great interest in designing robust systems. Forward Uncertainty propagation is the quantification of uncertainties in system output(s) propagated from uncertain inputs. It focuses on the influence on the outputs from the parametric variability listed in the sources of uncertainty. The targets of uncertainty propagation analysis can be: To evaluate low-order moments of the outputs, i.e. mean and variance. To evaluate the reliability of the outputs. This is especially useful in reliability engineering where outputs of a system are usually closely related to the performance of the system. To assess the complete probability distribution of the outputs. This is useful in the scenario of utility optimization where the complete distribution is used to calculate the utility. Inverse Given some experimental measurements of a system and some computer simulation results from its mathematical model, inverse uncertainty quantification estimates the discrepancy between the experiment and the mathematical model (which is called bias correction), and estimates the values of unknown parameters in the model if there are any (which is called parameter calibration or simply calibration). Generally this is a much more difficult problem than forward uncertainty propagation; however it is of great importance since it is typically implemented in a model updating process. There are several scenarios in inverse uncertainty quantification: Bias correction only Bias correction quantifies the model inadequacy, i.e. the discrepancy between the experiment and the mathematical model. The general model updating formula for bias correction is: where denotes the experimental measurements as a function of several input variables , denotes the computer model (mathematical model) response, denotes the additive discrepancy function (aka bias function), and denotes the experimental uncertainty. The objective is to estimate the discrepancy function , and as a by-product, the resulting updated model is . A prediction confidence interval is provided with the updated model as the quantification of the uncertainty. Parameter calibration only Parameter calibration estimates the values of one or more unknown parameters in a mathematical model. The general model updating formulation for calibration is: where denotes the computer model response that depends on several unknown model parameters , and denotes the true values of the unknown parameters in the course of experiments. The objective is to either estimate , or to come up with a probability distribution of that encompasses the best knowledge of the true parameter values. Bias correction and parameter calibration It considers an inaccurate model with one or more unknown parameters,