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arxiv: 2606.21375 · v1 · pith:VMRQ3N7Pnew · submitted 2026-06-19 · 🧮 math.NA · cs.NA· math.AP

Inverse Source Problems for a Class of Fractional Elliptic Equations with Singular Coefficients

Zewen Wang , Liang Guo , Shufang Qiu , Bin Wu This is my paper

Pith reviewed 2026-06-26 13:33 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords inverse source problemfractional elliptic equationsingular coefficientsconditional stability estimateTikhonov regularizationquasi-boundary regularizationHilbert scaleexponential operators
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The pith

Hölder stability in an exponential Hilbert scale underpins convergent regularization for recovering sources in fractional elliptic equations with singular coefficients.

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

This work studies the inverse source problem for fractional elliptic equations that carry singular coefficients. It first establishes well-posedness of the direct problem via a formal solution. For the inverse problem it proves a Hölder-type conditional stability estimate inside a Hilbert scale generated by exponential operators. Two regularization schemes, exponential Tikhonov and exponential quasi-boundary regularization, are then built on this estimate and shown to converge under both a priori and a posteriori parameter rules. Spectral truncation further extends the schemes to arbitrary L2 sources, with numerical tests confirming accuracy for smooth and piecewise-smooth sources at low signal-to-noise ratios.

Core claim

A Hölder-type conditional stability estimate is obtained in a Hilbert scale associated with exponential operators for the inverse source problem; based on this estimate, exponential-type Tikhonov regularization and exponential quasi-boundary value regularization yield convergent reconstructions of the unknown source under a priori and a posteriori parameter choices, and finite-dimensional spectral approximations show applicability to general square-integrable sources.

What carries the argument

Hölder-type conditional stability estimate in a Hilbert scale associated with exponential operators, which serves as the foundation for the two regularization methods.

If this is right

  • The regularized approximations converge to the exact source as the noise level tends to zero under the given parameter choices.
  • Both a priori and a posteriori strategies for selecting the regularization parameter produce convergent reconstructions.
  • Finite-dimensional spectral approximations allow the methods to reconstruct general square-integrable sources without exponential source conditions.
  • Numerical experiments indicate stable and accurate recovery for both smooth and piecewise smooth sources even when the signal-to-noise ratio is low.

Where Pith is reading between the lines

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

  • The same stability framework could be tested on time-fractional or space-time fractional variants of the equation.
  • Comparison with classical Tikhonov regularization without the exponential scale would clarify the benefit of the new Hilbert scale.
  • Extension to three-dimensional domains or more general singular coefficients would test the robustness of the well-posedness assumption for the direct problem.

Load-bearing premise

The direct problem admits a formal solution whose well-posedness can be established for the given class of fractional elliptic operators with singular coefficients.

What would settle it

Numerical computation showing that the regularized solutions fail to converge to the true source for a sequence of noisy data with noise level approaching zero, while the source satisfies the assumed bounds, would disprove the convergence claims.

Figures

Figures reproduced from arXiv: 2606.21375 by Bin Wu, Liang Guo, Shufang Qiu, Zewen Wang.

Figure 1
Figure 1. Figure 1: Numerical inversions by the exponential-type Tik [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical inversions by the exponential-type qua [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical inversions by the exponential-type Tik [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical inversions by the exponential-type qua [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
read the original abstract

An inverse source problem for a class of fractional elliptic equations with singular coefficients is investigated in this paper. For the corresponding direct problem, a formal solution is derived and the well-posedness of the solution is established. For the inverse problem, a H\"older-type conditional stability estimate is obtained in a Hilbert scale associated with exponential operators. Based on this stability framework, two regularization methods are proposed for reconstructing the unknown source term: the exponential-type Tikhonov regularization method and the exponential quasi-boundary value regularization method. Convergence estimates for the regularized solutions are derived under both a priori and a posteriori choices of the regularization parameter. In addition, finite-dimensional spectral approximation results show that the proposed methods are also applicable to general square-integrable source terms, without requiring the exact source to satisfy an exponential-type source condition. Numerical experiments demonstrate that the proposed methods provide stable and accurate reconstructions for both smooth and piecewise smooth sources even under low signal-to-noise ratio conditions.

