Identification and sampling of Bayesian posteriors of high-dimensional symmetric positive-definite matrices for data-driven updating of computational models

Abstract : We propose a probabilistic methodology for data-driven updating of non-Gaussian high-dimensional symmetric positive-definite matrices involved in computational models. We cast the data-driven updating as a Bayesian identification of the symmetric positive-definite matrices. The posterior thus obtained exhibits several hyperparameters that control the dispersion of the prior and the weight of the weighted distance that represents the model-data misfit in the likelihood function. Using an identification criterion that quantifies the agreement between the predictions and the data, we identify these hyperparameters so as to obtain not only improved predictions but also a probabilistic representation of model uncertainties. The numerical implementation of the Bayesian inversion by using a Markov chain Monte Carlo (MCMC) method is computationally challenging because the support of the posterior is restricted to a set of symmetric positive-definite matrices and the dimensionality of the problem grows with the square of the matrix dimension and hence can be high. We thus use a transformation of measure to set up the Markov chain in terms of real-valued state variables whose distribution is Gaussian under the prior but non-Gaussian under the posterior. This transformation of measure allows us to sample the posterior using an Itô-SDE-based MCMC method that inherits computational efficiency in high dimension from leveraging the gradient of the posterior. We apply this methodology to a problem of a data-driven updating of a reduced-order model in computational structural dynamics.