Covariance-based MCMC for high-dimensional Bayesian updating with sequential Monte Carlo序列蒙特卡罗下高维 Bayes 更新的协方差 Markov 链蒙特卡罗
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Carrera B, Papaioannou I, 2024. Covariance-based MCMC for high-dimensional Bayesian updating with sequential Monte Carlo. Probabilistic Engineering Mechanics, 77: 103667.DOI: 10.1016/j.probengmech.2024.103667
序列蒙特卡罗 (sequential Monte Carlo, SMC) 是 Bayes 更新下后验参数分布的有效抽样方法。该方法通过序列分布抽样，从先验分布逐渐逼近后验分布。通过重采样-移动方法进行分布序列抽样，其移动流程采用 Markov 链蒙特卡罗 (Markov chain Monte Carlo, MCMC) 算法。高维 Bayes 更新问题中 Markov 链蒙特卡罗流程常选用预条件 Crank-Nicolson (preconditioned Crank-Nicolson, pCN) ，因其对先验分布维数不敏感。本文提出了两类采用协方差信息的序列蒙特卡罗改进以给出 Markov 链蒙特卡罗分布，并将其表现与基于预条件 Crank-Nicolson的序列蒙特卡罗对比。特别是采用协方差信息的预条件 Crank-Nicolson 算法改进以及主成分 Metropolis-Hastings 算法。这些算法与目标分布协方差矩阵间歇性和递归更新格式相结合，基于当前 Markov 链蒙特卡罗结果进行更新。我们通过三个数值算例验证了算法表现：二维代数算例、悬臂梁柔性估计以及含水层水力传导场估计。结果表明，协方差 Markov 链蒙特卡罗算法的参数均值和方差误差更小，且相比于预条件 Crank-Nicolson 方法，模型证据估计也更精确。关键词: Markov 链蒙特卡罗, 序列蒙特卡罗, Bayes 更新, 反演, 水力断层成像Sequential Monte Carlo (SMC) is a reliable method to generate samples from the posterior parameter distribution in a Bayesian updating context. The method samples a series of distributions sequentially, which start from the prior distribution and gradually approach the posterior distribution. Sampling from the distribution sequence is performed through application of a resample-move scheme, whereby the move step is performed using a Markov Chain Monte Carlo (MCMC) algorithm. The preconditioned Crank–Nicolson (pCN) is a popular choice for the MCMC step in high dimensional Bayesian updating problems, since its performance is invariant to the dimension of the prior distribution. This paper proposes two other SMC variants that use covariance information to inform the MCMC distribution proposals and compares their performance to the one of pCN-based SMC. Particularly, a variation of the pCN algorithm that employs covariance information, and the principle component Metropolis Hastings algorithm are considered. These algorithms are combined with an intermittent and recursive updating scheme of the target distribution covariance matrix based on the current MCMC progress. We test the performance of the algorithms in three numerical examples; a two dimensional algebraic example, the estimation of the flexibility of a cantilever beam and the estimation of the hydraulic conductivity field of an aquifer. The results show that covariance-based MCMC algorithms are capable of producing smaller errors in parameter mean and variance and better estimates of the model evidence compared to the pCN approach.
Keywords: Markov chain Monte Carlo; Sequential Monte Carlo; Bayesian updating; Inversion; Hydraulic tomography.![]()
Fig. 1. Prior and posterior distributions in two-dimensional example
Fig. 2. Two-dimensional example
Fig. 3. Two-dimensional example with noise
图 4: 悬臂梁的真实柔度、真实挠度与 1e-3 标准差下的估计挠度Fig. 4. Cantilever beam (top), true F(x) (left), true w(x) and w~ based on σ_η = 1e-3 (right)
Fig. 5. Beam example, measure mean and standard deviation bounds
Fig. 6. Beam example, study of measurement noise for J = 3000
Fig. 7. Beam example, study of number of parameters for J = 3000
Fig. 8. Hydraulic tomography example
Fig. 9. Hydraulic tomography, low σ_lnK case
Fig. 10. Hydraulic tomography, high σ_lnK case
图 11: 水力断层成像样本接受率均值与标准差界随退火步数的变化Fig. 11. Hydraulic tomography, sample acceptance rate mean and standard deviation bounds over number of tempering steps
作者信息 | Authors
Barbara Carrera, 通讯作者 (Corresp.)美国 GSI 环境公司 (GSI Environmental Inc)Email: barbara.carrera@tum.de
德国慕尼黑工业大学 (Technische Universität München) 工程风险分析研究所
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
