Hamiltonian Monte Carlo and Borrowing Strength in Hierarchical Inverse Problems

Abstract

Bayesian approaches to uncertainty quantification and information acquisition in hierarchically defined inverse problems are presented. The techniques comprise simple updating, staged estimation, and multilevel model calibration. In particular, the estimation of material properties within an ensemble of identically manufactured structural elements is considered. It is shown how inferring the characteristics of an individual specimen can be accomplished by exhausting statistical strength from tests of other ensemble members. This is useful in experimental situations where evidence is scarce or unequally distributed. Hamiltonian Monte Carlo is proposed to cope with the numerical challenges of the devised approaches. The performance of the algorithm is studied and compared to classical Markov chain Monte Carlo sampling. It turns out that Bayesian posterior computations can be drastically accelerated.