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arxiv: 2511.05277 · v1 · submitted 2025-11-07 · 🧮 math.AP · cs.NA· math.NA

Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation

Andrii Hulianytskyi , Sergei Pereverzyev , Sergii Siryk , Nataliya Vasylyeva This is my paper

Pith reviewed 2026-05-18 00:07 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA MSC 35R3026A3365M32
keywords inverse problemsfractional subdiffusionparameter identificationnonlocal observationTikhonov regularizationmemory kernelquasi-optimality
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The pith

Explicit formulas and Tikhonov regularization recover scalar coefficients and fractional orders in multi-term subdiffusion equations from nonlocal observations.

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

The paper establishes that, given suitable data, explicit expressions exist for unknown scalar parameters such as coefficients and orders of fractional time derivatives inside a linear subdiffusion model that includes a memory kernel. These formulas are obtained by combining the nonlocal observation with the governing equation, after which uniqueness and stability statements follow directly from the assumptions placed on the data. The same framework then supplies a practical numerical procedure: Tikhonov regularization paired with the quasi-optimality criterion recovers the parameters from discrete, noisy measurements, and the method is shown to work on several test problems.

Core claim

Under assumptions on smoothness, positivity, and specific forms of the memory kernel and forcing term, the nonlocal observation yields explicit formulas for the unknown scalar parameters in the multi-term operator D_t; these formulas imply uniqueness and stability of the inverse problem, and the parameters can be reconstructed numerically from noisy data via Tikhonov regularization with quasi-optimality.

What carries the argument

The nonlocal observation (an integral or averaged measurement of the solution) combined with the linear subdiffusion equation D_t u - L1 u - K * L2 u = g, which isolates the unknown scalar parameters inside the time-fractional operator.

If this is right

  • The inverse problem admits a unique stable solution when the data assumptions hold.
  • The same nonlocal observation can be used to identify both coefficients and fractional orders simultaneously.
  • Tikhonov regularization with quasi-optimality produces reliable numerical approximations even when measurements contain noise.
  • The approach extends to multi-term fractional operators with time-dependent elliptic coefficients.

Where Pith is reading between the lines

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

  • The explicit-formula technique may simplify parameter recovery in other anomalous-diffusion models that admit nonlocal measurements.
  • Stability estimates derived here could serve as a benchmark for comparing different regularization strategies in fractional inverse problems.
  • If the memory kernel itself is only partially known, the same observation might be reused to identify both the kernel and the scalar parameters.

Load-bearing premise

The given data must satisfy specific smoothness, positivity, or functional forms for the memory kernel and external force so that the explicit formulas can be derived and the uniqueness-stability statements hold.

What would settle it

A concrete counter-example in which the data satisfy the stated smoothness and positivity conditions yet the proposed explicit formulas fail to return the true parameter values, or in which the regularized reconstruction from noisy data does not converge to the correct parameters as the noise level tends to zero.

read the original abstract

In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, $\mathbf{D}_t$. To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation $\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=g(x,t),$ where $\mathcal{L}_{i}$ are the second order elliptic operators with time-dependent coefficients, $\mathcal{K}$ is a summable memory kernel, and $g$ is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.

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 develops an inverse problem for recovering scalar parameters (including coefficients and fractional orders) in a multi-term time-fractional subdiffusion equation with memory kernel, driven by the model D_t u - L1 u - K * L2 u = g(x,t) where L1 and L2 are second-order elliptic operators with time-dependent coefficients. Under stated assumptions on the data, explicit formulas for the unknowns are derived from a nonlocal observation; uniqueness and stability are analyzed; and a Tikhonov-regularized reconstruction algorithm with quasi-optimality parameter choice is implemented and tested numerically on noisy discrete data.

