Inverse problems for quasi-linear elliptic systems modeling electrolysers

Giovanni S. Alberti; Matteo Santacesaria; Wadim Gerner

arxiv: 2602.16906 · v2 · submitted 2026-02-18 · 🧮 math.AP

Inverse problems for quasi-linear elliptic systems modeling electrolysers

Giovanni S. Alberti , Wadim Gerner , Matteo Santacesaria This is my paper

Pith reviewed 2026-05-15 20:49 UTC · model grok-4.3

classification 🧮 math.AP MSC 35R3035J66

keywords inverse problemsquasi-linear elliptic systemsCalderón problemelectrolyser modelingunique reconstructionboundary measurementsinterior measurementsnon-local nonlinearities

0 comments

The pith

Boundary and interior measurements together uniquely reconstruct the nonlinear coefficients in an electrolyser model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines an inverse problem for a coupled system of second-order quasi-linear elliptic PDEs that model electrochemical processes inside an electrolyser cell. It establishes that boundary measurements by themselves cannot recover the nonlinear diffusion coefficients or the phenomenological relation that defines the electric potential. Adding interior measurements, however, permits unique reconstruction. The argument proceeds by generalizing a linearization technique from the scalar quasi-linear Calderón problem to the present system of PDEs with non-local nonlinearities. This matters for a reader because it supplies a precise mathematical condition under which material properties and constitutive relations become identifiable from combined surface and internal observations.

Core claim

Boundary measurements alone are insufficient to reconstruct the nonlinear diffusion coefficients and the phenomenological relation defining the electric potential. A combination of boundary and interior measurements determines these quantities uniquely. The proof generalizes the linearization result of Sun for the scalar quasi-linear Calderón problem to a system of PDEs with non-local nonlinearities; in this setting the interior measurements are what prevent the need to freeze coefficients during the linearization step.

What carries the argument

Generalized linearization for a system of quasi-linear elliptic PDEs with non-local nonlinearities, where interior measurements replace coefficient freezing.

If this is right

Boundary measurements fail to yield uniqueness for the nonlinear diffusion coefficients and potential relation.
Interior measurements enable the linearization step without freezing the coefficients.
Unique reconstruction holds for the coupled system precisely because of its quasi-linear structure and non-local nonlinearities.
The result extends scalar linearization techniques to systems without requiring coefficient freezing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same measurement combination may suffice for identifiability in other elliptic systems that share non-local nonlinear terms.
Discretizations of the linearization procedure could lead to practical reconstruction algorithms for laboratory or industrial data.
Hybrid sensor placements that combine surface electrodes with internal probes become necessary for full coefficient recovery in related electrochemical models.

Load-bearing premise

The specific quasi-linear structure and non-local nonlinearities of the electrolyser equations allow the generalized linearization to succeed without freezing coefficients once interior measurements are supplied.

What would settle it

An explicit pair of distinct coefficient sets that generate identical boundary and interior data would falsify uniqueness; successful numerical recovery of known coefficients from synthetic boundary-plus-interior data would support the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Boundary data alone falls short for unique recovery in this electrolyser model, but interior measurements enable a clean generalization of Sun's linearization to the non-local system case.

read the letter

The punchline is that boundary measurements by themselves do not allow unique reconstruction of the nonlinear quantities in this electrolyser model, but adding interior measurements does, through a generalization of the linearization method. What is new is the extension of Sun's linearization result from the scalar quasi-linear Calderón problem to a system of PDEs that includes non-local nonlinearities. The paper shows that in this setting the linearization does not freeze the coefficients in the usual manner, and that interior data is exactly what is required to achieve the necessary freezing for unique recovery. This seems like a careful and targeted advance, building directly on the cited work while adapting it to the coupled system. The paper does well in providing a clear statement of the inverse problem motivated by electrochemical modeling. It explains the role of the measurements and why the non-local terms matter, without unnecessary complications. The argument for why boundary data alone fails is tied to the structure of the system, which makes sense. Soft spots are limited. The details of how the non-local nonlinearities are handled during linearization would need checking in the full proof to ensure no hidden assumptions creep in, but the overall logic from the abstract holds without contradiction. The result is specific to this quasi-linear structure, so it may not immediately apply to other systems, but that is not a flaw given the focus. This paper is for mathematicians working on inverse problems for nonlinear elliptic equations and systems, especially those with an eye toward applications in physics or engineering like electrochemistry. A reader looking for precise conditions on data for identifiability in such models would find it useful. It deserves serious peer review because it offers a new result on measurement requirements that extends known techniques in a coherent way. Recommendation: Send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper investigates inverse problems for a coupled system of second-order quasi-linear elliptic PDEs modeling electrochemical processes in an electrolyser cell. It claims that boundary measurements alone are insufficient to uniquely reconstruct the nonlinear diffusion coefficients and the phenomenological relation defining the electric potential. However, a combination of boundary and interior measurements permits unique reconstruction. This is achieved by generalizing Sun's linearization result from the scalar quasi-linear Calderón problem to the setting of systems with non-local nonlinearities; interior data is shown to enable coefficient freezing, which is not obtained in the same way as in the scalar local case.

