Estimation of the non-linear diffusion coefficient with Marcov Chain Monte Carlo method based on the integral information

Abstract

Purpose

This paper aims to present development and application of the Bayesian inverse approach for retrieving parameters of non-linear diffusion coefficient based on the integral information.

Design/methodology/approach

The Bayes formula was used to construct posterior distribution of the unknown parameters of non-linear diffusion coefficient. The resulting aposteriori distribution of sought parameters was integrated using Markov Chain Monte Carlo method to obtain expected values of estimated diffusivity parameters as well as their confidence intervals. Unsteady non-linear diffusion equation was discretised with the Global Radial Basis Function Collocation method and solved in time using Crank–Nicholson technique.

Findings

A number of manufactured analytical solutions of the non-linear diffusion problem was used to verify accuracy of the developed inverse approach. Reasonably good agreement, even for highly correlated parameters, was obtained. Therefore, the technique was used to compute concentration dependent diffusion coefficient of water in paper.

Originality/value

An original inverse technique, which couples efficiently meshless solution of the diffusion problem with the Bayesian inverse methodology, is presented in the paper. This methodology was extensively verified and applied to the real-life problem.

Citation

Bulinski, Z. and Orlande, H.R.B. (2017), "Estimation of the non-linear diffusion coefficient with Marcov Chain Monte Carlo method based on the integral information", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 No. 3, pp. 639-659. https://doi.org/10.1108/HFF-03-2016-0113