Tiejun Li

Chemical Reaction Kinetics: Currently we focus our research on the understanding of the tau-leaping algorithm and constructing more accurate and stabilized schemes based on solid mathematical analysis. I mainly collaborate with Assyr Abdulle (EPFL), Fangting Li (PKU) and my students Yucheng Hu and Bin Min (PKU) in this research field.

• Analysis of explicit tau-leaping schemes for simulating chemically reacting systems, Multi. Mod. Simul. 6 (2007), 417-436.
• S-ROCK methods for stiff Ito SDEs, with Assyr Abdulle, Comm. Math. Sci. 6 (2008), 845-868.
• Highly accurate tau-leaping methods for simulating chemically reacting systems, with Yucheng Hu, J. Chem. Phys 130 (2009), 124109.
• Chebyshev methods with discrete noise: the tau-ROCK methods, with Assyr Abdulle and Yucheng Hu, J. Comp. Math. 28 (2010), 195-217.
• The weak convergence analysis of tau-leaping methods: revisited, with Yucheng Hu and Bin Min, Comm. Math. Sci. 9 (2011), 965-996.
• A weak second order tau-leaping method for chemical kinetic systems, with Yucheng Hu and Bin Min, J. Chem. Phys. 135 (2011), 024113.
• Efficient simulation under a population genetics model of carcinogenesis, with Tianqi Zhu, Yucheng Hu, Zhiming Ma, Dexing Zhang, Ziheng Yang, Bioinformatics 27 (2011), 837-843.

Rare events: We aim to develop the mathematical theory and numerical methods for the rare events in material sciences and systems biology. I mainly collaborate with Wei Zhang (PKU), Pingwen Zhang (PKU) and Xiang Zhou (CityU HK) in this field.

• Numerical study for the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics, with Wei Zhang and Pingwen Zhang, Comm. Math. Sci. 10 (2012), 1105-1132.
• Nucleation rate calculations for the phase transition of diblock copolymers under stochastic Cahn-Hilliard dynamics, with Wei Zhang and Pingwen Zhang, SIAM Multi. Model. Simul. 11 (2013), 385-409.
• Transition path, quasi-potential energy landscape and robustness of genetic switches, with Cheng Lv, Xiaoguang Li and Fangting Li, Preprint.

Anderson Localization: The theme is to understand the effect of the noise (time-dependent for linear Schrodinger equation and time-independent for nonlinear Schrodinger equation) to the quantum system. I mainly collaborate with Weizhu Bao (NUS) and Bin Min (PKU) in this field.

• Subdiffusive spreading of a Bose-Einstein condensate in random potentials, with Weizhu Bao, Bin Min and M. Rosenkranz, Phys. Rev. A 86 (2012), 053612.
• Transport in dynamically disordered media: diffusion or nondiffusion?, with Bin Min, Submitted to Nature Physics.

Multiscale modeling of complex fluids: We have done some theoretical analysis for the multiscale models of complex fluids, such as the wellposedness of the equations, convergence analysis of the BCF method, and HMM methods. This is an exciting area. I mainly collaborate with Weinan E (Princeton), Eric Vanden-Eijnden (Courant), Hui Zhang (BNU) and Pingwen Zhang (PKU) in this field.

• Stochastic models of polymeric liquids at small Deborah number , with Eric Vanden-Eijnden, Pingwen Zhang, and Weinan E, J. Non-Newtonian Fluid Mech. 121 (2004), 117-125.
• Analysis of 1+1 dimensional stochastic models of liquid crystal polymer flows, with Pingwen Zhang and Xiang Zhou, Comm. Math. Sci. 2 (2004), 295-316.
• Local existence for the dumbbell model of polymeric fluids, with Hui Zhang and Pingwen Zhang, Comm. PDE, 29 (2004), 903-923.
• Well-posedness for the dumbbell model of polymeric liquids, with Weinan E and Pingwen Zhang, Comm. Math. Phys. 248 (2004), 409-427.
• Convergence analysis of BCF methods for Hookean dumbbell model , with Pingwen Zhang, Multi. Model. Simul. 5 (2006), 205-234.
• Mathematical analysis of multiscale models of complex fluids , with Pingwen Zhang, Comm. Math. Sci. 5 (2007), 1-51.

Statistical data analysis: The data is emerging each second in our time. How to extract the information from the data? We consider this problem from different interesting backgrounds such as the complex networks, quantum optics, Netflix problems etc.

• Model reduction of complex networks: Complex networks usually have huge amount of nodes and edges. How to optimally reduce this type of models to explore its essential behavior? This covers the problems from Sociology, Biology and dynamical system theory. We try to do this rationally based on Markov chain theory.
• Optimal partition and effective dynamics of complex networks, with Weinan E and Eric Vanden-Eijnden, Proc. Nat. Acad. Sci., 105 (2008), 7907-7912. SI
• Probabilistic framework for network partition, with Weinan E and Jian Liu, Phys. Rev. E 80 (2009), 026106.
• A validity index approach to network partitions, with Jian Liu, Phys. A, 390 (2011), 3579-3591.
• Other fields:
• Statistical method for resolving the photon-photoelectron-counting inversion problem, with Jinlong Wu, Xiang Peng and Hong Guo, J. Comp. Phys. 230 (2011) 726.
• A modified fuzzy C-means algorithm for collaborative filtering, with Jinlong Wu, The 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2008), Las Vegas, 2008.
• Effect of exposure to trace elements in the soil on the prevalence of neural tube defects in a high-risk area of China, with Jing Huang et al, Biomed. Environ. Sci. 24 (2011), 94-101.

Miscs:

• Localized basis of eigen-subspaces and operator compression, with Weinan E and Jianfeng Lu, Proc. Nat. Acad. Sci. 107 (2010), 1273.
• A note on the analysis of heterogeneous rod like chain in DNA modeling, with Yan Ding, Int. J. Mod. Phys. B, 22 (2008), 2213-2224.
• On the Poisson approximation for photon distribution for faint lasers, with Yucheng Hu, Xiang Peng and Hong Guo, Phys. Lett. A, 367 (2007), 173-176.
• Effectiveness of implicit methods for stiff stochastic differential equations, with Assyr Abdulle and Weinan E, Comm. Comp. Phys. 3 (2008), 295-307.

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