Modified Metropolis-Hastings Algorithm Research Articles

Pipelines reliability assessment considering corrosion-related failure modes and probability distributions characteristic using subset simulation

A methodology using the subset simulation and the modified Metropolis-Hastings algorithm with delayed rejection (SS-MMHDR) is developed to evaluate the time-dependent reliability of corroded pipelines under multiple failure modes. This methodology considers the competition between the small leak and burst failures and the correlated non-normal distribution of the defect variables, and characterizes the defect geometric probability distribution from extensive inspection data. Results demonstrate that SS-MMHDR has high accuracy and low variability in evaluating the pipeline failures, outperforming the SS-MMH and Monte Carlo simulation algorithms, especially under low probability levels, but with slightly increased computation time. The Generalized Extreme Value Type II (GEVII) distribution is the optimal distribution for internal corrosion defect lengths from the Kolmogorov-Smirnov test. Lognormal, Weibull and Normal distributions provide more conservative estimates on burst and composite failure probabilities and limiting maintenance times than GEVII. Effects of operating pressure and its variability, corrosion rate on failure probability, and pipeline limiting maintenance time are also revealed through numerical cases.

Estimating rare events in biochemical systems using conditional sampling.

The paper focuses on development of variance reduction strategies to estimate rare events in biochemical systems. Obtaining this probability using brute force Monte Carlo simulations in conjunction with the stochastic simulation algorithm (Gillespie's method) is computationally prohibitive. To circumvent this, important sampling tools such as the weighted stochastic simulation algorithm and the doubly weighted stochastic simulation algorithm have been proposed. However, these strategies require an additional step of determining the important region to sample from, which is not straightforward for most of the problems. In this paper, we apply the subset simulation method, developed as a variance reduction tool in the context of structural engineering, to the problem of rare event estimation in biochemical systems. The main idea is that the rare event probability is expressed as a product of more frequent conditional probabilities. These conditional probabilities are estimated with high accuracy using Monte Carlo simulations, specifically the Markov chain Monte Carlo method with the modified Metropolis-Hastings algorithm. Generating sample realizations of the state vector using the stochastic simulation algorithm is viewed as mapping the discrete-state continuous-time random process to the standard normal random variable vector. This viewpoint opens up the possibility of applying more sophisticated and efficient sampling schemes developed elsewhere to problems in stochastic chemical kinetics. The results obtained using the subset simulation method are compared with existing variance reduction strategies for a few benchmark problems, and a satisfactory improvement in computational time is demonstrated.

Damage identification in concrete structures with uncertain but bounded measurements

The major sources of error in the measurements of concrete structures are the gauge sensitivities, calibration accuracies, amplitude linearities, and temperature corrections to the gauge sensitivities, which are given in terms of plus–minus ranges, and the round off errors in the measured responses, which are better represented by interval bounds. An algorithm is proposed adapting the existing modified Metropolis Hastings algorithm for estimating the posterior probability of the damage indices as well as the posterior probability of the bounds of the interval parameters, when the measurements are given in terms of interval bounds. A damage index is defined for each element of the finite element model considering the parameters of each element are intervals. Methods are developed for evaluating response bounds in the finite element software ABAQUS, when the parameters of the finite element model are intervals. The proposed method is validated against reinforced concrete beams with three damage scenarios mainly (1) loss of stiffness, (2) loss of mass, and (3) loss of bond between concrete and reinforcement steel, which have been tested in our laboratory. Comparison of the prediction from the proposed method with those obtained from Bayesian analysis and interval optimization technique show improved accuracy and computational efficiency in addition to better representation of measurement uncertainties through interval bounds.

