Bayesian model calibration with interpolating polynomials based on adaptively weighted Leja nodes

TL;DR

This paper introduces an efficient Bayesian model calibration algorithm that uses interpolating polynomials and adaptively weighted Leja nodes to reduce the number of expensive model evaluations, with proven convergence properties.

Contribution

The paper presents a novel algorithm combining interpolating surrogate models with weighted Leja nodes for Bayesian calibration, improving efficiency and convergence in complex models.

Findings

The algorithm achieves accurate posterior estimates with fewer model evaluations.

Theoretical convergence rate matches the model's convergence rate, doubling in Kullback-Leibler divergence.

Validated on analytical, Burgers' equation, and turbulence model calibration cases.

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.