arXiv subjects

Son N. T. Tu

Publications and source records attributed to Son N. T. Tu.

13 recordsLinked to original sources

arXiv2026

Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. These estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

math.AP↗

arXiv2026

Quantitative homogenization for static contact Hamilton-Jacobi equations

We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton--Jacobi equations $$ H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilonΔu_\varepsilon+c \right) $$ for all $\varepsilon>0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=Δu+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$.

math.AP↗

arXiv2026

Operator Splitting, Policy Iteration, and Machine Learning for Stochastic Optimal Control

We propose a splitting approach to solve the second-order Hamilton--Jacobi equation, reducing it to a heat step and a purely first-order step. The latter is implemented using a gradient value policy iteration algorithm, enabling efficient characteristic-based machine learning methods. We establish convergence rates for the splitting method. In particular, with $h$ the splitting step, the $L^\infty$ error is bounded between $\mathcal{O}(h)$ and $\mathcal{O}(h^{1/5})$ for Lipschitz data, improving to $\mathcal{O}(h^{1/3})$ for semiconcave data. In the periodic setting, we also obtain an $L^1$ error of order $\mathcal{O}(h^{1/2})$. For the first-order step, we provide a weighted $L^2$ error analysis that shows exponential convergence. Each iteration solves linear characteristic equations and learns the value function by minimizing a weighted value gradient loss. The approach yields stable and accurate numerical results.

math.OC↗

arXiv2025

On the rate of convergence in superquadratic Hamilton--Jacobi equations with state constraints

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton--Jacobi equations with state constraints. For every $p>2$, we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order $ \mathcal{O}(\varepsilon^{1/2}) $ and obtain an improved upper rate of order $ \mathcal{O}\big(\varepsilon^{\frac{p}{2(p-1)}}\big)$ for semiconcave data.

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↗

arXiv2025

Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation $H(x, Du, λu) = 0$ in $\mathbb{R}^n$ as $λ\to 0^+$. Under the assumption that the Aubry set is localized, we employ a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.

math.AP↗

arXiv2024

Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates

In this paper, we demonstrate a polynomial convergence rate for homogenization of Hamilton-Jacobi equations with quasi-periodic potentials. We establish a connection between the convergence rate of homogenization and the regularity of the effective Hamiltonian, by using a new quantitative ergodic estimate for bounded quasi-periodic functions with Diophantine frequencies. As an application, we also study the convergent rate for Birkhoff average of unbounded quasi-periodic functions.

math.AP↗

arXiv2024

The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton--Jacobi equation

We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type $Ω_λ= (1+r(λ))Ω$ with a continuous function $r$ and a positive constant $λ$. We characterize all solutions to the ergodic problem on $Ω$ in terms of $r$. In addition, we demonstrate that the additive eigenvalue $λ\mapsto c_{Ω_λ}$ on a rescaled domain $Ω_λ= (1+λ)Ω$ possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of $λ\mapsto c_{Ω_λ}$. We provide examples where higher regularity is achieved.

math.AP↗

arXiv2022

State-constraint static Hamilton-Jacobi equations in nested domains

We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{Ω_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $Ω_k \subset Ω_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $Ω_k$, to $u$, the solution to the corresponding problem in $Ω= \bigcup_{k \in \mathbb{N}} Ω_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.

math.AP↗

arXiv2022

Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n = 1$.

math.AP↗

arXiv2022

FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network

Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent nonlinear differential equations. In this work, we analyze potential issues with the well-known Physic Informed Neural Network for differential equations with little constraints on the boundary (i.e., the constraints are only on a few points). This analysis motivates us to introduce a novel technique called FinNet, for solving differential equations by incorporating finite difference into deep learning. Even though we use a mesh during training, the prediction phase is mesh-free. We illustrate the effectiveness of our method through experiments on solving various equations, which shows that FinNet can solve PDEs with low error rates and may work even when PINNs cannot.

cs.LG↗

arXiv2022

Unequal Covariance Awareness for Fisher Discriminant Analysis and Its Variants in Classification

Fisher Discriminant Analysis (FDA) is one of the essential tools for feature extraction and classification. In addition, it motivates the development of many improved techniques based on the FDA to adapt to different problems or data types. However, none of these approaches make use of the fact that the assumption of equal covariance matrices in FDA is usually not satisfied in practical situations. Therefore, we propose a novel classification rule for the FDA that accounts for this fact, mitigating the effect of unequal covariance matrices in the FDA. Furthermore, since we only modify the classification rule, the same can be applied to many FDA variants, improving these algorithms further. Theoretical analysis reveals that the new classification rule allows the implicit use of the class covariance matrices while increasing the number of parameters to be estimated by a small amount compared to going from FDA to Quadratic Discriminant Analysis. We illustrate our idea via experiments, which show the superior performance of the modified algorithms based on our new classification rule compared to the original ones.

cs.LG↗

arXiv2021

Vanishing discount problem and the additive eigenvalues on changing domains

We study the asymptotic behavior, as $λ\rightarrow 0^+$, of the state-constraint Hamilton--Jacobi equation $ϕ(λ) u_λ(x) + H(x,Du_λ(x)) = 0$ in $(1+r(λ))Ω$ and the corresponding additive eigenvalues, or ergodic constant $H(x,Dv(x)) = c(λ)$ in $(1+r(λ))Ω$ with state-constraint. Here, $Ω$ is a bounded domain of $ \mathbb{R}^n$, $ϕ(λ), r(λ):(0,\infty)\rightarrow \mathbb{R}$ are continuous functions such that $ϕ$ is nonnegative and $\lim_{λ\rightarrow 0^+} ϕ(λ) = \lim_{λ\rightarrow 0^+} r(λ) = 0$. We obtain both convergence and non-convergence results in the convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue $c(λ)$ as $λ\rightarrow 0^+$. The main tool we use is a duality representation of solution with viscosity Mather measures.

math.AP↗