A comparison of Markov Chain Monte Carlo algorithms for Bayesian inference of constitutive models

2026-04-01

AI Summary

This study addresses the lack of systematic criteria for selecting Markov chain Monte Carlo (MCMC) algorithms in Bayesian calibration of constitutive model parameters. Focusing on heat conduction and viscous flow systems, it introduces Kullback–Leibler (KL) divergence as a quantitative measure of posterior approximation accuracy and systematically evaluates the performance of Metropolis–Hastings, Affine Invariant Stretch Move, and No-U-Turn Sampler (NUTS) using complementary diagnostics such as the Gelman–Rubin statistic and effective sample size. The results reveal a qualitative consistency between heuristic convergence metrics and KL divergence. NUTS demonstrates superior efficiency in viscous flow due to its high sampling performance, yet underperforms in heat conduction problems where model evaluations are computationally expensive, highlighting gradient computation cost as a decisive factor in algorithm selection.

Abstract

Employing Bayesian inference to calibrate constitutive model parameters has grown substantially in recent years. Among the available techniques, Markov Chain Monte Carlo (MCMC) sampling remains one of the most widely used approaches for estimating the posterior distribution. Nevertheless, the selection of a specific MCMC algorithm is often driven by practical considerations, such as software availability or prior user experience. To support sampler selection, we present a comparison of three prominent samplers in the context of two distinct physical systems: a thermal conduction system and a viscous flow system. Calibration data are obtained through tailor-made experimental setups. We use the Kullback-Leibler (KL) divergence, which quantifies the statistical distance between the sampled posterior and the reference ('true') posterior, as a measure of convergence to compare the performance of the following MCMC sampling methods: the Metropolis-Hastings (MH) sampler, the Affine Invariant Stretch Move (AISM) sampler, and the No-U-Turn Sampler (NUTS). We study how this metric correlates to heuristic indicators such as the Gelman-Rubin diagnostic and the effective sample size. In addition, we assess the samplers' computational effort in terms of required number of model evaluations. Based on the results, we find that the heuristic convergence and performance indicators provide a good qualitative measure for KL-divergence for both systems. Regarding computational effort, the NUTS is net beneficial for the viscous flow system, as the high effective sample size outweighs the additional effort required for gradient-based proposal generation. For the thermal conduction system, which involves more expensive model evaluations, the NUTS is not advantageous. Thus, the computational efficiency of gradient evaluations is an important argument in sampler selection.

Problem

• Bayesian inference
• Markov Chain Monte Carlo
• constitutive models
• sampler selection
• computational efficiency

Innovation

• Markov Chain Monte Carlo
• Bayesian inference
• Kullback-Leibler divergence
• No-U-Turn Sampler
• constitutive model calibration