arxiv: 2605.06354 · v1 · submitted 2026-05-07 · 🧮 math.AP

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H\"older Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems

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

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classification 🧮 math.AP

keywords analytic inverse problemsHölder stabilityexact uniquenessfinite-dimensional parametersLojasiewicz inequalityboundary measurementsconductivity inverse problemelasticity inverse problem

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The pith

If real analytic boundary measurements uniquely determine a finite-dimensional parameter, they determine it with Hölder stability on compact sets.

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

The paper shows that in finite-dimensional parameter spaces, if a real analytic operator F mapping parameters to boundary measurements satisfies that equal measurements imply equal recovered quantities R, then R can be recovered with Hölder stability from F on any compact set of parameters. This is significant because it turns qualitative uniqueness results into quantitative stability estimates without needing additional assumptions. The argument scalarizes the equality F(p) = F(q) using a Hilbert-Schmidt product and invokes the Lojasiewicz inequality to control the distance. It further establishes that the stability persists when only finitely many scalar data elements are used, drawn from dense families of inputs and tests. The theorem is illustrated with examples from anisotropic conductivity and elasticity inverse problems.

Core claim

If the boundary measurement operator F is real analytic on an open set U in R^m and satisfies the uniqueness condition that F(p) equals F(q) only when R(p) equals R(q), then the map R satisfies a Hölder stability estimate on every compact subset of U. The proof proceeds by reducing the operator equality to a scalar equation via Hilbert-Schmidt inner product and then using the Lojasiewicz distance inequality for real analytic functions. Additionally, after fixing countable dense families of inputs and tests, only finitely many matrix elements of the data are needed to achieve the same Hölder recovery.

What carries the argument

The exact uniqueness condition for the real analytic operator F, scalarized via Hilbert-Schmidt product and controlled by the Lojasiewicz inequality to produce Hölder stability for R.

Load-bearing premise

The operator F must be real analytic and satisfy exact uniqueness with respect to R, in a finite-dimensional parameter space.

What would settle it

A counterexample consisting of a real analytic F on a finite-dimensional U where F(p)=F(q) implies R(p)=R(q) but the difference |R(p)-R(q)| is not bounded by C |F(p)-F(q)|^alpha for any alpha>0 on some compact set.

read the original abstract

We prove a stability theorem for finite-dimensional analytic inverse problems. Let \(U\subset\R^m\) be an open parameter set, let \(F(p)\) be a boundary measurement operator, and let \(R(p)\) be the finite-dimensional quantity to be recovered. If \(F\) is real analytic and \[ F(p)=F(q)\quad\Longrightarrow\quad R(p)=R(q), \] then \(R\) satisfies a H\"older stability estimate on every compact subset of \(U\). The proof uses a Hilbert--Schmidt scalarization of the operator equation \(F(p)=F(q)\) and the \L{}ojasiewicz distance inequality. We also prove that, after fixing countable dense families of boundary inputs and tests, finitely many scalar matrix elements of the data give the same H\"older recovery on compact parameter sets. This finite-measurement conclusion is qualitative: the proof does not give an effective measurement list, exponent, or constant. The finite-measurement statement follows from finite determinacy of real analytic zero sets. We apply the result to local Neumann-to-Dirichlet data for piecewise constant anisotropic conductivities and to localized Dirichlet-to-Neumann data for piecewise homogeneous anisotropic elasticity.

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 paper turns uniqueness into Hölder stability via scalarization and Łojasiewicz but the general claim needs an explicit analyticity assumption on R.

read the letter

The central result here is a general theorem: when the forward map F is real analytic on a finite-dimensional parameter space and exact uniqueness holds for the quantity R, then R obeys a Hölder stability estimate on compact sets. The proof scalarizes the operator equation F(p)=F(q) with Hilbert-Schmidt products to produce an analytic function g whose zero set matches the equality set, then invokes the Łojasiewicz inequality to bound the distance in R by a power of g.

Referee Report

1 major / 0 minor

Summary. The paper proves that if the forward measurement operator F is real analytic on an open set U ⊂ ℝ^m and satisfies the exact uniqueness condition F(p) = F(q) ⇒ R(p) = R(q), then the finite-dimensional recovery map R satisfies a Hölder stability estimate on every compact subset of U. The argument scalarizes the operator equation via Hilbert–Schmidt inner products to produce an analytic function g(p, q) whose zero set coincides with the set where F(p) = F(q), then invokes the Łojasiewicz inequality. A finite-measurement version follows from finite determinacy of real-analytic zero sets. The result is applied to local Neumann-to-Dirichlet data for piecewise-constant anisotropic conductivities and to localized Dirichlet-to-Neumann data for piecewise-homogeneous anisotropic elasticity.

