A note on several inverse problems with generally random coefficients

C\u{a}t\u{a}lin I. C\^arstea

arxiv: 2605.20004 · v2 · pith:M4UREFRWnew · submitted 2026-05-19 · 🧮 math.AP · math.PR

A note on several inverse problems with generally random coefficients

C\u{a}t\u{a}lin I. C\^arstea This is my paper

Pith reviewed 2026-05-25 06:18 UTC · model grok-4.3

classification 🧮 math.AP math.PR

keywords inverse problemsrandom coefficientsDirichlet-to-Neumann mapGreen's operatorSchrödinger equationconductivity equationstatistical recovery

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The pith

The averaged Schrödinger Green's operator determines the pointwise mean and variance of a random potential, while the expected Dirichlet-to-Neumann map does not determine even the mean.

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

For elliptic equations with random coefficients that have no special probabilistic structure, the full law of the Dirichlet-to-Neumann map determines the law of the random potential. Its expectation, however, or any finite set of moments of the boundary bilinear form, fails to recover even the mean of the potential. The averaged interior Green's operator, in contrast, determines the pointwise mean and variance of the potential. In a simple two-atom model it further determines all moments of the two-point law. The same pattern of results holds for the conductivity equation.

Core claim

The paper establishes that the full law of the Dirichlet-to-Neumann map determines the law of the random potential, but the expected Dirichlet-to-Neumann map and any fixed finite hierarchy of its boundary moments need not determine even the mean potential. In contrast, the averaged Schrödinger Green's operator determines the pointwise mean and variance of the potential, and in a two-atom model it determines all pointwise moments of the two-point law.

What carries the argument

The averaged Schrödinger Green's operator, which provides interior averaged information that encodes the pointwise statistical moments of the random potential.

If this is right

The full law of the Dirichlet-to-Neumann map recovers the entire law of the random potential.
No fixed finite number of moments of the Dirichlet-to-Neumann map recovers the mean potential.
The averaged Green's operator recovers the mean and variance at each interior point.
In a two-atom model the averaged Green's operator recovers all moments of the two-point law.
Analogous positive and negative results hold for the conductivity equation.

Where Pith is reading between the lines

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

Interior averaged data may be essential for recovering statistics in random media inverse problems where boundary data alone is insufficient.
The results suggest testing whether similar distinctions appear in nonlinear or time-dependent elliptic problems with randomness.
Practical recovery algorithms could prioritize averaged Green's functions over boundary maps for statistical identification.
Extensions to unbounded domains or other boundary conditions could reveal if the interior advantage persists.

Load-bearing premise

The elliptic equations are well-posed on a bounded domain with the Dirichlet-to-Neumann map and Green's operator well-defined, and the coefficients are random without imposed probabilistic structure.

What would settle it

Finding two different random potentials whose averaged Green's operators coincide but whose pointwise means or variances differ would falsify the recovery result for the Green's operator.

read the original abstract

We consider several inverse problems for elliptic equations whose coefficients are random, without imposing a special probabilistic structure on the randomness. The main body treats the Schr\"odinger equation. We compare what can be recovered from the full law of the Dirichlet-to-Neumann map, from its expectation, from finitely many joint moments of its boundary bilinear form, and from the averaged interior Green's operator. We obtain both positive and negative results. That the full law of the Dirichlet-to-Neumann map determines the law of the random potential is almost trivial. However, the expected Dirichlet-to-Neumann map and, more generally, any fixed finite hierarchy of its boundary moments need not determine even the mean potential. In contrast, the averaged Schr\"odinger Green's operator determines the pointwise mean and variance of the potential. In a two-atom model it determines all pointwise moments of the two-point law. The appendices contain the corresponding results for the conductivity equation.

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

The note cleanly shows that finite moments of the DN map fail to recover even the mean potential while the averaged Green's operator recovers mean and variance, but the arbitrary-randomness setup risks ill-posedness.

read the letter

The paper compares what the full law of the Dirichlet-to-Neumann map, its expectation, finite moments, and the averaged interior Green's operator can recover for the Schrödinger equation with a random potential that has no special probabilistic structure. The full law determines the law of the potential in a straightforward way. Finite moments of the boundary data do not determine the mean. The averaged Green's operator does recover the pointwise mean and variance, and in the two-atom case it gets the full two-point law. The appendices give the same comparisons for the conductivity equation. These distinctions look new relative to the cited literature on stochastic inverse problems. The results are stated directly as positive and negative statements without extra assumptions on the randomness, which keeps the note short and focused. The main soft spot is the well-posedness question raised in the stress test. If the random coefficient can fall outside L^∞ (or the appropriate integrability class) on a set of positive probability, then the DN map and Green's operator are not defined for those realizations, so the statistical comparisons lose their meaning. The negative result also needs explicit counterexamples that remain well-posed while matching low-order moments. The paper is a short theoretical note aimed at people working on inverse problems for elliptic equations with randomness. It clarifies the relative strength of different data types in a setting without extra structure. It deserves peer review because the claims are specific, the distinctions are sharp, and the counterexamples and positive results can be checked directly once the well-posedness is pinned down.

