Conventional Markov Chain Monte Carlo Research Articles

A Study on the parameter estimation for crack growth prediction under variable amplitude loading

Bayesian formulation is presented to address the parameters estimation under uncertainty in the crack growth prediction subjected to variable amplitude loading. Huang's model is employed to describe the retardation and acceleration of the crack growth during the loadings. Model parameters are estimated in probabilistic way and updated conditional on the measured data by Bayesian inference. Markov Chain Monte Carlo (MCMC) method is employed for efficient sampling of the parameter distributions. As the model under variable amplitude loading is more complex, the conventional MCMC often fails to converge to the equilibrium distribution due to the increased number of parameters and correlations. An improved MCMC is introduced to overcome this failure, in which marginal PDF is employed as a proposal density function. A center- cracked panel under a mode I loading is considered for the feasibility study. Parameters are estimated based on the data from specimen tests. Prediction is carried out afterwards under variable amplitude loading for the same specimen, and validated by the ground truth data.

An Efficient Two-Stage Sampling Method for Uncertainty Quantification in History Matching Geological Models

Summary The paper presents a novel method for rapid quantification of uncertainty in history matching reservoir models using a two-stage Markov Chain Monte Carlo (MCMC) method. Our approach is based on a combination of fast linearized approximation to the dynamic data and the MCMC algorithm. In the first stage, we use streamline-derived sensitivities to obtain an analytical approximation in a small neighborhood of the previously computed dynamic data. The sensitivities can be conveniently obtained using either a finite-difference or streamline simulator. The approximation of the dynamic data is then used to modify the instrumental proposal distribution during MCMC. In the second stage, those proposals that pass the first stage are assessed by running full flow simulations to assure rigorousness in sampling. The uncertainty analysis is carried out by analyzing multiple models sampled from the posterior distribution in the Bayesian formulation for history matching. We demonstrate that the two-stage approach increases the acceptance rate, and significantly reduces the computational cost compared to conventional MCMC sampling without sacrificing accuracy. Finally, both 2D synthetic and 3D field examples demonstrate the power and utility of the two-stage MCMC method for history matching and uncertainty analysis.

A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: easy Bayesian computing for real parameter spaces

Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. This paper integrates the essential ideas of DE and MCMC, resulting in Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a fixed multiple of the differences of two random parameter vectors that are currently in the population. The selection process of DE-MC works via the usual Metropolis ratio which defines the probability with which a proposal is accepted. In tests with known uncertainty distributions, the efficiency of DE-MC with respect to random walk Metropolis with optimal multivariate Normal jumps ranged from 68% for small population sizes to 100% for large population sizes and even to 500% for the 97.5% point of a variable from a 50-dimensional Student distribution. Two Bayesian examples illustrate the potential of DE-MC in practice. DE-MC is shown to facilitate multidimensional updates in a multi-chain "Metropolis-within-Gibbs" sampling approach. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities.

Efficient MCMC estimation of discrete distributions

In this paper we propose an efficient Markov chain Monte Carlo (MCMC) method for estimation of discrete distributions by solving an appropriate system of linear equations. We call the estimator the equation-solving estimator. Our numerical results show that the new estimator makes significant improvements over the conventional frequency MCMC estimator in terms of accuracy of the estimates. The new estimator can be used in Bayesian model comparison problems.

System identification using evolutionary Markov chain Monte Carlo

System identification involves determination of the functional structure of a target system that underlies the observed data. In this paper, we present a probabilistic evolutionary method that optimizes system architectures for the identification of unknown target systems. The method is distinguished from existing evolutionary algorithms (EAs) in that the individuals are generated from a probability distribution as in Markov chain Monte Carlo (MCMC). It is also distinguished from conventional MCMC methods in that the search is population-based as in standard evolutionary algorithms. The effectiveness of this hybrid of evolutionary computation and MCMC is tested on a practical problem, i.e., evolving neural net architectures for the identification of nonlinear dynamic systems. Experimental evidence supports that evolutionary MCMC (or eMCMC) exploits the efficiency of simple evolutionary algorithms while maintaining the robustness of MCMC methods and outperforms either approach used alone.