A hierarchical Bayesian approach to regularization with application to the inference of relaxation spectra

The inference of the spectral function representing the relaxation process of a material is an ill-posed problem and regularization is key to solving such problems. Significant information about molecular structure can be found from the relaxation time spectra of materials, such as polymers and soft materials. Various deterministic data-driven methods including L-curve and generalized cross-validation in Tikhonov regularization have been employed in the literature to find the optimal regularization parameter. The application of Bayesian techniques for solving such ill-posed inverse problems has gained significant interest in recent years due to the increasing availability of computational resources. In this work, we formulate the inverse problem in a hierarchical Bayesian framework and consider the degree of regularization as a stochastic quantity and the relaxation spectra as the high-dimensional model parameters. Using synthetic and real data (frequency-dependent storage and loss moduli), this novel approach is employed to characterize the linear viscoelastic response of materials and obtain sparse probabilistic solutions to regression. The computations are carried out using Metropolis-Hastings-within-Gibbs sampling. Numerical results demonstrating the performance of the hierarchical Bayesian approach and comparisons with the deterministic L-curve approach are presented. (c) 2021 The Society of Rheology.