Calculating Principal Eigen-Functions of Non-Negative Integral Kernels: Particle Approximations and Applications
针对非负积分核的主特征函数和特征值难以精确计算的问题，提出一种交互粒子算法进行数值近似，并应用于稀有事件概率估计和随机最优控制中的值函数计算。
Often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigenfunction and eigenvalue of a nonnegative integral kernel. Except in the finite-dimensional case, usually neither the principal eigenfunction nor the eigenvalue can be computed exactly. In this paper, we develop numerical approximations for these quantities. We show how a generic interacting particle algorithm can be used to deliver numerical approximations of the eigenquantities and the associated so-called “twisted” Markov kernel as well as how these approximations are relevant to the aforementioned applications. In addition, we study a collection of random integral operators underlying the algorithm, address some of their mean and pathwise properties, and obtain error estimates. Finally, numerical examples are provided in the context of importance sampling for computing tail probabilities of Markov chains and computing value functions for a class of stochastic optimal control problems.