Welcome to Tiejun Li's homepage

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) 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.
• Local truncation error analysis of tau-leaping methods: revisited, with Yucheng Hu and Bin Min, submitted to Comm. Math. Sci., 2009.
• Chebyshev methods with discrete noise: the tau-ROCK methods, with Assyr Abdulle and Yucheng Hu, submitted to J. Comp. Math., 2009.

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.

• Convergence of a stochastic method for the modeling of polymeric fluids, with Weinan E and Pingwen Zhang, Acta Math. Appl. Sin. 18 (2002), 529-536.
• 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.

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
• A probabilistic framework for network partition, with Weinan E and Jian Liu, submitted to Phys. Rev. E

Miscs: Though diversified, most of them have one central theme: STOCHASTICS.

• Localized basis of eigen-subspaces and operator compression, with Weinan E and Jianfeng Lu, submitted to Proc. Nat. Acad. Sci.
• Image segmentation by using the localized subspace iteration algorithm, with Jinlong Wu, Science in China, series A, 51 (2008), 1495-1509.
• 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.
• 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.

If you want to apply as my graduate student, or ask me to write recommendation letters for you, please read this at first.