Referee Report

1 major / 0 minor

Summary. The paper considers inverse problems for the Schrödinger equation (and conductivity equation in appendices) with random coefficients lacking special probabilistic structure. It claims that the full law of the Dirichlet-to-Neumann map determines the law of the random potential (almost trivially), but the expectation of the DN map or any fixed finite collection of its boundary moments need not determine even the mean potential. In contrast, the averaged interior Green's operator determines the pointwise mean and variance of the potential, and in a two-atom model determines all pointwise moments of the two-point law.

Significance. If the results hold, they usefully distinguish the informational content of full distributional boundary data versus low-order moments versus averaged interior data in random-coefficient inverse problems. The positive result on the averaged Green's operator and the negative result on finite DN moments provide a clear contrast that could inform future work on stochastic inverse problems.

major comments (1)

[Introduction, §2 (well-posedness assumptions)] The manuscript states results for random coefficients 'without imposing a special probabilistic structure,' yet all claims presuppose that the Schrödinger equation is well-posed and that the DN map (as a bounded operator on H^{1/2}(∂Ω)) and Green's operator exist for a.e. realization. Standard theory requires q ∈ L^∞(Ω) (or at least q ∈ L^{n/2+ε}) a.s. for this; nothing in the stated generality prevents realizations outside this class on a set of positive probability. This is load-bearing for the positive and negative results in the abstract and main body (§2 and following). The negative result on finite moments further requires explicit counterexamples that remain inside the well-posed regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on well-posedness. We address it directly below.

read point-by-point responses

Referee: [Introduction, §2 (well-posedness assumptions)] The manuscript states results for random coefficients 'without imposing a special probabilistic structure,' yet all claims presuppose that the Schrödinger equation is well-posed and that the DN map (as a bounded operator on H^{1/2}(∂Ω)) and Green's operator exist for a.e. realization. Standard theory requires q ∈ L^∞(Ω) (or at least q ∈ L^{n/2+ε}) a.s. for this; nothing in the stated generality prevents realizations outside this class on a set of positive probability. This is load-bearing for the positive and negative results in the abstract and main body (§2 and following). The negative result on finite moments further requires explicit counterexamples that remain inside the well-posed regime.

Authors: We agree that the results presuppose well-posedness almost surely and that this requires the random coefficient q to lie in L^∞(Ω) (or the appropriate space L^{n/2+ε}) almost surely. Although the phrase 'without imposing a special probabilistic structure' was intended only to exclude assumptions such as Gaussianity or independence, the referee is correct that the current wording does not explicitly record the necessary integrability. In the revision we will add a precise statement in the introduction and at the beginning of §2 that all realizations satisfy q ∈ L^∞(Ω) a.s., thereby guaranteeing the existence of the DN map and Green's operator almost everywhere. This is a clarification rather than a restriction on the probabilistic law. For the negative results on finite moments of the DN map, the counterexamples are constructed with bounded potentials (hence inside the well-posed regime); we will insert a short remark confirming that each realization satisfies the L^∞ bound used in the existence theory. revision: yes

Circularity Check

0 steps flagged

No circularity; results are direct comparisons using external deterministic uniqueness

full rationale

The paper's central claims rest on (1) the standard deterministic uniqueness for the Calderón problem (external to this work), making the 'full law determines law of q' statement a direct transfer; (2) explicit counterexample constructions for the negative results on moments of the DN map; and (3) direct integral identities or kernel expansions for the averaged Green's operator recovering pointwise moments. None of these steps reduce by definition or self-citation to the paper's own inputs. The well-posedness assumption is stated but does not create a definitional loop. This matches the expected non-circular case for a short note comparing data types.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are visible. Standard PDE assumptions are implicit.

axioms (2)

standard math Elliptic equations are well-posed on bounded domains with smooth boundary
Required for Dirichlet-to-Neumann map and Green's operator to be defined.
domain assumption Random coefficients have no special probabilistic structure
Explicitly stated as the setting for the results.