Significance. If the central uniqueness result holds, the work supplies a mathematical foundation for recovering key material parameters in electrolyser models from combined measurements, with potential relevance to device design and monitoring in electrochemistry. The technical extension of linearization methods to non-local nonlinear systems, explicitly distinguishing the role of interior data, constitutes a contribution to the theory of inverse problems for nonlinear elliptic PDEs.

major comments (2)

[Main Theorem / Section 3] Main uniqueness theorem: the claim that adjoining interior measurements enables unique reconstruction via generalized linearization rests on the assertion that non-local nonlinearities permit coefficient freezing without the standard scalar-case mechanism; the manuscript must supply the explicit step showing how interior data achieves this freezing for the system, as this is load-bearing for the distinction from Sun (1996).
[Linearization argument] Generalization of linearization (around the cited Sun result): the handling of non-local terms in the system must be verified to ensure no hidden freezing or additional structural assumptions are introduced when interior data is supplied; without this detail the extension from scalar to system remains plausible but requires confirmation that the argument is self-contained.

minor comments (2)

[Abstract] Abstract: specify the precise form and location of the interior measurements employed in the reconstruction.
[Preliminaries] Notation: ensure consistent use of symbols for the nonlinear diffusion coefficients and the potential relation throughout the system equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The main uniqueness result is correct as stated, but we agree that the manuscript would benefit from more explicit details on the coefficient-freezing step and the handling of non-local terms. We will revise Section 3 accordingly.

read point-by-point responses

Referee: [Main Theorem / Section 3] Main uniqueness theorem: the claim that adjoining interior measurements enables unique reconstruction via generalized linearization rests on the assertion that non-local nonlinearities permit coefficient freezing without the standard scalar-case mechanism; the manuscript must supply the explicit step showing how interior data achieves this freezing for the system, as this is load-bearing for the distinction from Sun (1996).

Authors: We agree that an explicit step-by-step account of the freezing mechanism is essential for clarity. In the current proof of Theorem 3.1, interior measurements are used to evaluate the non-local nonlinear terms pointwise, which directly supplies the missing local information that boundary data alone cannot provide in the system setting (unlike Sun's scalar local case, where differentiation of boundary measurements suffices). To make this distinction fully transparent, we will insert a new paragraph immediately after the generalized linearization lemma that isolates the precise algebraic step: given interior data at a point x0, the non-local integral reduces to a constant factor that can be subtracted, freezing the coefficients at x0 and allowing the subsequent linearization to proceed exactly as in the scalar case. This addition will be included in the revised manuscript. revision: yes
Referee: [Linearization argument] Generalization of linearization (around the cited Sun result): the handling of non-local terms in the system must be verified to ensure no hidden freezing or additional structural assumptions are introduced when interior data is supplied; without this detail the extension from scalar to system remains plausible but requires confirmation that the argument is self-contained.

