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Mean field study of a propagation-turnover lattice model for the dynamics of histone marking

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Science China(Physics,Mechanics & Astronomy) 2017年第2期60卷 49-63页

作者： Fan Yao FangTing li tiejun li LMAM and School of Mathematical Sciences Peking University Beijing 100871 China Schooi of Physics Peking University Beijing 100871 China Center for Quantitative Biology Peking University Beijing 100871 China

We present a mean field study of a propagation-tumover lattice model, which was proposed by Hodges and Crabtree [Proc. Nat. Acad. Sci. 109, 13296 （2012）] for understanding how posttranslational histone marks modulate gene expression in mammalian ceils. The kinetics of the lattice model consists of nucleation, propagation and turnover mechanisms, and exhibits second-order phase transition for the histone marking domain. We showed rigorously that the dynamics essentially depends on a non-dimensional parameter k = k＋/k-, the ratio between the propagation and turnover rates, which has been observed in the simulations. We then studied the lowest order mean field approximation, and observed the phase transition with an analytically obtained critical parameter. The boundary layer analysis was utilized to investigate the structure of the decay profile of the mark density. We also studied the higher order mean field approximation to achieve sharper estimate of the critical transition parameter and more detailed features. The comparison between the simulation and theoretical results shows the validity of our theory.

关键词： histone marking contact process mean field approximation boundary layer analysis

An Parallel FPGA SAT Solver Based on Multi-Thread and Pipeline

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Chinese Journal of Electronics 2021年第6期30卷 1008-1016页

作者： li tiejun MA Kefan ZHANG Jianmin College of Computer National University of Defense Technology

The Boolean Satisfiability(SAT) problem is the key problem in computer theory and application. A parallel multi-thread SAT solver named pprob SAT+ on a configurable hardware is proposed. In the algorithm,multithreads are executed simultaneously to hide the circuit stagnate. In order to improve the working frequency and throughput of the SAT solver, the deep pipeline strategy is adopted. When all data stored in block random access memory of the field programmable gate array, the solver can achieve maximum performance. If partial data are stored in the external memory, the size of the problem instances the SAT solver can be greatly improved. The experimental results show that the speedup of three-thread SAT solver is approximately 2.4 times with single thread,and shows that the pprob SAT+ have achieved substantial improvement while a solution is found.

关键词： field programmable gate arrays configurable hardware parallel multithread SAT solver Boolean satisfiability problem deep pipeline strategy block random access memory random-access storage pprobSAT three-thread SAT solver parallel FPGA SAT solver field programmable gate array computer theory

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation

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National Science Review 2024年第7期11卷 135-147页

作者： Yue Zhao Wei Zhang tiejun li Center for Data Science Peking University Zuse Institute Berlin Department of Mathematics and Computer Science Freie Universit?t Berlin Laboratory of Mathematics and Applied Mathematics(LMAM)and School of Mathematical Sciences Peking University Center for Machine Learning Research Peking University

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state ***-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product ***, our loss function has an intimate connection with the steady entropy production rate(EPR),enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional(8D)limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

关键词： high-dimensional potential landscape non-equilibrium system entropy production rate dimensionality reduction deep learning

CHEBYSHEV METHODS WITH DISCRETE NOISE:THE T-ROCK METHODS

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Journal of Computational Mathematics 2010年第2期28卷 195-217页

作者： Assyr Abdulle Yucheng Hu tiejun li Mathematics Section Ecole Polytechnique Federale de Lausanne CH-1015 Lausanne Switzerland Laboratory of Mathematics and Applied Mathematics and School of Mathematical Sciences Peking University Beijing 100871 China

Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345（10）, 2007 and Commun. Math. Sci, 6（4）, 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.

关键词： Stiff stochastic differential equations Runge-Kutta Chebyshev methods Chem-ical reaction systems tau-leaping method

Accelerating an FPGA-Based SAT Solver by Software and Hardware Co-design

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Chinese Journal of Electronics 2019年第5期28卷 953-961页

作者： MA Kefan XIAO liquan ZHANG Jianmin li tiejun College of Computer National University of Defense Technology

The Boolean Satisfiability(SAT) problem is the key problem in computer theory and ***-programmable gate array(FPGA) has been addressed frequently to accelerate the SAT solving process in the last few years owing to its parallelism and *** have proposed a novel SAT solver based on an improved local search algorithm on the reconfigurable hardware platform. The new software preprocessing procedure and hardware architecture are involved to solve large-scale SAT problems instances. As compared with the past solvers, the proposed solver has the following advantages: the preprocessing technology can strongly improve the efficiency of solver; the strategy of strengthening the variable selection can avoid the same variable flipped continuously and repeatedly. It reduces the possibility of search falling into local minima. The experimental results indicate that the solver can solve problems of up to 32 K variables/128 K clauses without off-chip memory banks, and has better performance than previous works.

