A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem

1. Introduction

• The method first reconstructs exterior conductivity from noisy, finite-dimensional DN data, then uses regularized interior continuation.Mesh-localized exterior inputs and mass-lumped H1-Tikhonov regularization support the exterior determination; the Liouville reduction and Cauchy continuation support interior recovery.Source: §1 · pp. 2–3
• Exterior reconstruction has L2(W0) and L∞(W0) error bounds that separate concentration, approximation, regularization, and data-discrepancy contributions.The mass-lumped H1-Tikhonov system is symmetric positive definite.Source: §1 · p. 3
• Separated measurements recover the Schrödinger states and interior potential, while overlapping measurements recover the exterior potential tail for Liouville–Cauchy continuation.The tail, potential, and exterior value errors enter the final Cauchy comparison.Source: §1 · pp. 3–4
• The complete exterior-to-interior reconstruction converges under the stated approximation and parameter conditions.For full observation spaces, the paper also proves logarithmic state-bias and conductivity estimates.Source: §1 · p. 3
• At the critical Hölder exponent βc = 2s, the ideal-profile concentration error includes a logarithmic factor.The stated rate is h^λc(1 + |log h|) when βc = 2s, and h^λc otherwise.Source: §1 · p. 3
• Under the stated assumptions and α(h) ≤ cαh2, exterior reconstruction converges in L∞(W0) as h and the data discrepancy tend to zero.When δ = O(h^λδ), the error is O(ch + h^λδ).Source: §1 · p. 3

2. Preliminaries and continuous background

• The analysis assumes a bounded Lipschitz domain, 0 < s < min{1, d/2}, and a bounded nonempty exterior set W; the continuous results also cover domains bounded in one direction.The setup uses low-regularity theory, fractional Poincaré inequalities, and concentrating-sequence constructions.Source: §2 · p. 4

2.1. Function spaces and fractional operators.

• Fractional Sobolev embedding, compactness of the zero-exterior space into L2 on bounded Lipschitz sets, and unique continuation provide analytic tools used in the analysis.The stated unique continuation property says that if w and (−Δ)^s w vanish on a nonempty open set, then w vanishes throughout R^d.Source: §2.1 · p. 5
• The Liouville potential is treated as a multiplier from H^s to H^-s, while the exterior factor γ^-1/2 uses a local multiplier space; the Liouville reduction additionally requires mγ in H^s,d/s(R^d).That extra Bessel-potential regularity is not required for the forward conductivity problem.Source: §2.1 · pp. 5–6

2.2. The forward problem and the exterior Dirichlet-to-Neumann map.

• For uniformly positive bounded conductivity, the exterior value problem has a unique weak solution, and its DN map is bounded, self-adjoint, and nonnegative.The exterior energy estimate applies to inputs supported in a bounded exterior set, with its constant depending on the stated geometry and coefficient bounds.Source: §2.2 · p. 6

3.1. Finite element spaces and localized exterior data.

• Uniformly centered concentrating functions supported in W are interpolated into finite element exterior data that are well defined for sufficiently small h.The construction uses exterior concentrating functions uniform over centers in W0, and its constants are independent of h, reconstruction nodes, and the concentration index.Source: §3.1 · pp. 7–8

3.2. Measured DN map.

• The exterior determination uses only diagonal DN samples with sample-wise noise bounded by δ; total discrepancy must also include vanishing assembly and outer-tail errors.A stronger finite-dimensional DN noise model implies the sample-wise bound when the exterior inputs are normalized.Source: §3.2 · p. 9

3.3. Regularized reconstruction of the conductivity on the exterior domain.

• A mass-lumped Tikhonov reconstruction fits the measured samples on the reconstruction mesh and has a unique minimizer for every α > 0.The reconstruction uses the H1(W0) norm and a row-sum lumped mass matrix; its coefficient system is symmetric positive definite.Source: §3.3 · p. 10

