sensitive safety specifications for stochastic systems using Conditional Value-at-Risk

This paper proposes a safety analysis method that facilitates a tunable balance between the worst-case and risk-neutral perspectives. First, we define a risk-sensitive safe set to specify the degree of safety attained by a stochastic system. This set is defined as a sublevel set of the solution to an optimal control problem that is expressed using the Conditional Value-at-Risk (CVaR) measure. This problem does not satisfy Bellman's Principle, thus our next contribution is to show how risk-sensitive safe sets can be under-approximated by the solution to a CVaR-Markov Decision Process. We adopt an existing value iteration algorithm to find an approximate solution to the reduced problem for a class of linear systems. Then, we develop a realistic numerical example of a stormwater system to show that this approach can be applied to non-linear systems. Finally, we compare the CVaR criterion to the exponential disutility criterion. The latter allocates control effort evenly across the cost distribution to reduce variance, while the CVaR criterion focuses control effort on a given worst-case quantile--where it matters most for safety.

eess.SY↗