Bayesian identification of material parameters in viscoelastic structures as an inverse problem in a semigroup setting

This paper considers the nonlinear inverse problem of identifying the material parameters in viscoelastic structures based on a generalized Maxwell model. The aim is to reconstruct the model parameters from stress data acquired from a relaxation experiment, where the number of Maxwell elements, and thus the number of material parameters themselves, is assumed to be unknown. This implies that the forward operator acts on a Cartesian product of a semigroup (of integers) and a Hilbert space, and thus demands an extension of existing regularization theory. We develop a stable reconstruction procedure by applying Bayesian inversion to this setting. We use an appropriate binomial prior that takes the integer setting for the number of Maxwell elements into account, and at the same time computes the underlying material parameters. We extend the regularization theory for inverse problems to this special setup, and prove the existence, stability, and convergence of the computed solution. The theoretical results are evaluated by extensive numerical tests.