子集模拟中马尔可夫链蒙特卡洛抽样算法比较

Comparison of Markov Chain Monte Carlo sampling algorithms in subset simulation

文章着重研究子集模拟中马尔可夫链蒙特卡洛(MCMC)抽样算法的抽样效率与计算精度。首先,阐述可靠度子集模拟的基本原理与中间状态样本生成的各种MCMC抽样算法,在稳态马尔可夫链构造基础上提出延迟拒绝MMH(Modified Metropolis-Hasting)算法,通过在MMH算法上增加备选样本的延迟拒绝步提高MMH算法的抽样效率;阐述基于随机游走与基于扩散方程MCMC方法中建议分布的差异,进一步对备选样本接受率为1的preconditioned Crank-Nicolson(pCN)算法和条件抽样(Conditional sampling, CS)算法开展研究,证明两种算法的等价性;推导有效样本量的计算方法,提出采用有效样本量与总样本量的比值定义MCMC方法的抽样效率。通过复杂目标分布的样本生成研究不同MCMC抽样算法建议分布及其参数对备选样本接受率与抽样效率的影响,最后通过计算实例研究子集模拟过程采用不同MCMC抽样算法得到失效概率的相对误差及其变异性,揭示不同MCMC抽样算法对失效概率计算精度的影响。研究表明:不同MCMC抽样算法生成备选样本的接受率及其自相关性受建议分布及其参数影响较大,对于复杂的目标分布,pCN算法和CS算法的抽样效率较高,延迟拒绝MMH算法次之;采用CS算法和延迟拒绝MMH算法进行子集模拟得到的失效概率精度较高且变异性较低;增加子集模拟中间状态样本量可以提高失效概率计算精度并降低其变异性。

The sampling efficiency and accuracy for the Markov Chain Monte Carlo(MCMC) methods in the subset simulation are evaluated in this work.The principles of the subset simulation for structural reliability and different MCMC algorithms of generating samples in the intermediate state are elaborated.Based on the construction of a steady-state Markov chain, a modified Metropolis-Hasting(MMH) with delayed rejection(MMHDR) is proposed, in which the step of delayed rejection for the candidate sample is added over the current MMH to improve the sampling efficiency.The differences in the proposed distribution for MCMC methods based on random walk and diffusion equation are illustrated.Especially, the preconditioned Crank-Nicolson(pCN) algorithm and the conditional sampling(CS) algorithm with acceptance rates of candidate samples equal to 1 have been proved to be equivalent in the standard normal space.In addition, the method of evaluating the effective sample size is deduced, and the sampling efficiency is defined as the ratio of effective sample size to total sample size.Subsequently, to evaluate the influences of the proposed distribution and its parameters on the acceptance rate of candidate samples and the sampling efficiency, the samples are generated by the different algorithms and compared with the target distribution.The relative errors and variabilities for the failure probability calculated by different MCMC algorithms are studied to reveal the influences of sampling algorithms on the computing accuracy.The results indicate that the acceptance rate and the autocorrelation of candidate samples are influenced significantly by the proposed distribution and its parameters.For the complicated target distributions, the samplingefficiencies of the pCN and the CS algorithms are comparatively higher, and that of the MMHDR comes second.By using the CS algorithm and the MMHDR in the subset simulation, the higher accuracy and the lower variability for failure probabilities can be obtained.Also, the computing accuracy and the variability for failure probability can be improved with the increase in the sample size of intermediate states.