Tina Mai's Homepage

Research Overview 

A central goal of my research is to develop mathematically grounded multiscale modeling frameworks that combine analysis, structure-preserving numerical methods, and data-assisted techniques for nonlinear partial differential equations arising in applications.

My work bridges analytical and computational approaches for complex multiscale systems in physics, biology, and engineering.

My recent research establishes rigorous analysis of hydrodynamic limits and entropic convergence for kinetic equations, alongside multiscale numerical methods for nonlinear PDEs.

Research Interests

• Analysis of nonlinear PDEs and kinetic theory

• Multiscale and structure-preserving numerical methods

• Scientific machine learning and data-driven techniques for multiscale systems

Note.  In the publication lists below, an asterisk (*) denotes corresponding authorship, and my name is shown in bold.

Selected Publications

• Tina Mai*.  Hydrodynamic limit of the Kuramoto–Sakaguchi equation with inertia and noise effects.
  arXiv: 2410.05113, 2024.
  https://arxiv.org/abs/2410.05113 

(submitted for publication, 2025; under review)

Rigorously derives the hydrodynamic limit of the kinetic Kuramoto equation with inertia and noise using generalized collision invariants, supported by a Hardy-type inequality.

• Tina Mai*.  Entropic convergence and the linearized limit for the Boltzmann equation with external force.
  Preprint, arXiv:1612.05096v2, 2025.
  https://arxiv.org/abs/1612.05096v2

Extends Levermore’s entropic convergence framework to the nonlinear Boltzmann equation with external forces, establishing the strong linearized limit via entropy methods, together with semigroup theory, dissipation equality, and compactness arguments.

• Dmitry Ammosov, Tina Mai*, Juan Galvis.  Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model, Journal of Computational Physics, 519 (2024) 113428. 
  https://doi.org/10.1016/j.jcp.2024.113428     
  https://arxiv.org/abs/2403.14178

Develops a generalized multiscale finite element framework for nonlinear strain-limiting Cosserat elasticity, establishing convergence and demonstrating efficiency and robustness in heterogeneous media.

• Denis Spiridonov, Sergei Stepanov, Tina Mai*.  Prediction of discretization of online GMsFEM using deep learning for Richards equation, Journal of Computational and Applied Mathematics, 454 (2025) 116167.
  https://doi.org/10.1016/j.cam.2024.116167
  https://arxiv.org/abs/2403.14177

Combines deep learning with online GMsFEM to rapidly predict multiscale basis functions and efficiently compute coarse-scale solutions for the nonlinear Richards equation in heterogeneous, non-periodic media with uncertainty.

Publications 

Multiscale Numerical Methods

1. Dmitry Ammosov, Tina Mai*, Juan Galvis.  Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model, Journal of Computational Physics, 519 (2024) 113428. 
   https://doi.org/10.1016/j.jcp.2024.113428     
   https://arxiv.org/abs/2403.14178

2. Tina Mai, Siu Wun Cheung, and Jun Sur Richard Park.  Constraint energy minimizing generalized multiscale finite element method for multi-continuum Richards equations, Journal of Computational Physics, 477 (2023) 111915.
   https://doi.org/10.1016/j.jcp.2023.111915
   https://arxiv.org/abs/2205.11294

3. Jun Sur Richard Park, Siu Wun Cheung, Tina Mai*.  Multiscale simulations for multi-continuum Richards equations, Journal of Computational and Applied Mathematics, 397 (2021) 113648. 
   https://doi.org/10.1016/j.cam.2021.113648
   https://arxiv.org/abs/2010.09181

4. Jun Sur Richard Park, Siu Wun Cheung, Tina Mai*, Viet Ha Hoang.  Multiscale simulations for upscaled multi-continuum flows, Journal of Computational and Applied Mathematics, 374 (2020) 112782.
   https://doi.org/10.1016/j.cam.2020.112782
   https://arxiv.org/abs/1909.04722

5. Shubin Fu, Eric Chung, Tina Mai*.  Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity, Journal of Computational Physics, 417 (2020) 109569.
   https://doi.org/10.1016/j.jcp.2020.109569
   https://arxiv.org/abs/1909.13267

6. Shubin Fu, Eric Chung, Tina Mai*.  Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model, Journal of Computational and Applied Mathematics, 359 (2019) 153–165. 
   https://doi.org/10.1016/j.cam.2019.03.047
   https://arxiv.org/abs/1812.09347

Scientific Machine Learning and Data-Driven Multiscale Methods

1. Denis Spiridonov, Sergei Stepanov, Tina Mai*.  Prediction of discretization of online GMsFEM using deep learning for Richards equation, Journal of Computational and Applied Mathematics, 454 (2025) 116167.
   https://doi.org/10.1016/j.cam.2024.116167
   https://arxiv.org/abs/2403.14177

2. Sergei Stepanov, Denis Spiridonov, Tina Mai*.  Prediction of numerical homogenization using deep learning for the Richards equation, Journal of Computational and Applied Mathematics, 424 (2023) 114980.
   https://doi.org/10.1016/j.cam.2022.114980
   https://arxiv.org/abs/2208.12161

Kinetic Theory and Nonlinear PDEs

1. Tina Mai*.  Hydrodynamic limit of the Kuramoto–Sakaguchi equation with inertia and noise effects.
   arXiv: 2410.05113, 2024.
   https://arxiv.org/abs/2410.05113 

    (submitted for publication, 2025; under review)

2. Tina Mai*.  Entropic convergence and the linearized limit for the Boltzmann equation with external force.
   Preprint, arXiv:1612.05096v2, 2025.
   https://arxiv.org/abs/1612.05096v2

Optimization / Computational Mathematics

1. Tina Mai*, Daniele Mortari.  Theory of functional connections applied to quadratic and nonlinear programming under equality constraints, Journal of Computational and Applied Mathematics, 406 (2022) 113912.
   https://doi.org/10.1016/j.cam.2021.113912
   https://arxiv.org/abs/1910.04917

Continuum Mechanics

1. Tina Mai*, Jay R. Walton.  On monotonicity for strain-limiting theories of elasticity, Journal of Elasticity, 120 (I) (2015) 39–65.
   https://doi.org/10.1007/s10659-014-9503-4

2. Tina Mai, Jay R. Walton.  On strong ellipticity for implicit and strain-limiting theories of elasticity, Mathematics and Mechanics of Solids, 20 (II) (2015) 121–139. 
   https://doi.org/10.1177/1081286514544254

Research Grants

RFBR Research Grant, 2021–2022: Russian Foundation for Basic Research (RFBR) 

on Cooperation with Vietnam Academy of Science and Technology (VAST),

a principal investigator (co-PI), grant number RFBR-VAST 21-51-54001, 

“New mathematical models for multiscale infiltration process using machine learning approaches”, 

with Aleksandr Grigorev (co-PI), Denis Spiridonov, Sergei Stepanov, and Hieu Nguyen (members)

NAFOSTED Research Grant, 2020–2022: Vietnam National Foundation for Science and Technology Development (NAFOSTED), 

a principal research member, grant number 101.99-2019.326, 

“Methods for solving multiscale partial differential equations”, 

with Nguyen Trung Hieu (PI) and Chu Van Tiep (principal research member)