关键词： FPGA SAT Preprocessing Enhanced constraint Incomplete algorithm Co-design

Image segmentation by using the localized subspace iteration algorithm

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Science China Mathematics 2008年第8期51卷 1495-1509页

作者： WU JinLong li tiejun LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China

An image segmentation algorithm called"segmentation based on the localized subspace iterations"(SLSI)is proposed in this *** basic idea is to combine the strategies in Ncut algorithm by Shi and Malik in 2000 and the LSI by E,li and Lu in *** LSI is applied to solve an eigenvalue problem associated with the affinity matrix of an image,which makes the overall algorithm linearly *** choices of the partition number,the supports and weight functions in SLSI are *** experiments for real images show the applicability of the algorithm.

关键词： image segmentation localized subspace iteration ncut graph Laplacian 15A18 68U10

Energy landscape decomposition for cell differentiation with proliferation effect

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National Science Review 2022年第8期9卷 144-152页

作者： Jifan Shi Kazuyuki Aihara tiejun li Luonan Chen Research Institute of Intelligent Complex Systems Fudan University International Research Center for Neurointelligence The University of Tokyo Institutes for Advanced Study The University of Tokyo Key Laboratory of Mathematics and Its Applications (LMAM) and School of Mathematical Sciences Peking University Key Laboratory of Systems Biology Shanghai Institute of Biochemistry and Cell Biology Center for Excellence in Molecular Cell Science Chinese Academy of Sciences Key Laboratory of Systems Health Science of Zhejiang Province Hangzhou Institute for Advanced Study University of Chinese Academy of Sciences Chinese Academy of Sciences School of life Science and Technology Shanghai Tech University Guangdong Institute of Intelligence Science and Technology

Complex interactions between genes determine the development and differentiation of cells. We establish a landscape theory for cell differentiation with proliferation effect, in which the developmental process is modeled as a stochastic dynamical system with a birth-death term. We find that two different energy landscapes, denoted U and V, collectively contribute to the establishment of non-equilibrium steady differentiation. The potential U is known as the energy landscape leading to the steady distribution, whose metastable states stand for cell types, while V indicates the differentiation direction from pluripotent to differentiated cells. This interpretation of cell differentiation is different from the previous landscape theory without the proliferation effect. We propose feasible numerical methods and a mean-field approximation for constructing landscapes U and V. Successful applications to typical biological models demonstrate the energy landscape decomposition’s validity and reveal biological insights into the considered processes.

关键词： energy landscape cell differentiation stochastic systems

The graph limit of the minimizer of the Onsager-Machlup functional and its computation

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Science China Mathematics 2021年第2期64卷 239-280页

作者： Qiang Du tiejun li Xiaoguang li Weiqing Ren Department of Applied Phgysics and Applied Mathematics Columbia UniversityNeu YorkNY 10027USA LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Beijing Computational Science Research Certer Beijing 100193China MOE-LCSM School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China Department of Mathematics National University of SingaporeSingapore 119076Singapore

The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing ***,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to *** hinders the interpretation of the results obtained by minimizing the OM *** provide a new perspective on this *** mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of *** graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal *** further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path *** algorithm is successfully applied to several typical examples in rare event *** interesting comparisons with the Freidlin-Wentzell action functional are also made.

关键词： Onsager-Machlup functional Freidlin-Wentzell functional graph limit geometric minimization Maupertuis principle

Effectiveness of Implicit Methods for Stiff Stochastic Differential Equations

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Communications in Computational Physics 2008年第2期3卷 295-307页

作者： tiejun li Assyr Abdulle Weinan E LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China School of Mathematics and Maxwell Institute for Mathematical Sciences King’s BuildingsEdinburghEH93JZUK Department of Mathematics and PACM Princeton UniversityPrincetonNJ 08544USA.

In this paper we study the behavior of a family of implicit numerical methods applied to stochastic differential equations with multiple time *** show by a combination of analytical arguments and numerical examples that implicit methods in general fail to capture the effective dynamics at the slow time *** is due to the fact that such implicit methods cannot correctly capture non-Dirac invariant distributions when the time step size is much larger than the relaxation time of the system.

关键词： Implicit methods stiff ODE stiff SDE invariant distribution multiscale.