Large deviations of Markov chains with multiple time-scales

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in Kang and Kurtz (2013) and Kang et al. (2014). We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.     

Continuum percolation for Cox point processes

We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.     

系统数值优化方法的研究与应用

大系统具有结构复杂、功能综合、因素众多等特点,相应的规划和决策问题的求解是非常困难的。我们针对大系统工程规划和决策的特点,结合运筹学的原理,提出了大系统数值优化概念,建立了优化模型,给出了优化问题的求解方法;并选择了中石油的DS-1油田作为应用实例,进行了油田采油工程数值优化,取得了很好的效果。为一般的复杂工程系统的规划和决策提供了一个新方法。     

A curvilinear method based on minimal-memory BFGS updates

We present a new matrix-free method for the computation of negative curvature directions based on the eigenstructure of minimal-memory BFGS matrices. We determine via simple formulas the eigenvalues of these matrices and we compute the desirable eigenvectors by explicit forms. Consequently, a negative curvature direction is computed in such a way that avoids the storage and the factorization of any matrix. We propose a modification of the L-BFGS method in which no information is kept from old iterations, so that memory requirements are minimal. The proposed algorithm incorporates a curvilinear path and a linesearch procedure, which combines two search directions; a memoryless quasi-Newton direction and a direction of negative curvature. Results of numerical experiments for large scale problems are also presented.     

On consistency of stationary points of stochastic optimization problems in a Banach space

Recently, Balaji and Xu studied the consistency of stationary points, in the sense of the Clarke generalized gradient, for the sample average approximations to a one-stage stochastic optimization problem in a separable Banach space with separable dual. We present an alternative approach, showing that the restrictive assumptions that the dual space is separable and the Clarke generalized gradient is a (norm) upper semicontinuous and compact-valued multifunction can be dropped. For that purpose, we use two results having independent interest: a strong law of large numbers and a multivalued Komlós theorem in the dual to a separable Banach space, and a result on the weak* closedness of the expectation of a random weak* compact convex set.     

Comparison of Krylov subspace methods on the PageRank problem

PageRank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions. The problem incorporates a parameter that determines the difficulty of the problem. In this paper, the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of . We see that stationary methods are very reliable and more competitive when the problem is well conditioned, that is for small values of . However, for large values of the parameter the problem becomes more difficult and methods such as preconditioned BiCGStab or restarted preconditioned GMRES become competitive with stationary methods in terms of Mflops count as well as in number of iterations necessary to reach convergence.     

Variational representations of Varadhan functionals

Motivated by the theory of large deviations, we introduce a class of non-negative non-linear functionals that have a variational ``rate function" representation.