  An efficient algorithm is proposed for Bayesian model calibration, which is
commonly used to estimate the model parameters of non-linear, computationally
expensive models using measurement data. The approach is based on Bayesian
statistics: using a prior distribution and a likelihood, the posterior
distribution is obtained through application of Bayes' law. Our novel algorithm
to accurately determine this posterior requires significantly fewer discrete
model evaluations than traditional Monte Carlo methods. The key idea is to
replace the expensive model by an interpolating surrogate model and to
construct the interpolating nodal set maximizing the accuracy of the posterior.
To determine such a nodal set an extension to weighted Leja nodes is
introduced, based on a new weighting function. We prove that the convergence of
the posterior has the same rate as the convergence of the model. If the
convergence of the posterior is measured in the Kullback-Leibler divergence,
the rate doubles. The algorithm and its theoretical properties are verified in
three different test cases: analytical cases that confirm the correctness of
the theoretical findings, Burgers' equation to show its applicability in
implicit problems, and finally the calibration of the closure parameters of a
turbulence model to show the effectiveness for computationally expensive
problems.
