26 January, 2017
In this work, we employ statistical and Bayesian techniques to analyze mathematical forward models. The forward models usually arise from phenomenological and physical phenomena and are expressed through regression-based models or partial differential equations (PDEs) associated with uncertain parameters and input data. One of the critical challenges in real life applications is to estimate and quantify uncertainties of such parameters using observations. Two main statistical inferential approaches are considered to find estimates of the model unknown parameters: likelihood-based methods and Bayesian techniques, the latter providing a full probability distribution for the parameters of interest, named the posterior distribution. In the classical approach, bootstrap confidence intervals procedures are used to assess the estimates variability. To generate accurate fatigue life predictions, competing forward models are ranked according to several classical information-based measures, such as the Akaike information criterion (AIC). A different set of predictive information criteria, relying on cross-validation techniques, is then used to compare the candidate Bayesian models. We also propose a novel marginalization method for inverse problems in linear PDEs using a hierarchical Bayesian framework. This method accounts for uncertainties in the input data and therefore reduces the bias error of the maximum posterior estimate. Moreover, we generalize the marginalization technique to a sequential framework by deriving and implementing an ensemble marginalized Kalman filter.