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

Rodrigo L. S. Silva, Clemens Verhoosel, Erik Quaeghebeur

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.

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

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.
Paper Structure (23 sections, 16 equations, 10 figures, 1 table)

This paper contains 23 sections, 16 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Transient heat conduction problem
  3. Physical system
  4. Mathematical model
  5. Numerical simulation
  6. Bayesian estimation and uncertainty quantification
  7. Bayesian modeling
  8. Stochastic simulation with Markov chain Monte Carlo methods
  9. Application of the Bayesian framework to the heat conduction problem
  10. Definition of simulated temperature measurements
  11. Project design using sensitivity analysis
  12. The likelihood function
  13. Temperature-dependent thermal conductivity models
  14. Third-degree polynomial model class
  15. Coefficients parametrization
  16. ...and 8 more sections

Figures (10)

  • Figure 1: On the left, an illustration of the transient heat conduction problem. On the right, a typical modeling result, showing the temperature distribution along the slab at different times.
  • Figure 2: The temperature-dependent conductivity function $\kappa(\theta)$ represented by two models.
  • Figure 3: On the left, sensitivity coefficients $J_{0, mn}$ with respect to $\kappa_n$ evaluated at $X = 0$. On the right, sensitivity coefficients $J_{1, mn}$ with respect to $\kappa_n$ evaluated at $X = 1$.
  • Figure 4: Markov chains of the conductivities $\kappa_n$ obtained with the uniform prior and the third-degree polynomial model parametrized with the conductivity values.
  • Figure 5: Histograms of the limit distributions regarding the conductivities $\kappa_n$ obtained with the uniform prior and the third-degree polynomial model parametrized with the conductivity values.
  • ...and 5 more figures