Relaxation spectra using nonlinear Tikhonov regularization with a Bayesian criterion

Abstract

Nonlinear Tikhonov regularization within a Bayesian framework is incorporated into a computer program called pyReSpect, which infers the continuous and discrete relaxation spectra from oscillatory shear experiments. It uses Bayesian inference to provide uncertainty estimates for the continuous spectrum h(τ) by propagating the uncertainty in the regularization parameter λ. The new algorithm is about 6–9 times faster than an older version of the program (ReSpect) in which the optimal λ was determined by the L-curve method. About half of the speedup arises from the Bayesian formulation by restricting the window of λ explored. The other half arises from the nonlinear formulation for which the spectrum is a weak function of λ, allowing us to use a coarse mesh for λ. The program is tested and validated on three examples: a synthetic spectrum, a H-polymer, and an elastomer with a nonzero terminal plateau.