Inverse problems for quasi-linear elliptic systems modeling electrolysers

TL;DR

This paper studies an inverse problem in electrolysis modeling, demonstrating that boundary and interior measurements together enable unique reconstruction of nonlinear diffusion coefficients and potential relations in coupled elliptic PDE systems.

Contribution

It generalizes a linearisation technique to coupled PDE systems with non-local nonlinearities, showing interior measurements are essential for unique reconstruction.

Findings

Boundary measurements alone are insufficient for reconstruction.

Interior measurements enable unique determination of nonlinear coefficients.

Generalized linearisation does not 'freeze' coefficients unlike in scalar cases.

Abstract

We investigate the electrochemical processes within an electrolyser cell, which are modelled by a coupled system of second-order quasi-linear elliptic PDEs. In this context, we study an inverse problem aiming to reconstruct both the non-linear diffusion coefficients and the phenomenological relation defining the electric potential. Our main results state that boundary measurements alone are not enough to reconstruct these non-linear quantities. However, we show that a combination of boundary and interior measurements allow for their unique reconstruction. To achieve this result we generalise a linearisation result in the context of the scalar quasi-linear Calder\'{o}n problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calder\'{o}n case, the generalised linearisation does not "freeze" the coefficients. We show that interior measurements are precisely what is required to achieve this freezing and thus enable the unique reconstruction.