The applications of an importance sampling method to reliability analysis of the inside flap of an aircraft

Based on the cross-entropy method and modified Metropolis–Hastings algorithm, an efficient importance sampling method is proposed to perform reliability analysis for the inside flap of an aircraft. In the proposed method, the cross-entropy method is utilized to estimate the optimal importance sampling probability density function. The derivation shows that the distribution parameters of the optimal importance sampling probability density function can be estimated by the failure sample points generated according to the original probability density function. The crude Monte Carlo method can be employed to simulate the failure sample points but its computational cost is prohibitive. In order to circumvent the difficulties, the modified Metropolis–Hasting algorithm is employed to generate failure sample points distributed as the original probability density function. After the distribution parameters of the optimal importance sampling probability density function have been obtained, the importance sampling probability density function can be constructed, then the proposed importance sampling method combined with the quasi-Monte Carlo simulation can be utilized to perform the reliability analysis. Meanwhile, the confidence interval and its length for the results have also been derived. Various numerical examples demonstrate the efficiency and accuracy of the presented importance sampling method. Finally, the reliability analysis and sensitivity analysis of an inside flap are solved by the proposed importance sampling method.

Effects of soil spatial variability on rainfall-induced landslides

This paper presents a probabilistic framework to assess the stability of unsaturated slope under rainfall. The effects of soil spatial variability on the probability of rainfall-induced slope failure (landslides) are investigated. Soil spatial variability is considered by modeling the saturated hydraulic conductivity of the soil ( k s ) as a stationary lognormal random field. Subset simulation with a modified Metropolis–Hastings algorithm is used to estimate the probability of slope failure. It is demonstrated numerically that probabilistic analysis accounting for spatial variability of k s can reproduce a shallow failure mechanism widely observed in real rainfall-induced landslides. This shallow failure is attributed to positive pore-water pressures developed in layers near the ground surface. In contrast, analysis assuming a homogeneous profile cannot reproduce a shallow failure except for the extreme case of infiltration flux being almost equal to k s . Therefore, ignoring spatial variability leads to unconservative estimates of failure probability. The correlation length of k s affects the probability of slope failure significantly. The applicability of subset simulation with a modified Metropolis–Hastings algorithm to assess the reliability of problems involving spatial variability is highlighted.

Modified Metropolis–Hastings algorithm with delayed rejection

The development of an efficient MCMC strategy for sampling from complex distributions is a difficult task that needs to be solved for calculating the small failure probabilities encountered in the high-dimensional reliability analysis of engineering systems. Usually different variations of the Metropolis–Hastings algorithm (MH) are used. However, the standard MH algorithm does not generally work in high dimensions, since it leads to very frequent repeated samples. In order to overcome this deficiency one can use the Modified Metropolis–Hastings algorithm (MMH) proposed in Au and Beck (2001) [1]. Another variation of the MH algorithm, called the Metropolis–Hastings algorithm with delayed rejection (MHDR) has been proposed by Tierney and Mira (1999) [7]. The key idea behind the MHDR algorithm is to reduce the correlation between states of the Markov chain. In this paper we combine the ideas of MMH and MHDR and propose a novel modification of the MH algorithm, called the Modified Metropolis–Hastings algorithm with delayed rejection (MMHDR). The efficiency of the new algorithm is demonstrated with a numerical example where MMHDR is used together with Subset simulation for computing small failure probabilities in high dimensions.

Modified Metropolis–Hastings algorithm with reduced chain correlation for efficient subset simulation

Simulation of Markov chain samples using the Metropolis–Hastings algorithm is useful for reliability estimation. Subset simulation is an example of the reliability estimation method utilizing this algorithm. The efficiency of the simulation is governed by the correlation between the simulated Markov chain samples. The objective of this study is to propose a modified Metropolis–Hastings algorithm with reduced chain correlation. The modified algorithm differs from the original in terms of the transition probability. It has been verified that the modified algorithm satisfies the reversibility condition and therefore the simulated samples follow the target distribution for the correct theoretical reasons. When applied to subset simulation, the modified algorithm produces a more accurate estimate of failure probability as indicated by a lower coefficient of variation and a lower mean square error. The advantage is more significant for small failure probability. Examples of soil slope with spatially variable properties were presented to demonstrate the applicability of the proposed modification to reliability estimation of engineering problems. It was found that the modified algorithm produces a more accurate estimator over the range of random dimensions studied.

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f**x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.