A new preprint reports explicit high-probability error bounds for recovering an unknown conductivity from noisy measurements taken at finitely many deterministic scattered locations. The analysis covers both a continuous regularized reconstruction and a finite-element version, giving bounds that show how regularization, sampling and mesh-related quantities enter the estimate.
The paper asks whether the conductivity coefficient in a bounded elliptic problem can be recovered from those observations. Its measurement model adds independent, identically distributed, zero-mean sub-Gaussian noise, with the noise level represented by σ.
A controlled reconstruction
The proposed continuous reconstruction minimizes a least-squares objective over the admissible conductivity set while applying a W 1,4 (Ω) penalty, a mathematical regularizer used in the analysis to provide the needed regularity and Lipschitz stability. The paper also reports that at least one global minimizer exists for the continuous problem, almost surely with respect to the data.
For computation, the reconstruction is discretized with the Galerkin finite element method using continuous piecewise linear elements. The discrete theorem assumes the target conductivity has the stated H 2 (Ω) and W 1,∞ (Ω) regularity, belongs to the admissible set, and is paired with a right-hand side f in Lp (Ω) for p greater than the spatial dimension. Under those conditions, the paper likewise reports existence of at least one global minimizer for the finite-element problem, almost surely.
What the theorems cover
For the continuous solution, Theorem 2.4 gives a high-probability error bound under Assumptions 2.1 and 2.2. The result also yields an L2 (Ω) conclusion when Condition 2.3 holds, but that conclusion is conditional rather than automatic.
The finite-element counterpart, Theorem 2.6, adds mesh-dependent terms to the high-probability estimate. Its displayed bound depends on η, h, n and ω, and it includes a conditional L2 (Ω) conclusion under Condition 2.3. The paper gives guidance for choosing the mesh size h and regularization parameter γ together, and states that the resulting convergence rate agrees with the continuous case.
Those results do not apply to every possible scattered layout. The theoretical analysis assumes the observation points are quasi-uniformly distributed: the largest spacing must stay within a fixed multiple of the smallest spacing for all sufficiently large n. The unweighted L2 conclusion also depends on Condition 2.3, so the theorem’s scope is tied to that extra condition as well.
Synthetic tests show the sampling signal
The numerical illustrations use admissible bounds c0 = 1.0 and c1 = 3.0, a unit-square domain, and uniformly distributed observation points. They compare a single-bump conductivity profile with a multiple-bump profile while varying the number of sampling points and testing two noise strengths.
The synthetic observations add standard-normal noise scaled by σ∥u†∥L∞ (Ω). The discrete optimization is carried out by steepest descent, with gradients computed using an adjoint-state approach and Newton-based iterations.
The reported optimal tested regularization values were 10−8.5 when σ was 5.0% and 10−9 when σ was 1.0%. The automatically determined a priori values were 3.32e-9 and 3.89e-10, respectively, and were close to the tested optima.
Across both conductivity profiles and both tested noise strengths, the reported conductivity and state error metrics decreased steadily toward zero as the number of sampling points increased. The examples therefore illustrate the theory under the reported setup rather than establish performance for other sampling designs or noise laws.
The document is labeled arXiv:2608.25749v1 [math.NA] and dated 26 Aug 2026.
The paper reports support from Hong Kong RGC, the ANR / Hong Kong RGC Joint Research Scheme, The Chinese University of Hong Kong, the University of Macau, the National Natural Science Foundation of China and the Guangdong Provincial Key Laboratory, with the grants listed in the paper.
Paper data and sources
Original title: Error Analysis of the Inverse Conductivity Problem with Scattered Measurements
Authors: Bangti Jin, Qimeng Quan, Wenlong Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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