3.4. Error analysis of the reconstructed conductivity.

• Uniform quantitative concentration holds for bounded positive conductivity with continuous square root, and Hölder regularity sharpens the estimate through annular integration.The concentration bound applies for t in [4r_h, d_W]; its constant is independent of the mesh, reconstruction node, and t.Source: §3.4 · pp. 11–12
• The measured-sample error is bounded through ideal-profile concentration, finite element concentration, interpolation, and measurement noise.The support-based estimate also covers normalized finite element sources supported in balls of radius at most Ch; assembly and outer-tail errors remain in the total discrepancy.Source: §3.4 · p. 13
• The leading concentration exponent is λc = θN min{βc, 2s}, with the critical logarithmic correction unchanged by the forward-space degree.At maximal interpolation gain σI = k+1−s, the profile interpolation contribution is ch h^(1−θN)(k+1−s); forward approximation error is included separately in δ.Source: §3.4 · pp. 12–13
• The a priori L2 reconstruction estimate controls averaged error, and choosing regularization relative to the combined discrepancy yields L2 convergence.The combined discrepancy includes concentration, data error, and the h^2 approximation error of the piecewise affine conductivity space.Source: §3.4 · p. 14
• Under αCBh^-2 ≤ q < 1, maximum-norm consistency combines regularized-operator stability with nodal interpolation control.The resulting consistency estimate includes an O(h^βc) interpolation contribution; the maximum-norm bound is derived independently rather than from the L2 estimate via an inverse inequality.Source: §3.4 · pp. 14, 16

Standing Liouville assumptions for the interior analysis.

• The analysis assumes bounded positive conductivity, supported Liouville deviation with specified Sobolev regularity, an L∞ interior potential, and exterior C1 regularity for discrete multiplier consistency.The transformed Schrödinger problem is well-posed through the conductivity problem, so zero need not be separately assumed to avoid being a Dirichlet eigenvalue.Source: Standing Liouville assumptions for the… · p. 17

4. Interior reconstruction by discrete Liouville–Cauchy continuation

• Separated exterior measurements recover Schrödinger states and fit the interior potential; overlapping measurements then recover the exterior potential tail for regularized Cauchy continuation of the conductivity.The exterior conductivity supplies mγ on the overlap region, while the tail supplies (−∆)^s mγ; the reconstructed square root is clipped to a positive admissible interval before squaring.Source: Standing Liouville assumptions for the… · pp. 19–20

Standing Liouville assumptions for the interior analysis.

• Local mollification of the exterior conductivity approximation yields a consistent discrete Liouville multiplier on the observation regions under the stated regularity assumptions.The construction smooths the approximation inside W0 before applying the inverse-square-root transform, ensuring the required multiplier convergence for finite-element exterior data.Source: Standing Liouville assumptions for the… · pp. 20–21

4.1. Discrete reconstruction of the Schrödinger potential.

• Galerkin–Tikhonov continuation recovers Schrödinger states from exterior observations, after which regularized weak fractional Laplacians support stabilized residual fitting of the potential.The cited single-measurement logarithmic potential estimate is not used because its smoothness, support, and continuous single-source discrepancy requirements do not match the L∞ potential and finite-dimensional observations here.Source: §4.1 · p. 21

4.1. Discrete reconstruction of the Schr¨odinger potential.

• The reduced separated data become consistent as measurement noise and exterior reconstruction errors vanish, supporting Tikhonov recovery of the state and regularized weak fractional Laplacian.The consistency estimate gives ηab(h, δ) → 0 under the stated vanishing-error conditions; exact-data state convergence is established through minimality and weak lower semicontinuity.Source: §4.1 · pp. 22, 24, 25
• With fixed sources and full fitted observation spaces, the reconstructed Schrödinger state has a logarithmic bias bound, with noisy-data error added through the state estimate.The proposition assumes shape-regular, quasi-uniform meshes and fixed Ja.Source: §4.1 · pp. 24–25
• The discrete interior-potential error is controlled under bounded state norms, but convergence conditions require reconstruction and regularization errors to decay relative to the visibility constant cQ,h.For Qγ in W^{m,d/s}(Ω), the coefficient-approximation term is bounded by C h^m |Qγ|_{W^{m,d/s}(Ω)}; the required error decay relative to visibility remains an additional condition.Source: §4.1 · p. 27

4.2. Recovery of the exterior Liouville tail.

• Overlapping measurements yield moments of qγ on Wc, and fitted cell averages of the exterior tail converge in Lp as the mesh is refined, without an asserted algebraic rate.The moments arise by comparing the measured pairing with the pairing approximated using the reconstructed interior potential.Source: §4.2 · p. 27
• For fixed sources, the clipped interior-potential reconstruction converges strongly in L2 when its stated error terms and regularization parameter vanish, without a condition involving cQ,h.The estimate also bounds the error using Bh and the measure of the set where the illuminating state is at most t.Source: §4.2 · p. 28

