Tina Mai

Multiscale Modeling and Computation

Research Interests

• Analysis of nonlinear PDEs and kinetic theory

• Multiscale numerical methods (GMsFEM and CEM-GMsFEM)

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

Research Overview 

My research develops multiscale modeling and computation for complex systems arising in physics, biology, and engineering.

My analytical work includes a sole-authored paper in Kinetic and Related Models on the hydrodynamic limit of the Kuramoto–Sakaguchi equation with inertia and noise (a kinetic model of collective dynamics incorporating stochastic effects, with applications to coupled oscillators and biological synchronization) and a substantially revised preprint on entropic convergence and the linearized limit for the Boltzmann equation with external force.

In parallel, my computational research develops and analyzes frameworks based on the generalized multiscale finite element method (GMsFEM) and its constraint energy minimizing variant (CEM-GMsFEM) for strain-limiting models of nonlinear elasticity, Cosserat (micropolar) elasticity, poroelasticity, and multi-continuum Richards equations describing soil-moisture dynamics in heterogeneous porous media.  I have also integrated deep learning with homogenization and online GMsFEM to predict effective coefficients and multiscale basis functions under input uncertainty, enabling efficient computation of coarse-scale solutions.

Across both strands, I bring analysis, computation, and data together: analytical insights guide numerical design, while multiscale methods and scientific machine learning make it possible to study heterogeneous and stochastic systems for which direct fine-scale simulation is prohibitively expensive.

My current work advances this program in three directions: asymptotic limits for spatially heterogeneous kinetic models, multiscale discretizations for nonlinear and coupled PDE systems, and operator learning of coefficient-to-basis maps supported by approximation and stability analysis.

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.   Kinetic and Related Models, 24 (2026) 75–108.

Journal: https://www.aimsciences.org/article/doi/10.3934/krm.2026020
arXiv: https://arxiv.org/abs/2410.05113 

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 and evolution-family constructions, 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 a sample-informed model-reduction approach for coupled nonlinear multi-continuum Richards equations, enabling efficient simulation of flow in heterogeneous fractured porous media and providing 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 

Kinetic Theory and Nonlinear PDEs

1. Tina Mai*.  Hydrodynamic limit of the Kuramoto–Sakaguchi equation with inertia and noise effects.  Kinetic and Related Models, 24 (2026) 75–108.

Journal: https://www.aimsciences.org/article/doi/10.3934/krm.2026020
arXiv: https://arxiv.org/abs/2410.05113 

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

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
   †Graduate student collaborator at the time of the work.

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
   †Graduate student collaborators at the time of the work.

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
   †Graduate student collaborators at the time of the work.

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

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

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

Welcome
August 2 – 7, 2027, The 22nd International Congress of Mathematical Physics (ICMP) will take place at Duy Tan University in Da Nang, Vietnam.  I look forward to welcoming colleagues from around the world to Da Nang.  

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 key research team member, grant number 101.99-2019.326, 

“Methods for solving multiscale partial differential equations”, 

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

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)

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 jointly offered by DTU and Troy University)

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

Primary instructor: teaching in person, co-teaching some sessions            Spring 2026

(15 students)

Selected students' evaluations Spring 2026                                                        

Class Picture

Primary instructor: teaching in person, co-teaching some sessions            Spring 2025
(19 students)

Selected students' evaluations Spring 2025         

Primary instructor: course design, assessment, and grading,                     Spring 2024
with colleagues delivering lectures (29 students)                                        

Primary instructor: course design, assessment, and grading,                  Summer 2022
with colleagues delivering lectures (15 students)                                     
         

Lecturer at DTU: STA 151 Probability and Statistics, Fall 2018,
four sections, approximately 100 students each 
(bilingual lectures in Vietnamese and English)   

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        

Research collaborations with graduate students:

• Jun Sur Richard Park (PhD 2020, Texas A&M University (TAMU); now an Assistant Professor, Division of Big Data Science, Korea University Sejong Campus, South Korea); three joint papers (JCAM 2020, 2021; JCP 2023)

• Siu Wun Cheung (PhD 2020, TAMU; now a Research Scientist in the Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, USA); three joint papers (JCAM 2020, 2021; JCP 2023)

• Dmitry Ammosov (PhD 2023, North-Eastern Federal University (NEFU); now a Postdoctoral Fellow in the Chemical and Petroleum Engineering Department, Khalifa University of Science and Technology, Abu Dhabi, UAE); joint paper (JCP 2024)

Service and Outreach (Selected)

Journal Referee, SIAM's Multiscale Modeling & Simulation, 2023

Invited Session Chair, contributed talks session, UNC Greensboro PDE Conference (virtual),
University of North Carolina at Greensboro, USA, June 9–11, 2023

Weekly Volunteer, Texas A&M Math Circle (enrichment program for school students in grades 5–8), Fall 2012

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.