Son Tu

Từ Nguyễn Thái Sơn

Research

I am interested in Partial Differential Equations. In particular, I am working on problems related to asymptotic behavior of solutions to Hamilton-Jacobi equations. I am recently interested in applied probability, statistics, and machine learning, especially the mathematical foundations to data analysis and machine learning.

My current projects concentrate on the nonlinear behavior of solutions to certain types of Hamilton–Jacobi equations and their applications to the free boundary problem in some models of fluid dynamics.

Preprints and Publications

  1. Polynomial convergence rate for quasiperiodic homogenization of Hamilton-Jacobi equations
    (with Bingyang Hu and Jianlu Zhang),
    preprint, submitted arxiv   ·     ·  

    In this paper, we demonstrate a polynomial convergence rate for quasi-periodic homogenization of Hamilton--Jacobi equations in one-dimensional with n-frequency potentials. The proof relies on a connection between optimal control theory, regularity of the effective Hamiltonian, and new quantitative ergodic estimates based on Diophantine approximations. 
            @misc{hu2024polynomial,
title={Polynomial convergence rate for quasiperiodic homogenization of Hamilton-Jacobi equations}, 
author={Bingyang Hu and Son N. T. Tu and Jianlu Zhang},
year={2024},
eprint={2405.11516},
archivePrefix={arXiv},
primaryClass={math.AP}
      }

  2. Well-posedness for viscosity solutions of the one-phase Muskat problem in all dimensions
    (with Russell Schwab and Olga Turanova),
    preprint, submitted arxiv   ·     ·  

    In this article, we apply the viscosity solutions theory for integro-differential equations to the $\textit{one-phase}$ Muskat equation (also known as the Hele-Shaw problem with gravity). We prove global well-posedness for the corresponding Hamilton-Jacobi-Bellmann equation with bounded, uniformly continuous initial data, in all dimensions. 
    
@misc{schwab2024wellposedness,
    title         = {Well-posedness for viscosity solutions of the one-phase 
                    Muskat problem in all dimensions}, 
    author        = {Russell Schwab and Son Tu and Olga Turanova},
    year          = {2024},
    eprint        = {2404.10972},
    archivePrefix = {arXiv},
    primaryClass  = {math.AP}
}

  3. On the regularity of stochastic effective Hamiltonian
    (with Jianlu Zhang),
    preprint, submitted arxiv   ·     ·  

    In this paper, we study the regularity of the ergodic constants for the viscous Hamilton--Jacobi equations. We also estimate the convergent rate of the ergodic constant in the vanishing viscosity process. 
    
@misc{tu_regularity_2024,
    title = {On the regularity of stochastic effective {Hamiltonian}},
    copyright = {All rights reserved},
    url = {http://arxiv.org/abs/2312.15649},
    doi = {10.48550/arXiv.2312.15649},
    abstract = {In this paper, we study the regularity of the ergodic 
                constants for the viscous Hamilton--Jacobi equations. 
                We also estimate the convergent rate of the ergodic 
                constant in the vanishing viscosity process.},
    urldate = {2024-01-25},
    publisher = {arXiv},
    author = {Tu, Son and Zhang, Jianlu},
    month = jan,
    year = {2024},
    note = {arXiv:2312.15649 [math]},
    keywords = {Mathematics - Analysis of PDEs, 
                Mathematics - Dynamical Systems, 
                35D40, 70H20, 35J60, 37J40, 49L25, 37K99},
    annote = {Comment: Updated to 8 pages, errors and typos fixed},
    }

  4. Generalized convergence of solutions for nonlinear Hamilton-Jacobi equations with state-constraint
    (with Jianlu Zhang),
    Journal of Differential Equations 406 (2024), 87-125 pdf   ·   arxiv   ·     ·  

