# Estimating a pressure dependent thermal conductivity coefficient with applications in food technology

## TL;DR

This paper presents a novel Bayesian approach using hierarchical Gaussian Markov random fields and a specialized MCMC algorithm to estimate pressure-dependent thermal conductivity in food technology, addressing smoothing effects and sensitivity issues.

## Contribution

It introduces a hierarchical GMRF model and a Single Variable Exchange Metropolis-Hastings algorithm for inverse estimation of thermal conductivity under pressure.

## Key findings

- Large integration times improve parameter identification
- A signal-to-noise ratio of about 1000 is needed for reliable estimates
- The method effectively estimates pressure-dependent thermal conductivity

## Abstract

In this paper we introduce a method to estimate a pressure dependent thermal conductivity coefficient arising in a heat diffusion model with applications in food technology. To address the known smoothing effect of the direct problem, we model the uncertainty of the conductivity coefficient as a hierarchical Gaussian Markov random field (GMRF) restricted to uniqueness conditions for the solution of the inverse problem established in Fraguela et al. Furthermore, we propose a Single Variable Exchange Metropolis-Hastings algorithm to sample the corresponding conditional probability distribution of the conductivity coefficient given observations of the temperature. Sensitivity analysis of the direct problem suggests that large integration times are necessary to identify the thermal conductivity coefficient. Numerical evidence indicates that a signal to noise ratio of roughly 1000 suffices to reliably retrieve the thermal conductivity coefficient.