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discussion (0)

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Works this paper leans on

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Inverse random source scatter- ing problems in several dimensions.SIAM/ASA Journal on Uncertainty Quantification, 4(1):1263–1287, 2016

Gang Bao, Chuchu Chen, and Peijun Li. Inverse random source scatter- ing problems in several dimensions.SIAM/ASA Journal on Uncertainty Quantification, 4(1):1263–1287, 2016

work page 2016

[2] [2]

Identifi- cation of Green’s functions singularities by cross correlation of noisy signals.Inverse Problems, 24(1):015011, 2008

Claude Bardos, Josselin Garnier, and George Papanicolaou. Identifi- cation of Green’s functions singularities by cross correlation of noisy signals.Inverse Problems, 24(1):015011, 2008

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Detecting stochastic inclusions in electrical impedance tomography

Andrea Barth, Bastian Harrach, Nuutti Hyv¨ onen, and Lauri Mustonen. Detecting stochastic inclusions in electrical impedance tomography. Inverse Problems, 33(11):115012, 2017

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Inverse scattering for a random potential.Analysis and Applications, 17(4):513–567, 2019

Pedro Caro, Tapio Helin, and Matti Lassas. Inverse scattering for a random potential.Analysis and Applications, 17(4):513–567, 2019

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C. I. Cˆ arstea, Ali Feizmohammadi, and Lauri Oksanen. Remarks on the anisotropic Calder´ on problem.Proceedings of the American Mathematical Society, 151(10):4461–4473, 2023

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de Hoop and Knut Solna

Maarten V. de Hoop and Knut Solna. Estimating a Green’s function from field-field correlations in a random medium.SIAM Journal on Applied Mathematics, 69(4):909–932, 2009

work page 2009

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Recovering singularities of a potential from singularities of scattering data.Communications in Mathematical Physics, 157(3):549–572, 1993

Allan Greenleaf and Gunther Uhlmann. Recovering singularities of a potential from singularities of scattering data.Communications in Mathematical Physics, 157(3):549–572, 1993

work page 1993

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An inverse problem for the wave equation with one measurement and the pseudorandom source

Tapio Helin, Matti Lassas, and Lauri Oksanen. An inverse problem for the wave equation with one measurement and the pseudorandom source. Analysis & PDE, 5(5):887–912, 2012

work page 2012

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Inverse problem for the wave equation with a white noise source.Communications in Mathematical Physics, 332(3):933–953, 2014

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work page 2014

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Cor- relation based passive imaging with a white noise source.Journal de Math´ ematiques Pures et Appliqu´ ees, 116:132–160, 2018

Tapio Helin, Matti Lassas, Lauri Oksanen, and Teemu Saksala. Cor- relation based passive imaging with a white noise source.Journal de Math´ ematiques Pures et Appliqu´ ees, 116:132–160, 2018. 31

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Kechris.Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics

Alexander S. Kechris.Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995

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Inverse problem for a random potential

Matti Lassas, Lassi P¨ aiv¨ arinta, and Eero Saksman. Inverse problem for a random potential. InPartial Differential Equations and Inverse Problems, volume 362 ofContemporary Mathematics, pages 277–288. American Mathematical Society, Providence, RI, 2004

work page 2004

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Inverse scattering problem for a two dimensional random potential.Communications in Mathematical Physics, 279:669–703, 2008

Matti Lassas, Lassi P¨ aiv¨ arinta, and Eero Saksman. Inverse scattering problem for a two dimensional random potential.Communications in Mathematical Physics, 279:669–703, 2008

work page 2008

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Inverse random source problems for time-harmonic acoustic and elastic waves.Communications in Partial Differential Equations, 45(10):1335–1380, 2020

Jianliang Li, Tapio Helin, and Peijun Li. Inverse random source problems for time-harmonic acoustic and elastic waves.Communications in Partial Differential Equations, 45(10):1335–1380, 2020

work page 2020

[16] [16]

Inverse elastic scattering for a random source

Jianliang Li and Peijun Li. Inverse elastic scattering for a random source. SIAM Journal on Mathematical Analysis, 51(6):4570–4603, 2019

work page 2019

[17] [17]

Inverse elastic scattering for a random potential.SIAM Journal on Mathematical Analysis, 54(5):5126– 5159, 2022

Jianliang Li, Peijun Li, and Xu Wang. Inverse elastic scattering for a random potential.SIAM Journal on Mathematical Analysis, 54(5):5126– 5159, 2022

work page 2022

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Inverse random potential scattering for the polyharmonic wave equation using far-field patterns.SIAM Journal on Applied Mathematics, 85(3):1237–1260, 2025