    For a continuous Hamiltonian $H : (x, p, u) \in T^*\mathbb{R}^n \times \mathbb{R}\rightarrow \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton-Jacobi equations with state-constraint $$ \begin{cases} \begin{aligned} H(x, Du, \lambda u)\leq C_\lambda, \quad x\in\Omega_\lambda\subset \mathbb{R}^n,\\ H(x, Du, \lambda u)\geq C_\lambda, \quad x\in\overline{\Omega}_\lambda\subset \mathbb{R}^n, \end{aligned} \end{cases} $$ as $\lambda\rightarrow 0^+$. When $H$ satisfies certain convex, coercive and monotone conditions, the domain $\Omega_\lambda:=(1+r(\lambda))\Omega$ keeps bounded, star-shaped for all $\lambda>0$ with $\lim_{\lambda\rightarrow 0^+}r(\lambda)=0$, and $\lim_{\lambda\rightarrow 0^+}C_\lambda=c(H)$ equals the ergodic constant of $H(\cdot,\cdot,0)$, we prove the convergence of solutions $u_\lambda$ to a specific solution of the critical equation $$ \begin{cases} \begin{aligned} H(x, Du, 0)\leq c(H), \quad x\in\Omega\subset\mathbb{R}^n,\\ H(x, Du, 0)\geq c(H), \quad x\in\overline{\Omega}\subset\mathbb{R}^n. \end{aligned} \end{cases} $$ We also discuss the generalization of such a convergence for equations with more general $C_\lambda$ and $\Omega_\lambda$. 
    
@misc{tu2023generalized,
    title={Generalized convergence of solutions for nonlinear 
            Hamilton-Jacobi equations with state-constraint}, 
    author={Son Tu and Jianlu Zhang},
    year={2023},
    eprint={2303.17058},
    archivePrefix={arXiv},
    primaryClass={math.AP}
}

  5. The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton-Jacobi equation
    (with Farid Bozorgnia and Dohyun Kwon),
    Journal of Differential Equations 402 (2024), 518-553 pdf   ·   arxiv   ·     ·  

    Let $c(\lambda)$ be the additive eigenvalue with respect to $(1+r(\lambda)\Omega)$, we show that $\lambda \mapsto c(\lambda)$ is differentiable except at most a countable set, while one-sided derivatives exist everywhere. The convergence of the vanishing discount problem on changing domains is also considered, and furthermore, the limiting solution can be parametrized by a real function. Finally, we connect the regularity of this real function to the regularity of $\lambda \mapsto c(\lambda)$. Some examples are given where higher regularity is achieved. 
    

@article{bozorgnia_regularity_2024,
    title = {The regularity with respect to domains of the additive eigenvalues of superquadratic {Hamilton}–{Jacobi} equation},
    volume = {402},
    issn = {0022-0396},
    url = {https://www.sciencedirect.com/science/article/pii/S002203962400295X},
    doi = {https://doi.org/10.1016/j.jde.2024.05.013},
    abstract = {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 Ωλ=(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↦cΩλ on a rescaled domain Ωλ=(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↦cΩλ. We provide examples where higher regularity is achieved.},
    journal = {Journal of Differential Equations},
    author = {Bozorgnia, Farid and Kwon, Dohyun and Tu, Son N. T.},
    year = {2024},
    keywords = {Optimal control theory, Rate of convergence, Second-order Hamilton–Jacobi equations, Semiconcavity, State-constraint problems, Viscosity solutions},
    pages = {518--553},
}

  6. Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations
    (with Yuxi Han),
    Applied Mathematics & Optimization 86, 1 (2022), 3 pdf   ·   arxiv   ·     ·  

    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 $\mathcal{O}(\varepsilon^{1/p})$ (where $1 < p < 2$ is the exponent of the gradient term) for nonnegative data $f\in \mathrm{C}^2(\overline{\Omega})$ 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. 
    
@article {hantu2022,
    AUTHOR = {Han, Yuxi and Tu, Son N. T.},
     TITLE = {Remarks on the vanishing viscosity process of state-constraint
              {H}amilton-{J}acobi equations},
   JOURNAL = {Appl. Math. Optim.},
  FJOURNAL = {Applied Mathematics and Optimization},
    VOLUME = {86},
      YEAR = {2022},
    NUMBER = {1},
     PAGES = {Paper No. 3, 42},
      ISSN = {0095-4616,1432-0606},
   MRCLASS = {35B40 (35D40 49J15 49L25 70H20 93E20)},
  MRNUMBER = {4436612},
       DOI = {10.1007/s00245-022-09874-z},
       URL = {https://doi.org/10.1007/s00245-022-09874-z},
}