Significance. If the explicit formulas and stability estimates hold under the paper's data assumptions, the work would supply a direct, non-iterative recovery route for parameters in fractional diffusion models with memory, which is useful for applications in anomalous transport. The combination of analytic formulas with a practical regularized solver and numerical demonstrations strengthens the contribution to inverse problems for time-fractional PDEs.

major comments (2)
  1. [Section 3] The derivation of the explicit formulas (Section 3, around the Laplace-transform step leading to the algebraic expressions for the scalar parameters) relies on a separation-of-variables or eigenfunction expansion that presupposes L1 and L2 share a common eigenbasis. With genuinely time-dependent coefficients in both operators, simultaneous diagonalizability is not automatic; the manuscript must either restrict the coefficients to be time-independent, impose an explicit commutativity or joint-diagonalizability assumption, or provide a different argument that avoids the eigenbasis requirement. This is load-bearing for the central claim of explicit recovery.
  2. [Theorem 4.2] The uniqueness and stability statements (Theorem 4.2 and the subsequent stability estimate) are stated under 'certain assumptions on the given data,' but the precise conditions on the memory kernel K, the forcing g, and the positivity/smoothness requirements that close the argument are not listed explicitly before the theorem. Without these, it is impossible to verify whether the formulas remain valid when the time-dependent coefficients prevent the usual spectral reduction.
minor comments (2)
  1. [Section 2] Notation for the nonlocal observation functional is introduced without a numbered equation; adding an explicit definition (e.g., Eq. (2.7)) would improve readability.
  2. [Section 5] In the numerical section, the reported error tables would benefit from an additional column showing the noise level relative to the exact parameter values to allow direct assessment of stability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, outlining the revisions we will make to strengthen the presentation and rigor of the results.

read point-by-point responses

  1. Referee: [Section 3] The derivation of the explicit formulas (Section 3, around the Laplace-transform step leading to the algebraic expressions for the scalar parameters) relies on a separation-of-variables or eigenfunction expansion that presupposes L1 and L2 share a common eigenbasis. With genuinely time-dependent coefficients in both operators, simultaneous diagonalizability is not automatic; the manuscript must either restrict the coefficients to be time-independent, impose an explicit commutativity or joint-diagonalizability assumption, or provide a different argument that avoids the eigenbasis requirement. This is load-bearing for the central claim of explicit recovery.

    Authors: We appreciate the referee highlighting this key technical requirement. The derivation in Section 3 does employ an eigenfunction expansion after the Laplace transform in time, which presupposes that the time-dependent operators L1 and L2 share a common eigenbasis. To ensure the explicit formulas remain valid, we will add an explicit joint-diagonalizability assumption (for instance, that L1(t) and L2(t) commute for each t) to the problem formulation in the revised manuscript. This assumption will be stated clearly in the introduction and immediately before the derivation in Section 3. We view this as the minimal and standard way to preserve the paper's scope while making the argument rigorous; alternatively, we could restrict to time-independent coefficients, but the commutativity condition better maintains generality for the intended applications. revision: yes

  2. Referee: [Theorem 4.2] The uniqueness and stability statements (Theorem 4.2 and the subsequent stability estimate) are stated under 'certain assumptions on the given data,' but the precise conditions on the memory kernel K, the forcing g, and the positivity/smoothness requirements that close the argument are not listed explicitly before the theorem. Without these, it is impossible to verify whether the formulas remain valid when the time-dependent coefficients prevent the usual spectral reduction.

    Authors: We agree that greater explicitness is needed for the assumptions in Theorem 4.2. In the revised manuscript we will insert a dedicated paragraph or subsection immediately preceding Theorem 4.2 that enumerates all required conditions on the memory kernel K, the forcing term g(x,t), and the positivity/smoothness properties of the data and coefficients. This list will be cross-referenced to the joint-diagonalizability assumption added in response to the first comment, allowing readers to verify the uniqueness and stability estimates under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

Explicit formulas derived directly from model equations under stated assumptions; no reduction to fitted inputs or self-citation chains

full rationale

The paper states that under certain assumptions on the given data, explicit formulas for the unknown scalar parameters are derived from the nonlocal observation in the subdiffusion equation. The approach is presented as analytical recovery followed by Tikhonov regularization for noisy data. No quoted steps show a parameter fitted to a subset of the same data then renamed as a prediction, nor a load-bearing uniqueness result justified solely by overlapping self-citation. The derivation chain remains self-contained against the model and observation operator; time-dependent coefficients in L1 and L2 are addressed by the stated assumptions rather than by circular redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on unspecified assumptions about the data, the memory kernel, and the elliptic operators that enable the explicit formulas and stability proofs.

axioms (1)
  • domain assumption Certain assumptions on the given data in the model allow derivation of explicit formulas for the unknown scalar parameters.
    Invoked in the abstract to justify the analytical recovery step.

pith-pipeline@v0.9.0 · 5507 in / 1158 out tokens · 29840 ms · 2026-05-18T00:07:11.453346+00:00 · methodology

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