Significance. If the central implication holds, the manuscript supplies a general, abstract route from qualitative uniqueness to quantitative Hölder stability in finite-dimensional analytic inverse problems. The approach is noteworthy for its use of the Łojasiewicz inequality and the finite-determinacy argument that yields a qualitative finite-measurement conclusion without explicit constants or exponents. The applications to conductivity and elasticity illustrate concrete settings where the hypotheses can be verified.

major comments (1)

[Abstract / Theorem 1.1] Abstract and main theorem statement: the claimed implication (analyticity of F plus exact uniqueness ⇒ Hölder stability of R) is stated without any regularity hypothesis on R. The proof route via Łojasiewicz requires that R(p) − R(q) itself be real analytic in (p, q) so that it vanishes on the zero set of the analytic scalarization g and therefore satisfies |R(p) − R(q)| ≤ C |g(p, q)|^β. Without this assumption the statement is false (counter-example: F(p) = p on an interval and R any continuous but non-Hölder function). The applications treat R as the vector of piecewise-constant parameter values, which is analytic, but the general theorem must state the analyticity requirement on R explicitly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to explicitly state the analyticity of the recovery map R in the main theorem. We agree that this is a necessary hypothesis for the application of the Łojasiewicz inequality and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / Theorem 1.1] Abstract and main theorem statement: the claimed implication (analyticity of F plus exact uniqueness ⇒ Hölder stability of R) is stated without any regularity hypothesis on R. The proof route via Łojasiewicz requires that R(p) − R(q) itself be real analytic in (p, q) so that it vanishes on the zero set of the analytic scalarization g and therefore satisfies |R(p) − R(q)| ≤ C |g(p, q)|^β. Without this assumption the statement is false (counter-example: F(p) = p on an interval and R any continuous but non-Hölder function). The applications treat R as the vector of piecewise-constant parameter values, which is analytic, but the general theorem must state the analyticity requirement on R explicitly.

Authors: We agree with this comment. The proof indeed relies on R being real analytic so that R(p) - R(q) is analytic and the Łojasiewicz inequality can be applied to bound |R(p) - R(q)| by a power of |g(p,q)|. The counterexample is correct and shows that without analyticity of R the conclusion fails. In our applications, R(p) coincides with the parameter vector p itself (the piecewise constant values), which is analytic. We will revise the abstract, the statement of Theorem 1.1, and relevant parts of the introduction to explicitly include the assumption that R is real analytic on U. This is a straightforward clarification and does not affect the validity of the results. revision: yes

Circularity Check

0 steps flagged

No circularity; proof relies on external theorems without self-referential reduction

full rationale

The derivation claims that real-analyticity of the forward map F together with the exact uniqueness implication F(p)=F(q) ⇒ R(p)=R(q) yields a Hölder stability estimate for R on compact subsets of the finite-dimensional domain U. The argument proceeds by Hilbert-Schmidt scalarization to produce an auxiliary analytic function g(p,q) whose zero set coincides with the level set of F, followed by an application of the Łojasiewicz inequality. No step reduces by construction to the paper's own inputs: there are no self-definitional equivalences, no fitted parameters renamed as predictions, and no load-bearing self-citations whose content is itself unverified within the manuscript. The central implication is supported by external, independently established results (Łojasiewicz inequality and finite determinacy of real-analytic zero sets) rather than by any tautological renaming or internal fit. The finite-measurement corollary is likewise derived from the same external analytic properties and does not collapse to the uniqueness assumption alone. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard theorems in real analytic geometry and operator theory, with no new free parameters or invented entities introduced by the paper.

axioms (3)

standard math The Łojasiewicz inequality holds for real analytic functions
Invoked to obtain the distance estimate from the scalarized equation.
domain assumption Hilbert-Schmidt scalarization of the operator equation preserves the zero set properties
Used to reduce the vector-valued operator equation to scalar analytic functions.
standard math Real analytic zero sets are finitely determined
Supports the finite-measurement conclusion.

pith-pipeline@v0.9.0 · 5514 in / 1561 out tokens · 87392 ms · 2026-05-08T07:04:25.787709+00:00 · methodology

discussion (0)

Reference graph

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