Itô-SDE MCMC method for Bayesian characterization of errors associated with data limitations in stochastic expansion methods for uncertainty quantification

• Published in: Journal of computational physics Vol. 349; pp. 59 - 79
• Main Authors: Arnst, M., Abello Álvarez, B., Ponthot, J.-P., Boman, R.
• Format: Journal Article Web Resource
• Language: English
• Published: Cambridge Elsevier Inc 15.11.2017; Elsevier Science Ltd; Elsevier
• Subjects: Bayesian analysis; Bayesian inference; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational physics; Computer simulation; DIFFERENTIAL EQUATIONS; Engineering, computing & technology; Error budget; ERRORS; Ingénierie, informatique & technologie; Itô stochastic differential equation; Limited data; Markov Chain Monte Carlo; Markov chains; MARKOV PROCESS; MONTE CARLO METHOD; Monte Carlo simulation; NONLINEAR PROBLEMS; Probability theory; Stochastic models; Uncertainty; Itô stochastic differential equation; Markov Chain Monte Carlo; Bayesian inference; Error budget; Limited data
• ISSN: 0021-9991; 1090-2716; 1090-2716
• DOI: 10.1016/j.jcp.2017.08.005

This paper is concerned with the characterization and the propagation of errors associated with data limitations in polynomial-chaos-based stochastic methods for uncertainty quantification. Such an issue can arise in uncertainty quantification when only a limited amount of data is available. When the available information does not suffice to accurately determine the probability distributions that must be assigned to the uncertain variables, the Bayesian method for assigning these probability distributions becomes attractive because it allows the stochastic model to account explicitly for insufficiency of the available information. In previous work, such applications of the Bayesian method had already been implemented by using the Metropolis–Hastings and Gibbs Markov Chain Monte Carlo (MCMC) methods. In this paper, we present an alternative implementation, which uses an alternative MCMC method built around an Itô stochastic differential equation (SDE) that is ergodic for the Bayesian posterior. We draw together from the mathematics literature a number of formal properties of this Itô SDE that lend support to its use in the implementation of the Bayesian method, and we describe its discretization, including the choice of the free parameters, by using the implicit Euler method. We demonstrate the proposed methodology on a problem of uncertainty quantification in a complex nonlinear engineering application relevant to metal forming.