Bayesian Model Calibration with Interpolating Polynomials Based on Adaptively Weighted Leja Nodes (Communications in Computational Physics, 2020)
An efficient algorithm is proposed for Bayesian model calibration, whichis commonly used to estimate the model parameters of non-linear, computationallyexpensive 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 determinethis posterior requires significantly fewer discrete model evaluations than traditionalMonte Carlo methods. The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing theaccuracy of the posterior. To determine such a nodal set an extension to weightedLeja 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 theconvergence of the posterior is measured in the Kullback–Leibler divergence, the ratedoubles. The algorithm and its theoretical properties are verified in three different testcases: analytical cases that confirm the correctness of the theoretical findings, Burgers'equation to show its applicability in implicit problems, and finally the calibration ofthe closure parameters of a turbulence model to show the effectiveness for computationally expensive problems.
Keywords: Bayesian model calibration; interpolation; Leja nodes; surrogate modeling.