Cyclical Variational Bayes Monte Carlo for efficient multi-modal posterior distributions evaluation

Multi-modal distributions of some physics-based model parameters are often encountered in engineering due to different situations such as a change in some environmental conditions, and the presence of some types of damage and non-linearity. In statistical model updating, for locally identifiable parameters, it can be anticipated that multi-modal posterior distributions would be found. The full characterization of these multi-modal distributions is important as methodologies for structural condition monitoring in structures are frequently based in the comparison of the damaged and healthy models of the structure. The characterization of posterior multi-modal distributions using state-of-the-art sampling techniques would require a large number of simulations of expensive-to-run physics-based models. Therefore, when a limited number of simulations can be run, as it often occurs in engineering, the traditional sampling techniques would not be able to capture accurately the multi-modal distributions. This could potentially lead to large numerical errors when assessing the performance of an engineering structure under uncertainty. Therefore, an approach is proposed for drastically reducing the number of models runs while yielding accurate estimates of highly multi-modal posterior distributions. This approach introduces a cyclical annealing schedule into the Variational Bayes Monte Carlo (VBMC) method to improve the algorithm's phase of exploration and the finding of high probability areas in the multi-modal posteriors throughout the different cycles. Three numerical and one experimental investigations are used to compare the proposed cyclical VBMC with the standard VBMC algorithm, the monotonic VBMC and the Transitional Ensemble Markov Chain Monte Carlo (TEMCMC). It is shown that the standard VBMC fails in capturing multi-modal posteriors as it is unable to escape already found regions of high posterior density. In the presence of highly multi-modal posteriors, the proposed cyclical VBMC algorithm outperforms all the other approaches in terms of accuracy of the resulting posterior, and number of model runs required.