Tongtong Li

Department of Mathematics and Statistics
University of Maryland, Baltimore County
1000 Hilltop Circle
Baltimore, MD 21250

Office : MP 438
email : tongtong.li at umbc.edu

I am currently an assistant professor at the Department of Mathematics and Statistics, University of Maryland, Baltimore County.

My research interests lie in the broad areas of computational and applied mathematics, including numerical analysis and solution of partial differential equations (PDEs), data assimilation and Bayesian inverse problems. On one hand, I focus on the development, theoretical analysis and computational implementation of numerical methods that approximate solutions of PDEs arising from complex systems in various fields, including environmental sciences, petroleum engineering, hydrology and biomechanics engineering. While PDEs serve as an essential tool to understand and predict dynamics, judicious treatment of data is crucial due to the structural complexity and computational intensity within these dynamics. In this regard, I am interested in the study of computational methods that enhance the information we can extract from existing data and knowledge by employing data assimilation and Bayesian approaches. My research aims to design rigorous and comprehensive methods for meaningful descriptions and novel treatments of systems in scientific and engineering applications, by leveraging advanced tools developed in areas including numerical analysis, data assimilation and Bayesian inference, understanding the underlying philosophies of each area, and forging connections between them.

Before joining UMBC, I was a postdoctoral researcher, working with Professor Anne Gelb and Professor Yoonsang Lee, at the Department of Mathematics, Dartmouth College. I received my PhD in Mathematics under the supervision of Professor Ivan Yotov at the University of Pittsburgh.

For Ph.D. applications: I am actively seeking motivated Ph.D. students to join my research group. The admission process is managed by the Department of Mathematics and Statistics, and you can find the application details here. Additionally, you are encouraged to send me your latest CV, an unofficial transcript, and a brief note outlining our shared research interests via email.

Research Interests

Numerical Analysis: numerical solution of partial differential equations, finite element methods, numerical conservation laws, high order methods
Data Assimilation: sequential inference, ensemble learning
Bayesian Inverse Problems: hierarchical Bayesian learning, Bayesian inference
Applications: computational fluid dynamics, interaction of fluid flow and poroelastic media, sea ice modeling

News

I presented at the SIAM Conference on Mathematics of Data Science (MDS24), Atlanta, GA, Oct 21nd - Oct 25th, 2024. Registration and travel support for this presentation was provided by National Science Foundation.
I gave a presentation in the Finite Element Circus, University of Maryland, Baltimore County, Oct 18th - Oct 19th, 2024.
I attended the ICERM-SIAM Workshop on Empowering a Diverse Computational Mathematics Research Community, Brown University, July 22nd - Aug 2nd, 2024.
Professor Anne Gelb and I are featured in the VOX article Researchers Improve Models to Predict Changes in Sea Ice.
I gave a presentation in the North American High Order Methods Conference (NAHOMCon), Dartmouth College, June 18th, 2024.
I gave a talk at Conference on Mathematical Models and Numerical Methods for Multiphysics Systems, University of Pittsburgh, May 3rd, 2024.

Employment

(August 2024 - current) Assistant Professor, Department of Mathematics and Statistics, University of Maryland, Baltimore County
(August 2021 - July 2024) Research Associate, Department of Mathematics, Dartmouth College
Supervisor : Anne Gelb

Education

PhD in Mathematics, University of Pittsburgh, USA, July 2021
Dissertation title : Mixed Formulations for Fluid-poroelastic Structure Interaction
Advisor : Ivan Yotov
MS in Mathematical Finance, Rutgers, the State University of New Jersey, USA, June 2016
Thesis title : Pricing Finite-maturity European Put-Heston Option with Barrier Discontinuity by FDM
Advisor : David Eliezer
BS in Economics, Huazhong Agricultural University, China, June 2014
Thesis title : Research of Chinese Agricultural Commodity Futures Market Volatility Spillover Effect Based on BEKK-GARCH Model - Taking DCE Yellow Soybean as an Example
Advisor : Guang Zeng

Awards

SIAM Early Career Travel Award: SIAM Conference on Mathematics of Data Science (MDS24). Atlanta, GA, Oct 2024.
Travel Award: ICERM-SIAM Workshop on Empowering a Diverse Computational Mathematics Research Community. Brown University, July - Aug. 2024.
First Place Award for Lightning Talks: Dartmouth-UNH Postdoctoral Research Day, Dartmouth College, March 2024.
Travel Award: Hot Topics: Recent Progress in Deterministic and Stochastic Fluid-Structure Interaction. Simons Laufer Mathematical Sciences Institute, Dec. 2023.
Travel Award: 17th U. S. National Congress on Computational Mechanics (USNCCM17). Albuquerque, NM, July 2023.
First Place Award (Outstanding Graduate Student Research Poster) for AWM Graduate Student Poster Competition. Seattle, WA, May 2023.
Travel Award: AWM Workshop at SIAM Conference on Optimization (OP23). Seattle, WA, May 2023.
Travel Award: SIAM Convening on Climate Science, Sustainability, and Clean Energy. Tysons, VA, Oct. 2022.
Second Place Award for Poster Presentation: High Performance Computing (HPC) Day. University of Massachusetts Lowell, Sept. 2022.
SIAM Early Career Travel Award: SIAM Conference on Mathematical Planet Earth (MPE22). Pittsburgh, PA, July 2022.
Thomas C. Hales Distinguished Research Award, University of Pittsburgh, 2022.
Mathematics Teaching Assistant Excellence Award, University of Pittsburgh, 2019.
Arts and Sciences Graduate Fellowship (two times, 2020 and 2017).