  7. Vanishing discount problems for Hamilton-Jacobi equations on changing domains,
    Journal of Differential Equations 317 (2022), 32–69 pdf   ·   arxiv   ·     ·  

    We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the state-constraint Hamilton--Jacobi equation \begin{equation} \begin{cases} \phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) \leq 0 \qquad\text{in}\,\;(1+r(\lambda))\Omega,\\ \phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) \geq 0 \qquad\text{on}\;(1+r(\lambda))\overline{\Omega}. \end{cases} \tag{$S_\lambda$} \end{equation} and the corresponding additive eigenvalues, or ergodic constant \begin{equation} \begin{cases} H(x,Dv(x)) \leq c(\lambda) \qquad\text{in}\,\;(1+r(\lambda))\Omega,\\ H(x,Dv(x)) \geq c(\lambda) \qquad\text{on}\;(1+r(\lambda))\overline{\Omega}. \end{cases} \tag{$E_\lambda$} \end{equation} Here, $\Omega$ is a bounded domain of $ \mathbb{R}^n$, $\phi(\lambda), r(\lambda):(0,\infty)\rightarrow \mathbb{R}$ are continuous functions such that $\phi$ is nonnegative and $\lim_{\lambda\rightarrow 0^+} \phi(\lambda) = \lim_{\lambda\rightarrow 0^+} r(\lambda) = 0$. 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 $c(\lambda)$ as $\lambda\rightarrow 0^+$. The main tool we use is a duality representation of solution with viscosity Mather measures. 
    
    @article {tu2022vanishing,
        AUTHOR = {Tu, Son N. T.},
         TITLE = {Vanishing discount problem and the additive eigenvalues on
                  changing domains},
       JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
        VOLUME = {317},
          YEAR = {2022},
         PAGES = {32--69},
          ISSN = {0022-0396,1090-2732},
       MRCLASS = {35B40 (35D40 49J15 49L25 70H20)},
      MRNUMBER = {4379308},
           DOI = {10.1016/j.jde.2022.01.055},
           URL = {https://doi.org/10.1016/j.jde.2022.01.055},
    }

  8. State-Constraint Static Hamilton–Jacobi Equations in Nested Domains
    (with Yeoneung Kim and Hung V. Tran),
    SIAM Journal on Mathematical Analysis 52, 5 (2020), 4161–4184 pdf   ·   arxiv   ·     ·  

    We study state-constraint static Hamilton-Jacobi equations in a sequence of domains Ω=⋃k∈NΩk\Omega = \bigcup_{k\in \mathbb{N}} \Omega_k in $\mathbb{R}^n$ such that Ωk⊂Ωk+1\Omega_k\subset \Omega_{k+1} for all k∈Nk\in \mathbb{N}. We obtain rates of convergence of uku_k, the solution to the state-constraint problem in Ωk\Omega_k, to uu, the solution to the corresponding problem in Ω=⋃k∈NΩk\Omega = \bigcup_{k\in \mathbb{N}} \Omega_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. 
    
@article {kimtrantu2020,
    AUTHOR = {Kim, Yeoneung and Tran, Hung V. and Tu, Son N.},
     TITLE = {State-constraint static {H}amilton-{J}acobi equations in
              nested domains},
   JOURNAL = {SIAM J. Math. Anal.},
  FJOURNAL = {SIAM Journal on Mathematical Analysis},
    VOLUME = {52},
      YEAR = {2020},
    NUMBER = {5},
     PAGES = {4161--4184},
      ISSN = {0036-1410,1095-7154},
   MRCLASS = {35F21 (35B40 35D40 49K05 49L12 49L25 70H20)},
  MRNUMBER = {4143416},
MRREVIEWER = {Sayonita\ Ghosh Hajra},
       DOI = {10.1137/19M1292035},
       URL = {https://doi.org/10.1137/19M1292035},
}

  9. Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension,
    Asymptotic Analysis 121, 2 (2021), 171–194   ·   pdf   ·   arxiv   ·     ·  

    Let uεu^\varepsilon and uu 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 O(ε)\mathcal{O}(\varepsilon) of uε→uu^\varepsilon \to u as ε→0+\varepsilon \to 0^+. 
    
