Bayesian Model Calibration with Interpolating Polynomials Based on Adaptively Weighted Leja Nodes

Abstract

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.

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“…The resulting collocation scheme is able to address a moderately high number of random model parameters. Moreover, weighted Leja nodes can handle almost arbitrary input probability distributions [11,[26][27][28][29][30] and are ideally suited for adaptivity [9]. In order to efficiently steer the adaptivity, we derive an adjoint representation of the stochastic error.…”

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“…Additionally, interpolation and quadrature grids with respect to any continuous PDF can be constructed by using weighted Leja sequences. 39,[50][51][52] Moreover, due to the fact that Leja sequences are by definition nested, that is, {x i } j i=0 ⊂ {x i } j+1 i=0 , they allow for re-using readily available Leja points and model evaluations on those points in case the sequence is further expanded. Due to the nestedness property, Leja points are natural candidates for constructing sparse interpolation or quadrature grids.…”

“…With respect to interpolation in particular, the Lebesgue constant of Leja sequence based interpolation grids is known to grow subexponentially, 47‐49 thus resulting in stable interpolations. Additionally, interpolation and quadrature grids with respect to any continuous PDF can be constructed by using weighted Leja sequences 39,50‐52 . Moreover, due to the fact that Leja sequences are by definition nested, that is, $$$ {\left\{{x}_i\right\}}_{i=0}^j\subset {\left\{{x}_i\right\}}_{i=0}^{j+1} $$$, they allow for re‐using readily available Leja points and model evaluations on those points in case the sequence is further expanded.…”