Journal papers and preprints

L. Liu, T. Li, A. Gelb, and Y. Lee, Entropy stable conservative flux form neural networks. Arxiv. Submitted. [link]
T. Li and A. Gelb, A Bayesian framework for spectral reprojection. Arxiv. Submitted. [link]
T. Li, A. Gelb, and Y. Lee, A structurally informed data assimilation approach for nonlinear partial differential equations. J. Comput. Phys., 519: 113396, 2024. DOI: 10.1016/j.jcp.2024.113396. [link]
S. Caucao, A. Dalal, T. Li and I. Yotov, Mixed finite element methods for the Navier-Stokes-Biot model. In: Ivan Lirkov, Svetozar Margenov (eds) Large-Scale Scientific Computations. 14th International Conference, LSSC 2023, Sozopol, Bulgaria, June 5–9, 2023, Revised Selected Papers. Springer, Cham. DOI: 10.1007/978-3-031-56208-2. [link]
T. Li, A. Gelb, and Y. Lee, Improving numerical accuracy for the viscous-plastic formulation of sea ice. J. Comput. Phys., 487: 112184, 2023. DOI: 10.1016/j.jcp.2023.112184. [link]
T. Li, S. Caucao and I. Yotov, An augmented fully mixed formulation for the coupling of the quasi-static Navier-Stokes and Biot models. IMA J. Numer. Anal, 2023. [link]
S. Caucao, T. Li and I. Yotov, A multipoint stress-flux mixed finite element method for the Stokes-Biot model. Numer. Math., 2022. DOI: 10.1007/s00211-022-01310-2. [link]
T. Li, X. Wang and I. Yotov, Non-Newtonian and poroelastic effects in simulations of arterial flows. Arxiv. Preprint. [link]
T. Li and I. Yotov, A mixed elasticity formulation for fluid-poroelastic structure interaction. ESAIM Math. Model. Numer. Anal., 56(1): 1-40, 2022. DOI: 10.1051/m2an/2021083. [link]
S. Caucao, T. Li and I. Yotov, A cell-centered finite volume method for the Navier-Stokes/Biot model. In: Klofkorn R., Keilegavlen E., Radu F., Fuhrmann J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. DOI: 10.1007/978-3-030-43651-3_29. [link]

Publications

University of Maryland, Baltimore County

Lecturer

2024 Fall - Math 225 Introduction to Differential Equations (Undergraduate Level)

Dartmouth College

Lecturer

2024 Spring - Math 23 Differential Equations (Undergraduate Level)
2023 Summer - Math 117 First Year Graduate Seminar (Data Assimilation, Graduate Level, One Week Voluntary)
2023 Winter - Math 116 Applied Mathematics (Numerical Linear Algebra, Graduate Level)
2022 Spring - Math 76/146 Applied Mathematics (Finite Element Method, Mixed Undergraduate and Graduate Levels)

Undergraduate student research mentoring

Ethan M. Baird, Summer 2024
Mason F. King, Summer 2024
Gary Jiang, Summer 2024
Jack E. Friedman, 2022 - 2023
David J. Appleton, 2022 - 2023

Graduate student directed reading

Jessica Rattray (Numerical Solution of PDEs), Spring 2023

University of Pittsburgh

Lecturer

2021 Summer - Math 0230 Analytic Geometry and Calculus 2
2020 Summer - Math 0290 Applied Differential Equations
2017 Summer - Math 0220 Analytic Geometry and Calculus 1

Teaching Fellow / Teaching Assistant

2020 Fall - Math 0240 Analytic Geometry and Calculus 3 (2 sections)
2020 Spring - Math 0220 Analytic Geometry and Calculus 1
2019 Fall - Math 0230 Analytic Geometry and Calculus 2 (3 sections)
2019 Spring - Math 0413 Intro to Theoretical Mathematics (2 sections)
2018 Fall - Math 0240 Analytic Geometry and Calculus 3 (3 sections)
2018 Spring - Math 0450 UHC Intro to Analysis
2017 Fall - Math 0220 Analytic Geometry and Calculus 1 (3 sections)

PDE solvers

It is a good exercise to develop your own PDE solvers based on what you have learned in Numerical Analysis. However, for practical research computations, it is strongly recommended to use PDE solvers developed and refined by many researchers. There are many PDE solvers freely available online. Among others, I recommend the following PDE solvers - AMReX @ LBL and PETSc @ Argonne - for robust and efficient computations. Check out the following links.

AMReX
PETSc

Data Assimilation Codes

Data assimilation combines a numerical forecast model with observational data to improve the prediction skill. If you are interested in testing/running data assimilation, please check the following programs.

If you are interested in a MATLAB program for data assimilation, you can check the following book by Law, Stuart and Zygalakis, which contains MATLAB codes

Data Assimilation, Texts in Applied Mathematics Vol 62, Springer-Verlag, 2015