李铁军
博士 教授 博士生导师
北京大学 数学科学学院
随机模型及算法
个性化签名
 姓名：李铁军
 目前身份：在职研究人员
 担任导师情况：博士生导师
 学位：博士

学术头衔：
博士生导师
 职称：高级教授

学科领域：
计算数学
 研究兴趣：随机模型及算法
李铁军 北京大学数学科学学院科学与工程计算系 教授
教育经历:
2001 北京大学 博士
1998 清华大学 硕士
1995 清华大学 学士
工作经历:
2010 北京大学数学科学学院 教授
20052010 北京大学数学科学学院 副教授
20012005 北京大学数学科学学院 助理教授
研究领域：
随机模型及算法
获得2012年度国家优秀青年科学基金资助和2018年度国家杰出青年科学基金资助。

主页访问
15

关注数
0

成果阅读
36

成果数
37
【期刊论文】Analysis of explicit tauleaping schemes for simulating chemically reacting systems
Multiscale Model. Simul.，0001，6（2）：417–436
1年11月30日
This paper builds a convergence analysis of explicit tauleaping schemes for simulating chemical reactions from the viewpoint of stochastic differential equations. Mathematically, thechemical reaction process is a pure jump process on a lattice with statedependent intensity. Thestochastic differential equation form of the chemical master equation can be given via Poisson random measures. Based on this form, different types of tauleaping schemes can be proposed. In orderto make the problem wellposed, a modified explicit tauleaping scheme is considered. It is shownthat the mean square strong convergence is of order 1/2 and the weak convergence is of order 1 forthis modified scheme. The novelty of the analysis is to handle the nonLipschitz property of thecoefficients and jumps on the integer lattice.
tauleaping scheme,， jump process,， statedependent intensity,， convergence analysis,， nonLipschitz coefficient
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【期刊论文】SROCK methods for stiff Itô SDEs
Commun. Math. Sci.，0001，6（4）：845–868
1年11月30日
In this paper, we present a class of explicit numerical methods for stiff Itˆo stochasticdifferential equations (SDEs). These methods are as simple to program and to use as the wellknownEulerMaruyama method, but much more efficient for stiff SDEs.For such problems, it is wellknown that standard explicit methods face stepsize reduction. While semiimplicit methods canavoid these problems at the cost of solving (possibly large)nonlinear systems, we show that the stepsize reduction phenomena can be reduced significantly for explicit methods by using stabilizationtechniques. Stabilized explicit numerical methods calledSROCK (for stochastic orthogonal RungeKutta Chebyshev) have been introduced in [C. R. Acad. Sci. Paris, vol. 345, no. 10, 2007] asan alternative to (semi) implicit methods for the solutionof stiff stochastic systems. In this paperwe discuss a genuine Itˆo version of the SROCK methods whichavoid the use of transformationformulas from Stratonovich to Itˆo calculus. This is important for many applications. We present twofamilies of methods for onedimensional and multidimensional Wiener processes. We show that forstiff problems, significant improvement over classical explicit methods can be obtained. Convergenceand stability properties of the methods are discussed and numerical examples as well as applicationsto the simulation of stiff chemical Langevin equations are presented.
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【期刊论文】Highly accurate tauleaping methods with random corrections
J. Chem. Phys.，2009，130（12）：124109
2009年03月24日
We aim to construct higher order tauleaping methods for numerically simulating stochastic chemical kinetic systems in this paper. By adding a random correction to the primitive tauleaping scheme in each time step, we greatly improve the accuracy of the tauleaping approximations. This gain in accuracy actually comes from the reduction in the local truncation error of the scheme in the order of τ, the marching time step size. While the local truncation error of the primitive tauleaping method is
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【期刊论文】CHEBYSHEV METHODS WITH DISCRETE NOISE: THEτROCK METHODS
Journal of Computational Mathematics，2009，28（2）：195–217
2009年12月21日
Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiffordinary differential equations. Making use of special properties of Chebyshevlike polynomials, these methods have favorable stability properties compared to standard explicitmethods while remaining explicit. A new class of such methods, called ROCK, introducedin [Numer. Math., 90, 118, 2001] has recently been extended to stiff stochastic differentialequations under the name SROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun.Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the SROCK methodsto systems with discrete noise and propose a new class of methods for such problems, theτROCK methods. One motivation for such methods is the simulation of multiscale orstiff chemical kinetic systems and such systems are the focus of this paper, but our newmethods could potentially be interesting for other stiff systems with discrete noise. Twoversions of theτROCK methods are discussed and their stability behavior is analyzed ona test problem. Compared to theτleaping method, a significant speedup can be achievedfor some stiff kinetic systems. The behavior of the proposed methods are tested on severalnumerical experiments.
Stiff stochastic differential equations， RungeKutta Chebyshev methods， Chemical reaction systems， tauleaping method
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【期刊论文】The weak convergence analysis of tauleaping methods: revisited
Communications in Mathematical Sciences，2011，9（4）：965 – 996
2011年07月29日
There are two scalings for the convergence analysis of tauleaping methods in the literature. This paper attempts to resolve this debate in the paper. We point out the shortcomings of both scalings. We systematically develop the weak ItoTaylor expansion based on the infinitesimal generator of the chemical kinetic system and generalize the rooted tree theory for ODEs and SDEs driven by Brownian motion to rooted directed graph theory for the jump processes. We formulate the local truncation error analysis based on the large volume scaling. We find that even in this framework the midpoint tauleaping does not improve the weak local order for the covariance compared with the explicit tauleaping. We propose a procedure to explain the numerical order behavior by abandoning the dependence on the volume constant V from the leading error term. The numerical examples validate our arguments. We also give a general global weak convergence analysis for the explicit tauleaping type methods in the large volume scaling.