Jianliang Li, Peijun Li, Xu Wang, and Guanlin Yang. Inverse random potential scattering for the polyharmonic wave equation using far-field patterns.SIAM Journal on Applied Mathematics, 85(3):1237–1260, 2025

work page 2025

[19] [19]

Determining a random Schr¨ odinger equation with unknown source and potential.SIAM Journal on Mathematical Analysis, 51(4):3465–3491, 2019

Jingzhi Li, Hongyu Liu, and Shiqi Ma. Determining a random Schr¨ odinger equation with unknown source and potential.SIAM Journal on Mathematical Analysis, 51(4):3465–3491, 2019

work page 2019

[20] [20]

Determining a random Schr¨ odinger operator: both potential and source are random.Commu- nications in Mathematical Physics, 381(2):527–556, 2021

Jingzhi Li, Hongyu Liu, and Shiqi Ma. Determining a random Schr¨ odinger operator: both potential and source are random.Commu- nications in Mathematical Physics, 381(2):527–556, 2021

work page 2021

[21] [21]

Inverse random source scattering for the Helmholtz equation in inhomogeneous media.Inverse Problems, 34(1):015003, 2018

Ming Li, Chuchu Chen, and Peijun Li. Inverse random source scattering for the Helmholtz equation in inhomogeneous media.Inverse Problems, 34(1):015003, 2018

work page 2018

[22] [22]

Inverse random source scattering for the Helmholtz equation with attenuation.SIAM Journal on Applied Mathe- matics, 81(2):485–506, 2021

Peijun Li and Xu Wang. Inverse random source scattering for the Helmholtz equation with attenuation.SIAM Journal on Applied Mathe- matics, 81(2):485–506, 2021. 32

work page 2021

[23] [23]

Inverse scattering for the biharmonic wave equation with a random potential.SIAM Journal on Mathematical Analysis, 56(2):1959–1995, 2024

Peijun Li and Xu Wang. Inverse scattering for the biharmonic wave equation with a random potential.SIAM Journal on Mathematical Analysis, 56(2):1959–1995, 2024

work page 1959

[24] [24]

Inverse problem for a random Schr¨ odinger equation with unknown source and potential.Mathematische Zeitschrift, 304(2):Paper No

Hongyu Liu and Shiqi Ma. Inverse problem for a random Schr¨ odinger equation with unknown source and potential.Mathematische Zeitschrift, 304(2):Paper No. 28, 31 pp., 2023

work page 2023

[25] [25]

Inverse problems for stochastic partial differential equations: some progresses and open problems.Numerical Algebra, Control and Optimization, 14(2):227–272, 2024

Qi L¨ u and Xu Zhang. Inverse problems for stochastic partial differential equations: some progresses and open problems.Numerical Algebra, Control and Optimization, 14(2):227–272, 2024

work page 2024

[26] [26]

On recent progress of single-realization recoveries of random Schr¨ odinger systems.Electronic Research Archive, 29(3):2391–2415, 2021

Shiqi Ma. On recent progress of single-realization recoveries of random Schr¨ odinger systems.Electronic Research Archive, 29(3):2391–2415, 2021

work page 2021

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From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography

Petteri Piiroinen and Martin Simon. From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography. Annals of Applied Probability, 26(5):3001–3043, 2016

work page 2016

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Probabilistic interpretation of the Calder´ on problem.Inverse Problems and Imaging, 11(3):553–575, 2017

Petteri Piiroinen and Martin Simon. Probabilistic interpretation of the Calder´ on problem.Inverse Problems and Imaging, 11(3):553–575, 2017

work page 2017

[29] [29]

Reconstruction of the singularities of a potential from backscattering data in 2D and 3D.Inverse Problems and Imaging, 6(2):321–355, 2012

Juan Manuel Reyes and Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D.Inverse Problems and Imaging, 6(2):321–355, 2012

work page 2012

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Springer Spektrum, Wiesbaden, 2015

Martin Simon.Anomaly Detection in Random Heterogeneous Media: Feynman–Kac Formulae, Stochastic Homogenization and Statistical Inversion. Springer Spektrum, Wiesbaden, 2015

work page 2015

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A global uniqueness theorem for an inverse boundary value problem.Annals of Mathematics, 125:153–169, 1987

John Sylvester and Gunther Uhlmann. A global uniqueness theorem for an inverse boundary value problem.Annals of Mathematics, 125:153–169, 1987

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