  In this work, a Bayesian model calibration framework is presented that
utilizes goal-oriented a-posterior error estimates in quantities of interest
(QoIs) for classes of high-fidelity models characterized by PDEs. It is shown
that for a large class of computational models, it is possible to develop a
computationally inexpensive procedure for calibrating parameters of
high-fidelity models of physical events when the parameters of low-fidelity
(surrogate) models are known with acceptable accuracy. The main ingredients in
the proposed model calibration scheme are goal-oriented a-posteriori estimates
of error in QoIs computed using a so-called lower fidelity model compared to
those of an uncalibrated higher fidelity model. The estimates of error in QoIs
are used to define likelihood functions in Bayesian inversion analysis. A
standard Bayesian approach is employed to compute the posterior distribution of
model parameters of high-fidelity models. As applications, parameters in a
quasi-linear second-order elliptic boundary-value problem (BVP) are calibrated
using a second-order linear elliptic BVP. In a second application, parameters
of a tumor growth model involving nonlinear time-dependent PDEs are calibrated
using a lower fidelity linear tumor growth model with known parameter values.
