Estimate of convection-diffusion coefficients from modulated perturbative experiments as an inverse problem

TL;DR

This paper presents an explicit linear algebraic method for estimating convection-diffusion coefficients from experimental data, addressing issues of ill-conditioning and providing insights into accuracy and uncertainty in inverse problem solutions.

Contribution

It introduces a novel explicit solution approach for a specific class of inverse problems related to the convection-diffusion equation, improving parameter estimation accuracy.

Findings

Explicit solution reduces computational complexity.

Ill-conditioning linked to eigenvalue vanishing affects accuracy.

Comparison with numerical methods highlights advantages and limitations.

Abstract

The estimate of coefficients of the Convection-Diffusion Equation (CDE) from experimental measurements belongs in the category of inverse problems, which are known to come with issues of ill-conditioning or singularity. Here we concentrate on a particular class that can be reduced to a linear algebraic problem, with explicit solution. Ill-conditioning of the problem corresponds to the vanishing of one eigenvalue of the matrix to be inverted. The comparison with algorithms based upon matching experimental data against numerical integration of the CDE sheds light on the accuracy of the parameter estimation procedures, and suggests a path for a more precise assessment of the profiles and of the related uncertainty. Several instances of the implementation of the algorithm to real data are presented.