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Solving inverse problems using Bayesian methods can become prohibitivelyexpensive when likelihood evaluations involve complex and large scale numericalmodels. A common approach to circumvent this issue is to approximate theforward model or the likelihood function with a surrogate model. But alsothere, due to limited computational resources, only a few training points areavailable in many practically relevant cases. Thus, it can be advantageous tomodel the additional uncertainties of the surrogate in order to incorporate theepistemic uncertainty due to limited data. In this paper, we develop a novelapproach to approximate the log likelihood by a constrained Gaussian processbased on prior knowledge about its boundedness. This improves the accuracy ofthe surrogate approximation without increasing the number of training samples.Additionally, we introduce a formulation to integrate the epistemic uncertaintydue to limited training points into the posterior density approximation. Thisis combined with a state of the art active learning strategy for selectingtraining points, which allows to approximate posterior densities in higherdimensions very efficiently. We demonstrate the fast convergence of ourapproach for a benchmark problem and infer a random field that is discretizedby 30 parameters using only about 1000 model evaluations. In a practicallyrelevant example, the parameters of a reduced lung model are calibrated basedon flow observations over time and voltage measurements from a coupledelectrical impedance tomography simulation.

Solving Bayesian Inverse Problems With Expensive Likelihoods Using Constrained Gaussian Processes and Active Learning