chemical reaction kinetics,， large volume scaling,， convergence analysis,， rooted tree theory
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【期刊论文】A weak second order tauleaping method for chemical kinetic systems
J. Chem. Phys.，0001，135（2）：024113
1年11月30日
Recently Anderson and Mattingly [Comm. Math. Sci. 9, 301 (2011)] proposed a method which can solve chemical Langevin equations with weak second order accuracy. We extend their work to the discrete chemical jump processes. With slight modification, the method can also solve discrete chemical kinetic systems with weak second order accuracy in the large volume scaling. Especially, this method achieves higher order accuracy than both the Euler τleaping and midpoint τleaping methods in the sense that the local truncation error for the covariance is of order τ3V−1 when τ = V−β (0 < β < 1) and the system size V → ∞. We present the convergence analysis, numerical stability analysis, and numerical examples. Overall, in the authors’ opinion, the new method is easy to be implemented and good in performance, which is a good candidate among the highly accurate τleaping type schemes for discrete chemical reaction systems.
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【期刊论文】Efficient simulation under a population genetics model of carcinogenesis
Bioinformatics，2011，27（6）：837–843
2011年01月18日
Motivation: Cancer is well known to be the end result of somatic mutations that disrupt normal cell division. The number of such mutations that have to be accumulated in a cell before cancer develops depends on the type of cancer. The waiting time Tm until the appearance of m mutations in a cell is thus an important quantity in population genetics models of carcinogenesis. Such models are often difficult to analyze theoretically because of the complex interactions of mutation, drift and selection. They are also computationally expensive to simulate because of the large number of cells and the low mutation rate. Results: We develop an efficient algorithm for simulating the waiting time Tm until m mutations under a population genetics model of cancer development. We use an exact algorithm to simulate evolution of small cell populations and coarsegrained τleaping approximation to handle large populations. We compared our hybrid simulation algorithm with the exact algorithm in small populations and with available asymptotic results for large populations. The comparison suggested that our algorithm is accurate and computationally efficient. We used the algorithm to study the waiting time for up to 20 mutations under a Moran model with variable population sizes. Our new algorithm may be useful for studying realistic models of carcinogenesis, which incorporates variable mutation rates and fitness effects.
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【期刊论文】Numerical study for the nucleation of onedimensional stochastic CahnHilliard dynamics
Communications in Mathematical Sciences，2012，10（4）：1105 – 113
2012年07月23日
We consider the nucleation of onedimensional stochastic CahnHilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic CahnHilliard dynamics through finite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios.
CahnHilliard equation,， large deviation theory,， nucleation rate,， string method,， MetropolisHastings algorithm
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Multiscale Modeling & Simulation，2013，11（1）：385–409
2013年03月21日
We focus on the nucleation rate calculation for diblock copolymers by studying the twodimensional stochastic CahnHilliard dynamics with a LandauBrazovskii energy functional. To do this, we devise the string method to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space. Both methods are combined with the semiimplicit spectral method and hence are very effective. We derive the nucleation rate formula in the infinitedimensional case and prove the convergence under numerical discretizations. The computation of the determinant ratio is also discussed for obtaining the rate. The algorithm is successfully applied to investigate the nucleation from the lamellar phase to the cylinder phase in the mean field theory for diblock copolymer melts. The comparison with projected stochastic AllenCahn dynamics is also discussed.
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【期刊论文】Constructing the Energy Landscape for Genetic Switching System Driven by Intrinsic Noise
PLoS ONE ，2014，9（2）： e88167
2014年02月13日
Genetic switching driven by noise is a fundamental cellular process in genetic regulatory networks. Quantitatively characterizing this switching and its fluctuation properties is a key problem in computational biology. With an autoregulatory dimer model as a specific example, we design a general methodology to quantitatively understand the metastability of gene regulatory system perturbed by intrinsic noise. Based on the large deviation theory, we develop new analytical techniques to describe and calculate the optimal transition paths between the on and off states. We also construct the global quasipotential energy landscape for the dimer model. From the obtained quasipotential, we can extract quantitative results such as the stationary distributions of mRNA, protein and dimer, the noise strength of the expression state, and the mean switching time starting from either stable state. In the final stage, we apply this procedure to a transcriptional cascades model. Our results suggest that the quasipotential energy landscape and the proposed methodology are general to understand the metastability in other biological systems with intrinsic noise.
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