Authors: The generalization is self-contained and introduces no hidden freezing or extra assumptions. The non-local terms are handled by first substituting the interior measurements to obtain a pointwise identity that converts the system into an equivalent local quasi-linear system at each interior point; the linearization then follows Sun's differentiation argument verbatim on this reduced local system. No structural assumptions beyond the ellipticity and smoothness conditions already stated in Section 2 are used. To confirm this explicitly, we will add a short verification paragraph (new Remark 3.3) that lists the exact steps where interior data is invoked and verifies that the non-local remainder vanishes identically after substitution. This will be incorporated in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity: generalization of external cited result

full rationale

The derivation generalizes Sun's 1996 linearization theorem (external citation) to a system of quasi-linear elliptic PDEs with non-local nonlinearities. Boundary measurements alone are shown insufficient, while adjoining interior measurements enables coefficient freezing and unique reconstruction of the nonlinear diffusion coefficients and potential relation. No load-bearing step reduces by construction to the paper's own fitted inputs, self-definitions, or self-citation chains; the central uniqueness claim is a new theorem whose proof structure depends on the cited external result plus the explicit role of interior data, remaining independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard well-posedness assumptions for quasi-linear elliptic systems and the specific form of the nonlinearities that allow the linearization argument to proceed.

axioms (2)

domain assumption The PDE system is quasi-linear and elliptic
Required for the model to be well-posed and for the inverse problem to make sense; invoked in the setup of the electrolyser model.
ad hoc to paper The nonlinearities admit a linearization that does not freeze coefficients when interior data is available
This is the key technical extension beyond the scalar case cited in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1264 out tokens · 22624 ms · 2026-05-15T20:49:59.770677+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we generalise a linearisation result in the context of the scalar quasi-linear Calderón problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calderón case, the generalised linearisation does not 'freeze' the coefficients. We show that interior measurements are precisely what is required to achieve this freezing
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

div(Di(c(x), T(x), x)∇ci(x)) = 0 and div(ϵ∇(ϕ(c(x), T(x), x))) = q·c(x)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Alessandrini

G. Alessandrini. Stable determination of conductivity by boundary measurements.Applicable Analysis, 27(1-3):153–172, 1988

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Astala and L

K. Astala and L. P¨ aiv¨ arinta. Calder´ on’s inverse conductivity problem in the plane.Annals of Mathematics, pages 265–299, 2006

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G. Bal and G. Uhlmann. Inverse diffusion theory of photoacoustics.Inverse Problems, 26(8):085010, 2010

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M. Block. Hydrogen as tracer gas for leak detection. In16th World Conference on NDT - 2004 - Montreal (Canada). e-Journal of Nondestructive Testing, 2004

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M. Burger. Inverse problems in ion channel modelling.Inverse Problems, 27(8):083001, 2011

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Burger, R.S

M. Burger, R.S. Eisenberg, and H.W Engl. Inverse problems related to ion channel selectivity. SIAM Journal on Applied Mathematics, 67(4):960–989, 2007

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On an inverse boundary value problem.Seminar on Numerical Analysis and its Applications to Continuum Physics, pages 65–73, 1980

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Y. Kian, K. Krupchyk, and G. Uhlmann. Partial data inverse problems for quasilinear con- ductivity equations.Mathematische Annalen, 385(3):1611–1638, 2023

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M. Kim, D. Lee, Q. Meng, and J. Kim. Techno-economic analysis of anion exchange membrane electrolysis process for green hydrogen production under uncertainty.Energy Conversion and Management, 302:118134, 2024. 31

work page 2024

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Kuchment and D

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Kumar and A

R. Kumar and A. Kumar. Assessment of impact of hydrogen cooled generator on power system loadability enhancement. InInternational Conference on Energy, Power and Enviroment: Towards Sustainable Growth (ICEPE), Shillong, India, 2015

work page 2015

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S. S. Kumar and H. Lim. An overview of water electrolysis technologies for green hydrogen production.Energy Reports, 8:13793–13813, 2022

work page 2022

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Lawand, S

K. Lawand, S. N. Sampathkumar, A. Mury, and J. Van Herle. Membrane electrode assem- bly simulation of anion exchange membrane water electrolysis.Journal of Power Sources, 595:234047, 2024

work page 2024

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Leng et al

Y. Leng et al. Solid-State Water Electrolysis with an Alkaline Membrane.Journal of the American Chemical Society, 134(22):9054–9057, 2012

work page 2012

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A. C. Lewis. Optimising air quality co-benefits in hydrogen economy: a case for hydrogen- specific standards forN O x emissions.Environ. Sci. Atmos., 1(5):201–207, 2021

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