Department of Mathematics
Dartmouth College
27 N Main Street
Hanover, NH 03755
Office : 206 Kemeny Hall
email : yoonsang.lee at dartmouth.edu
I am an assistant professor in the Department of Mathematics at Dartmouth College. My research focuses on applied mathematics and computational issues in prediction and uncertainty quantification of complex dynamical systems. I am interested in particular in computational methods to efficiently combine numerical prediction models with data, which are scalable for big data and high-dimensional systems.
Research Interests
Computational Mathematics, Statistics and Physics
Data Analysis and Assimilation, Bayesian Inference, Uncertainty Quantification
Multiscale and Stochastic Modeling, Analysis, and Simulation
Neural network-based methods for solving PDEs
"The difficult is what takes a little time; the impossible is what takes a little longer." - Fridtjof Hansen, Nobel Peace Prize Laureate, 1922
I was invited to talk at International Conference on Machine Learning and PDEs, Seoul, South Korea, Aug 28, 2022.
I gave a seminar talk at Applied Mathematics seminar, Ewha Womans University, Jul 28, 2022.
Tongtong Li presented our joint work at NAHOMCon 22, San Diego, CA, Jul 18, 2022
Tongtong Li gave a talk about our joint work on sea ice numerics, SIAM Mathematics of Planet Earth, Pittsburgh, PA, July 2022.
Matt Partno and I co-organized a minisymposium at SIAM MPE 2022, Pittsburgh, PA, July 2022.
Jihun Han gave a talk about the joint work on "inhomogeneous regularization with limited and indirect data", SIAM Conference on Imaging Sciences. virtual, Mar 21, 2022.
I gave a talk at Applied and Computational Mathematics seminar. UC Riverside, Feb 2, 2022.
I presented at Machine Learning+X seminar. Brown University, Jan 7, 2022.
Employment
(August 2018 - current) Assistant Professor, Department of Mathematics, Dartmouth College
"The purpose of computing is insight, not numbers." - Richard Hamming
My research focuses on mathematical problems in prediction and uncertainty quantification of complex dynamical systems.
In particular I am intrested in robust and efficient computational methods to combine numerical prediction models with data,
which are scalable for big data and high-dimensional systems.
The mathematical framework of my research shares with research areas known as
data assimilation
Bayesian inverse problems
ensemble learning
reduced-order modeling
averaging and homogenization
Application areas of my research include but not limited to geophysical fluid systems and combustion models. You can find more details from my publications.
G. Pease, Y. Lee, A. Gelb, and B. Keller, A Bayesian formulation for estimating the composition of Earth's crust, AGU Geophysical Research Letters, in revision.
Z. Chen, A. Gelb, and Y. Lee, Designing Neural Networks for Hyperbolic Conservation Laws, arxiv:2211.14375
T. Li, A. Gelb, and Y. Lee, Improving numerical accuracy for the viscous-plastic formulation of sea ice, arXiv:2206.10061, in revision.
J. Han and Y. Lee, A Neural Network Approach for Homogenization of Multiscale Problems, SIAM Multiscale Modeling and Simulations, accepted.
J. Han and Y. Lee, Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks, arXiv:2112.01254. Will appear in Springer Lecture Notes in Computational Sciences
J. Han and Y. Lee, Inhomogenous Regularization in Inverse Problems, Journal of Computational and Applied Mathematics, Volume 428, 2023, 115193, ISSN 0377-0427
Y Lee, l_p regularization for ensemble Kalman inversibion, SIAM J. Sci. Comput., 43(5), A3417--A3437, 2021.
Y Lee, Fast time integration of parabolic equations with variable coefficients, submitted for publication [link]
I find fun in teaching mathematics, physics and computer science, especially their interdisciplinary applications in science and engineering. Applying theories to applications is one way to learn mathematics and science but I believe that it is more efficient to generalize ideas from examples (or experiences) and then apply to other applications. Thus motivating students from real-world examples is the primary goal of my teaching, which is followed by generalization of ideas and applications to other examples. Here are some comments from my students
Comments from my students
"You were so excited about the material and I liked that a lot! I enjoyed the part that you covered some basic stuff in the class when you realized that there are people from engineering and science, sitting there and they have absolutely no clue about what's going on! Thank you very much!"
"He did a great job of trying to incorporate real-world connections to help make what we were learning have value to us. He asked us questions to help us learn for ourselves and would take the time to show us alternative solutions or alternative problems so that we would be exposed to multiple ways of analyzing problems. Thank you for an enjoyable math experience from someone who hadn't been expecting one."
"He was amazing and gave helpful tips for not only this course but for future math and science courses."
"I hope you can find accomplishment in the fact that I have decided to double major in CS and Math because of what you have taught me."
"I really enjoyed the physics examples provided by Yoonsang. He created a very entertaining format to learn calculus."
"Extremely intelligent Yoonsang Lee gave lessons outside of math that pertained to many subjects. He related math to many applications, which at times complicated things."
"He will be a great professor one day because he is able to explain complex conceptual problems in simplistic terms!"
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.
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
