Bayesian estimation and uncertainty quantification of a temperature-dependent thermal conductivity

TL;DR

This paper develops a Bayesian framework for estimating temperature-dependent thermal conductivity from measurements, comparing polynomial and piecewise linear models, and analyzing the impact of prior information and parametrization on estimation robustness.

Contribution

It introduces a Bayesian estimation method that effectively incorporates prior knowledge and measurement errors for different model classes of thermal conductivity.

Findings

Piecewise linear models are more flexible but require more parameters.

Parametrizing with conductivity values is more natural for priors.

Robust estimation is achievable with accurate or weak priors.

Abstract

We consider the problem of estimating a temperature-dependent thermal conductivity model (curve) from temperature measurements. We apply a Bayesian estimation approach that takes into account measurement errors and limited prior information of system properties. The approach intertwines system simulation and Markov chain Monte Carlo (MCMC) sampling. We investigate the impact of assuming different model classes - cubic polynomials and piecewise linear functions - their parametrization, and different types of prior information - ranging from uninformative to informative. Piecewise linear functions require more parameters (conductivity values) to be estimated than the four parameters (coefficients or conductivity values) needed for cubic polynomials. The former model class is more flexible, but the latter requires less MCMC samples. While parametrizing polynomials with coefficients may feel more natural, it turns out that parametrizing them using conductivity values is far more natural for the specification of prior information. Robust estimation is possible for all model classes and parametrizations, as long as the prior information is accurate or not too informative. Gaussian Markov random field priors are especially well-suited for piecewise linear functions.