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Inverse problems in electrolysers

Giovanni S. Alberti, Wadim Gerner, Matteo Santacesaria

TL;DR

This work investigates inverse problems for a simplified electrochemical model of a PEM/AEM electrolyser, focusing on reconstructing the diffusion coefficients $D_i$, sources $g_i$, and electric potential $\phi$ from measurements of temperature and ion concentrations. It shows that boundary data alone are insufficient to recover interior $\phi$, but that incorporating interior temperature measurements (hybrid data) enables unique reconstruction of the state-dependent coefficients and the potential up to a constant, via a linearisation-based Calderón-type argument. The forward problem is established via a fixed-point approach, ensuring existence (and, under certain conditions, uniqueness) of weak solutions to the static coupled system. The results bridge classical Calderón-type inverse problems, quasilinear elliptic theory, and hybrid imaging, offering a rigorous pathway to identify solution-dependent parameters in coupled electrochemical PDEs and informing diagnostic strategies for electrolyser design and monitoring.

Abstract

We investigate the relationship between the electric potential within an electrolyser cell and its temperature and particle concentrations. Specifically, we identify the measurements required to uniquely reconstruct the potential $φ$ as a function of temperature and concentration.

Inverse problems in electrolysers

TL;DR

This work investigates inverse problems for a simplified electrochemical model of a PEM/AEM electrolyser, focusing on reconstructing the diffusion coefficients , sources , and electric potential from measurements of temperature and ion concentrations. It shows that boundary data alone are insufficient to recover interior , but that incorporating interior temperature measurements (hybrid data) enables unique reconstruction of the state-dependent coefficients and the potential up to a constant, via a linearisation-based Calderón-type argument. The forward problem is established via a fixed-point approach, ensuring existence (and, under certain conditions, uniqueness) of weak solutions to the static coupled system. The results bridge classical Calderón-type inverse problems, quasilinear elliptic theory, and hybrid imaging, offering a rigorous pathway to identify solution-dependent parameters in coupled electrochemical PDEs and informing diagnostic strategies for electrolyser design and monitoring.

Abstract

We investigate the relationship between the electric potential within an electrolyser cell and its temperature and particle concentrations. Specifically, we identify the measurements required to uniquely reconstruct the potential as a function of temperature and concentration.
Paper Structure (31 sections, 8 theorems, 135 equations)

This paper contains 31 sections, 8 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The inverse problem
  4. The equations and the unknowns
  5. The measurements
  6. The static source-free case
  7. The inverse problem.
  8. Main results
  9. The forward problem
  10. The inverse problem
  11. Boundary measurements
  12. Boundary and internal measurements
  13. Comparison to other inverse problems in the mathematical literature
  14. The Classical Calderón Problem.
  15. Quasilinear and Nonlinear Elliptic PDEs.
  16. ...and 16 more sections

Key Result

Theorem 2.1

Let $\Omega\subset\mathbb{R}^3$ be a bounded $C^1$-domain, $M\in \mathbb{N}$, $q\in \mathbb{R}^M$, $\phi\in \dot{C}_b^1(\mathbb{R}^{M+1}\times \overline{\Omega})$, $D_i\in C^0_b(\mathbb{R}^{M+1}\times \overline{\Omega})$, $g_i\in C^0_b(\mathbb{R}^{M+1}\times \overline{\Omega})$ for $1\leq i\leq M$, where $c=(c_1,\dots,c_M)$, $1\leq i\leq M$.

Theorems & Definitions (21)

  • Theorem 2.1: Existence of weak solutions
  • Theorem 2.2: Uniqueness of source-free weak solutions with constant boundary conditions
  • Theorem 2.3: Necessity of interior measurements
  • Remark 2.4
  • Theorem 2.5: Unique reconstruction for the inverse problem with boundary and internal data
  • Remark 2.6
  • Lemma 3.1: Existence of weak solutions I
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 11 more