@article{tu_rate_2021,
    title        = {Rate of convergence for periodic homogenization 
                    of convex {Hamilton}–{Jacobi} equations in one 
                    dimension},
    author       = {Tu, Son N. T.},
    year         = 2021,
    month        = jan,
    journal      = {Asymptotic Analysis},
    volume       = 121,
    number       = 2,
    pages        = {171--194},
    doi          = {10.3233/ASY-201599},
    issn         = {0921-7134},
    url          = {https://content.iospress.com/
                    articles/asymptotic-analysis/asy201599},
    urldate      = {2024-02-03},
    note         = {Publisher: IOS Press},
    abstract     = {Let u ε 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 O ( ε ) of u ε → u a},
    language     = {en},
}

  10. Some asymptotic problems on the theory of viscosity solutions of Hamilton–Jacobi equations,
    The UW-Madison ProQuest Dissertations Publishing,  2022. 29324377 pdf   ·     ·  


    Viscosity solutions arise naturally in many fields of study from engineering, physics, and operations research to economics. The study of viscosity solutions on its own has uncovered many new and interesting research problems, including the study of the asymptotic behavior of solutions with respect to the changing of parameters. In this dissertation, I present some new problems following the line of the asymptotic behavior of solutions. Each of the problems is related to the other through the old underlying theme of optimal control theory, yet presents many new problems on their own that are yet to be studied.

    The first direction is on homogenization of Hamilton--Jacobi equations. Using deep analysis of the dynamics of minimizers corresponding to the solution, I established in [1] the optimal rate of convergence under the multi-scale setting in one dimension, which could not be obtained by the previous pure PDEs technique.

    The second direction concerns various asymptotic problems for equations with state-constraint. In [2] , my co-authors and I established some *first quantitative results* on the rate of convergence of the solution to the Hamilton--Jacobi equations with state-constraint on a nested domain setting. Utilizing the weak KAM theory, in [3] , I established qualitatively various convergence results for the vanishing discount procedure with changing domains together with a new description of the regularity of the additive eigenvalues with respect to domain perturbation.

    Lastly, in [4] , my co-author and I established the rate of convergence for the vanishing viscosity procedure, concerning the viscous state-constraint viscosity *large* solution that blows on the boundary of the underlying domain. This is the first-rate established for blow-up solutions in the literature as far as we know.
    1. Tu, S. N. T. Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension. Asymptotic Analysis 121, 2 (2021), 171–194. ↩
    2. Kim, Y., Tran, H. V., and Tu, S. N. T. State-Constraint Static Hamilton–Jacobi Equations in Nested Domains. SIAM Journal on Mathematical Analysis 52, 5 (2020), 4161–4184. ↩
    3. Tu, S. N. T. Vanishing discount problem and the additive eigenvalues on changing domains. Journal of Differential Equations 317 (2022), 32–69. ↩
    4. Han, Y., and Tu, S. N. T. Remarks on the vanishing viscosity process of stateconstraint Hamilton–Jacobi equations. Applied Mathematics & Optimization 86, 1 (2022), 3. ↩

Refereed conference proceedings & papers.

  1. Unequal Covariance Awareness for Fisher Discriminant Analysis and Its Variants in Classification
    Thu Nguyen, Quang M. Le, Son N.T. Tu, Binh T. Nguyen
    2022 International Joint Conference on Neural Networks (IJCNN), (Jul. 2022) pdf   ·   arxiv   ·     ·  

    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 shows the superior performance of the modified algorithms based on our new classification rule compared to the original ones. 
    
@INPROCEEDINGS{9892588,
    author={Nguyen, Thu and Le, Quang M. and Tu, Son N.T. and Nguyen, Binh T.},
    booktitle={2022 International Joint Conference on Neural Networks (IJCNN)}, 
    title={Unequal Covariance Awareness for Fisher Discriminant Analysis 
            and Its Variants in Classification}, 
    year={2022},
    volume={},
    number={},
    pages={1-8},
    keywords={Neural networks;Feature extraction;
                Classification algorithms;
                Fisher Discrimiant Analysis;
                Linear Discriminant Analysis;
                Quadratic Discriminant Analysis;classification},
    doi={10.1109/IJCNN55064.2022.9892588}}

Invited Talks

Contributed Talks & Presentation

Research visits

Selected Conferences, Workshops, and Summer Schools attended

Reviewer for Mathematics Journals

Collaborators

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