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

1 major / 0 minor

Summary. The paper investigates an inverse source problem for a class of fractional elliptic equations with singular coefficients. For the direct problem, a formal solution is derived and well-posedness is established. For the inverse problem, a Hölder-type conditional stability estimate is obtained in a Hilbert scale associated with exponential operators. Two regularization methods are proposed: exponential-type Tikhonov regularization and exponential quasi-boundary value regularization, with convergence estimates derived under a priori and a posteriori parameter choices. Finite-dimensional spectral approximation results extend applicability to general square-integrable sources, and numerical experiments demonstrate stable reconstructions under noise.

Significance. If the foundational well-posedness holds, the work provides a stability framework and regularization schemes for inverse source problems in fractional elliptic equations with singular coefficients, using exponential Hilbert scales. The extension to general L2 sources without requiring exponential source conditions and the numerical validation for both smooth and piecewise smooth sources under low SNR are strengths that could support applications in models with anomalous diffusion or singular media.

major comments (1)
  1. [Abstract (direct problem paragraph)] Abstract (paragraph on direct problem): The well-posedness of the direct problem is load-bearing for the Hölder stability estimate and the convergence rates of both regularization methods. The abstract states only that a formal solution is derived and well-posedness established, without specifying the function space, the precise class of singular coefficients (e.g., whether in L^∞, weighted spaces, or weaker), or how the fractional power remains sectorial or admits a discrete spectrum suitable for the eigenfunction expansion after incorporating the singularity. This gap must be addressed explicitly to support the subsequent claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the recommendation for major revision. The single major comment concerns the level of detail in the abstract regarding the direct problem; we address it below and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract (direct problem paragraph)] Abstract (paragraph on direct problem): The well-posedness of the direct problem is load-bearing for the Hölder stability estimate and the convergence rates of both regularization methods. The abstract states only that a formal solution is derived and well-posedness established, without specifying the function space, the precise class of singular coefficients (e.g., whether in L^∞, weighted spaces, or weaker), or how the fractional power remains sectorial or admits a discrete spectrum suitable for the eigenfunction expansion after incorporating the singularity. This gap must be addressed explicitly to support the subsequent claims.

    Authors: We agree that the abstract would benefit from greater explicitness on these points to strengthen the presentation. In the revised manuscript we will expand the direct-problem sentence in the abstract to state that well-posedness is established in the space H¹₀(Ω) for coefficients belonging to L^∞(Ω) that satisfy a uniform positivity condition; under these assumptions the associated elliptic operator remains sectorial, generates a discrete spectrum, and permits the eigenfunction expansion used throughout the paper. The full technical details already appear in Section 2; the abstract revision will simply make this reference explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained

full rationale

The paper first derives a formal solution and establishes well-posedness for the direct problem, then obtains a Hölder conditional stability estimate in an exponential Hilbert scale from that foundation, and finally derives convergence rates for the two regularization methods under a priori/a posteriori parameter choices. No quoted step reduces by construction to a fitted input renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain. The central claims remain independent of the paper's own fitted quantities or prior self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of fractional elliptic operators and Hilbert-space theory; no free parameters are explicitly fitted in the abstract description, and no new entities are postulated.

axioms (2)
  • domain assumption The fractional elliptic operator with singular coefficients admits a formal eigenfunction expansion whose coefficients yield a well-posed direct problem.
    Invoked in the direct-problem paragraph of the abstract.
  • domain assumption The solution space can be equipped with a Hilbert scale generated by exponential operators that produces a Hölder-type conditional stability estimate for the inverse source map.
    Central to the stability framework stated in the abstract.

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Reference graph

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