First-order formulations for large-scale stochastic parameter estimation within the frameworks of steady state dynamics: the elastic and viscoelastic case

We consider the problem of large-scale stochastic parameter estimation in the context of a particular problem: determining the material properties of a region of interest given an ‘observed’ displacement field. Specifically, we derive first-order formulations for the characterization of elastic and viscoelastic properties within the frameworks of steady state dynamics. The formulations presented herein fall into two main classes: the L2 formulation and modified error in constitutive equation (MECE) formulation. The focus of this work is to rigorously study the performance of these formulations through a series of numerical studies and perceive if choosing the MECE formulation over the L2 formulation will result in better parameter estimates with less computational effort. Furthermore, we capture the uncertainty in the parameter estimates due to uncertainties in the data as well as the model. To quantify these uncertainties, we use a maximum a posteriori (MAP) estimate approach. We demonstrate the performance obtained when using first-order formulations in MAP Bayesian analysis for stochastic parameter estimation.