Efficient variational Bayesian model updating by Bayesian active learning

As a main task of inverse problem, model updating has received more and more attention in the area of inspection, sensing, and monitoring technologies during the recent decades, where the estimation of posterior probability density function (PDF) of unknown model parameters is still challenging for expensive-to-evaluate models of interest. In this paper, a novel variational Bayesian inference method is proposed to approximate the real posterior PDF of unknown model parameters by using Gaussian mixture model and measurement responses. A Gaussian process regression model is first trained for approximating the logarithm of the product of likelihood function and prior PDF, with which, another Gaussian process model is induced for approximating the expensive evidence lower bound (ELBO). Then, two Bayesian numerical methods, i.e., Bayesian optimization and Bayesian quadrature, are combined sequentially as a novel Bayesian active learning method for searching the global optima of the parameters of the variational posterior density. The proposed method inherits the advantages of both Bayesian numerical methods, which includes good global convergence, much less number of simulator calls, etc. Three examples, including the dynamic model of a two degrees of freedom structures, the lubrication model of a hybrid journal bearing, and the dynamic model of an airplane structure, are introduced for demonstrating the relative merits of the proposed method. Results show that, given desired requirement of numerical accuracy, the proposed method is more efficient than the parallel methods.