Tina Mai

Research Overview 

My research develops mathematically grounded multiscale modeling frameworks that combine rigorous analysis, structure-preserving numerical methods, and data-assisted techniques for nonlinear partial differential equations (PDEs) arising in applications across physics, biology, and engineering.

My published work has focused largely on generalized multiscale finite element methods (GMsFEM) and scientific machine learning approaches for nonlinear PDEs in heterogeneous systems.  My recent analytical work establishes rigorous results, including the hydrodynamic limit for the Kuramoto–Sakaguchi equation with inertia and noise, and entropic convergence for the Boltzmann equation with external force.

A unifying theme in my work is a two-way exchange between theory and computation.  Analytical structures, including asymptotic limits, entropy methods, and semigroup/evolution-family techniques, provide insight into the underlying PDE models and help guide stability and convergence analysis for numerical schemes.  In turn, multiscale numerical and machine learning techniques extend these frameworks to complex systems involving multiple scales, stochastic effects, and uncertainty.

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, showing that the strong linearized limit persists under physically meaningful force fields.  The key technical ingredients are new force-regularity assumptions and careful treatment of test-function regularity, which support the semigroup/evolution-family construction, enable the dissipation equality in the forced setting, and provide the compactness needed for the limit.

• 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.

• 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

Extends the CEM-GMsFEM to the coupled nonlinear multi-continuum Richards equations, enabling efficient simulation of flow in heterogeneous fractured porous media, complemented by a rigorous convergence analysis of the Picard iteration.

• 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)

Welcome
August 2 – 7, 2027, the 22nd ICMP (International Congress of Mathematical Physics),
taking place at Duy Tan University, Da Nang, Vietnam

Invited Talks

April 9, 2026, Invited Speaker, Stochastic Process Seminar, Department of Mathematics, Texas A&M University, College Station, USA

"Prediction of discretization of online GMsFEM using deep learning for Richards equation" 

(paper with Denis Spiridonov and Sergei Stepanov;  invited by Patricia Ning)

April 8, 2025, Invited Speaker, Nonlinear PDE Seminar, Department of Mathematics, Texas A&M University, College Station, USA

"Hydrodynamic limit of the Kuramoto–Sakaguchi equation with inertia and noise effects"

(slides;  paper;  invited by Edriss Titi)

Contributed Talks

February 27–28, 2026, Contributed Speaker, Finite Element Rodeo, 

"Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model"

The University of Texas at Austin, USA

(paper with Dmitry Ammosov and Juan Galvis)

April 13–14, 2023, Contributed Speaker, The 4th East Coast Optimization Meeting (Virtual), 

George Mason University, Fairfax, VA, USA

“Optimization and Game Theory”

Video of my talk: Prediction of numerical homogenization for Richards equation using deep learning

(paper with Sergei Stepanov and Denis Spiridonov)

Selected Activities 

April 10–12, 2025, Participant, Frontiers in Computational Mathematics, 

A conference in honor of Bjorn Engquist's 80th birthday,

The University of Texas at Austin, USA

February 6–10, 2023, Invited Participant, Workshop 1: Multiscale analysis and methods for PDEs: fluids and active matter dynamics, 

Program Multiscale Analysis and Methods for Quantum and Kinetic Problems,

Institute for Mathematical Sciences (IMS), National University of Singapore (NUS)

Group Picture

Teaching

I enjoy teaching across levels and mentoring students through guided reading and research projects

Selected teaching evaluations:
https://sites.google.com/view/tina-mai/mai_evaluations

At Duy Tan University (DTU), Da Nang, Vietnam
Adjunct Instructor of Troy University at DTU
(American Degree Program between Duy Tan and Troy University)

MTH 1126 Calculus II (lectures in English to Computer Science students)

In-person primary instructor, direct co-teaching              Spring 2026

Selected students' evaluations Spring 2026                                                           

Class Picture

In-person primary instructor, direct co-teaching              Spring 2025
Selected students' evaluations Spring 2025         

Class Picture

Online co-teaching and curriculum support             Summer 2022, Spring 2024

Lecturer at DTU: STA 151 Probability and Statistics, Fall 2018
(bilingual lectures in Vietnamese and English)     

Class Picture

At Texas A&M University (TAMU), College Station, USA

Instructor of Record at TAMU: Math 142 Business Mathematics II, Summer 2014

(lectures in English)
Selected students' evaluations Summer 2014        

Class Picture

Honors and Awards (Selected)

• SIAM Early Career Travel Award, Society for Industrial and Applied Mathematics (SIAM), USA, 2021.

Awarded to support my participation and contributed presentation at the Virtual SIAM Conference on Computational Science and Engineering (CSE21), March 1–5, 2021.

Talk: "Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Nonlinear Poroelasticity" 

(paper with Shubin Fu and Eric Chung).

• Mathematical Work Award (twice), National Program for the Development of Mathematics, 

Ministry of Education and Training (MOET) and Vietnam Institute for Advanced Study in Mathematics (VIASM),

2019,     2020.