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arxiv: 2605.20004 · v1 · 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 2026-05-20 03:22 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords inverse problemsrandom coefficientsDirichlet-to-Neumann mapGreen's operatorSchrödinger equationconductivity equationelliptic PDEsstatistical recovery
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The pith

The averaged interior Green's operator determines the pointwise mean and variance of a random potential, while its expected Dirichlet-to-Neumann map need 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.

The paper studies inverse problems for elliptic equations with completely general random coefficients that carry no assumed independence or ergodicity. It compares recovery from the full law of the Dirichlet-to-Neumann map, from its expectation or finite moments, and from the averaged interior Green's operator. The full law of the map recovers the law of the potential almost trivially, yet any fixed finite hierarchy of boundary moments fails to recover even the mean potential. In contrast, the averaged Green's operator recovers the pointwise mean and variance, and in a two-atom model it recovers all moments of the two-point law. A symmetric set of results holds for the conductivity equation.

Core claim

The full law of the Dirichlet-to-Neumann map determines the law of the random potential. However, the expected Dirichlet-to-Neumann map and any fixed finite hierarchy of its boundary moments need not determine even the mean potential. The averaged Schrödinger 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. Parallel statements hold for the conductivity equation in the appendices.

What carries the argument

the averaged Schrödinger Green's operator, which supplies averaged interior solution data sufficient to extract pointwise statistical moments of the random potential

If this is right

  • Knowledge of the full law of the Dirichlet-to-Neumann map recovers the entire distribution of the random potential.
  • The expected Dirichlet-to-Neumann map alone is insufficient to recover the mean potential.
  • The averaged Green's operator recovers both the pointwise mean and the pointwise variance of the potential.
  • In a two-atom discrete model the same averaged operator recovers every pointwise moment of the two-point law.

Where Pith is reading between the lines

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

  • Interior averaged data can succeed where boundary statistics fail, suggesting that measurement location matters more than volume of data in random-coefficient inverse problems.
  • The negative results for finite boundary moments indicate that statistical inverse problems may require qualitatively richer observations than their deterministic counterparts.
  • The two-atom model result hints that low-dimensional discrete approximations could serve as test cases for recovering higher-order statistics from averaged operators.

Load-bearing premise

The random coefficients remain bounded and uniformly elliptic so that the forward problems stay well-posed for arbitrary randomness.

What would settle it

Two distinct bounded elliptic random potentials that produce identical averaged Green's operators but different pointwise means or variances.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines inverse problems for the Schrödinger equation (and conductivity equation in appendices) with random coefficients that are bounded and elliptic but otherwise completely general, without independence, ergodicity, or other structure. It establishes that the full law of the Dirichlet-to-Neumann map determines the law of the random potential (via pushforward of the deterministic uniqueness result), while the expectation of the DN map or any fixed finite collection of its boundary moments fails to determine even the mean potential (via explicit counterexamples). In contrast, the averaged interior Green's operator recovers the pointwise mean and variance of the potential, and in a two-atom model recovers all pointwise moments of the two-point law.

Significance. If the results hold, the work usefully distinguishes the information content of different types of statistical boundary data for random-coefficient inverse problems. The negative results on moments of the DN map provide concrete limitations, while the positive recovery statements from the averaged Green's operator supply explicit, constructive information about moments without structural assumptions on the randomness. This clarifies what can and cannot be recovered in fully general random settings and may guide future work on stochastic inverse problems.

major comments (2)
  1. [§3] §3 (counterexamples for expected DN map): the explicit random potential used to show that the mean potential is not recovered should be stated with its support and the precise computation of the expectation of the DN map to allow direct verification that the mean is indeed lost.
  2. [§4] §4 (two-atom model): the expansion or identity relating the averaged Green's operator to the moments of the two-point law needs to be written out explicitly (including the role of the atom locations) so that the claim of recovering all moments is load-bearing and checkable.
minor comments (2)
  1. [Introduction] The introduction would benefit from a one-sentence roadmap indicating which sections contain the positive results, which contain the negative results, and where the two-atom model is treated.
  2. [§2] Notation for the averaged Green's operator G (versus the random Green's operator) should be introduced once and used consistently in all statements of the recovery results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which will improve its clarity. We address the major comments point by point below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (counterexamples for expected DN map): the explicit random potential used to show that the mean potential is not recovered should be stated with its support and the precise computation of the expectation of the DN map to allow direct verification that the mean is indeed lost.

    Authors: We agree that the counterexample in §3 would benefit from greater explicitness to facilitate direct verification. In the revised manuscript we will state the support of the random potential explicitly (as a two-point distribution supported on a pair of distinct deterministic potentials with identical means) and include the full computation of the expectation of the DN map, confirming that this expectation coincides while the underlying means differ. revision: yes

  2. Referee: [§4] §4 (two-atom model): the expansion or identity relating the averaged Green's operator to the moments of the two-point law needs to be written out explicitly (including the role of the atom locations) so that the claim of recovering all moments is load-bearing and checkable.

    Authors: We appreciate this suggestion for making the argument fully transparent. In the revision we will write out the explicit expansion of the averaged Green's operator in terms of the moments of the two-point law, explicitly indicating the contributions arising from the atom locations and showing how these terms permit isolation and recovery of all pointwise moments. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's main results compare what statistical information (full law of DN map, its expectation, finite moments, or averaged Green's operator) recovers about a general random potential. The 'almost trivial' positive result for the full law follows immediately from the standard deterministic uniqueness theorem for the DN map (an external fact, not derived or cited self-referentially here). Negative results use explicit counterexamples, while positive results for the averaged Green's operator follow from direct integral identities and moment expansions under the paper's bounded elliptic assumptions. No derivation reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the logic is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard well-posedness assumptions for elliptic equations with bounded random coefficients; no free parameters, new entities, or ad-hoc axioms are introduced beyond the general randomness assumption stated in the abstract.

axioms (1)
  • domain assumption Random coefficients are bounded and satisfy uniform ellipticity so that the forward Dirichlet problems are well-posed.
    Required for the Dirichlet-to-Neumann map and Green's operator to be defined for each realization.

pith-pipeline@v0.9.0 · 5690 in / 1289 out tokens · 65861 ms · 2026-05-20T03:22:49.779344+00:00 · methodology

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