4.2. Recovery of the exterior Liouville tail

• A positive nonzero source yields a positive interior state and a boundary-distance lower bound, which controls the measure of the region where the transformed state is small.The resulting estimates establish L2 and L^(d/(2s)) bounds for the indicated reconstruction quantity.Source: §4.2 · p. 28
• If the reconstructed interior potential error tends to zero, the discrete tail forward problem is uniquely solvable for sufficiently small h, with a uniform state-consistency estimate.The result assumes the stated fitted-mesh approximation property.Source: §4.2 · p. 31
• The reduced tail data and their elementwise L2 projections are consistent as the discretization and source errors vanish.The projected-tail consistency follows from reduced-data consistency together with the stated projection result.Source: §4.2 · p. 32
• A source family identifies elementwise-constant tails when one source has a nonzero moment against each element's positive interior bubble, making the discrete tail moment map injective.Equivalently, the smallest generalized singular value σ_t,h is positive.Source: §4.2 · p. 33
• When the tail source family is identifying (σ_t,h > 0), the regularized tail reconstruction satisfies an error estimate for every α_t > 0.The bubble criterion ensures σ_t,h > 0 for each fixed mesh size h and bounds the reconstructed tail norm by σ_t,h^-1.Source: §4.2 · p. 34

4.2. Recovery of the exterior Liouville tail.

• With fixed limiting sources, tail minimizers converge strongly when limiting-source coverage holds, an assumption not implied by the finite-mesh bubble criterion.The tail estimate retains quantitative dependence on σt,h.Source: §4.2 · pp. 34–35

4.3. Regularized Cauchy–Liouville continuation. Γδ

• The discrete continuation minimizes weighted interior Liouville and two exterior Cauchy residuals with a coercive Tikhonov penalty, yielding a unique minimizer.The residual weights are positive, and strict convexity also gives a unique minimizer on a prescribed reduced affine trial space.Source: §4.3 · p. 36
• The continuation stability estimate controls perturbations from the three reconstructed inputs, with constants independent of mesh size, noise, and regularization parameters.The remaining error is the exact-input discretization and regularization error.Source: §4.3 · pp. 37–38
• Under the stated hypotheses, the transformed conductivity converges strongly in W^s(R^d), and the admissible interior conductivity reconstruction converges in L2(Ω).The conductivity conclusion follows from the Lipschitz relation between the transformed variable and conductivity, together with the convergence conditions on the perturbation terms.Source: §4.3 · pp. 39–40
• For full fitted observation spaces on shape-regular, quasi-uniform meshes, Proposition 4.20 establishes a logarithmic estimate from discrete Cauchy data for bounded e.The stated bound is for e ∈ eHs(ΩR) with bounded W^s(R^d)-norm.Source: §4.3 · p. 40
• For small ϑ, the proof bounds ∥w∥L2(R) by CM[log(2+ϑ^-1)]^-µs and obtains (4.95); larger ϑ are covered by ∥e∥L2(Rd) ≤ M.The proof adds the bound ∥χe∥L2(Rd) ≤ CMϑ^1/2 for small ϑ.Source: §4.3 · p. 41
• Theorem 1.2’s convergence assertions follow from Theorem 4.19, while Proposition 4.20 and Lipschitz clipping establish estimate (1.9).Source: §4.3 · p. 41
• A rate in the original DN-map discrepancy δΛ requires a bound on the complete propagated error ηC,h, which is not proved here.For example, ηC,h ≤ C[log(2+δΛ^-1)]^-µ would yield a log–log rate.Source: §4.3 · p. 41
• The cited potential estimate [19] requires common compact support in Ω and a uniform C0,s bound, while continuous-data estimates [31] and [13] require additional data and admissibility assumptions.Those assumptions are not imposed here.Source: §4.3 · p. 41

5. Numerical experiments

• The experiments use manufactured conductivities and synthetic measurements from the corresponding discrete forward problem, with geometry, mesh, inverse spaces, regularization, and error evaluation specified for each study.The interior studies in one dimension use the exterior reconstruction computed for the same h and δ; in two dimensions, a fixed exterior reconstruction is transferred across interior grids.Source: §5 · p. 42, §5.1 · p. 43, §5.2 · p. 45

5.1. Example 5.1. One-dimensional exterior-to-interior reconstruction.

• The one-dimensional interior mesh errors are lower at h = 0.004 than at h = 0.032, but the trend is nonmonotone.These interior studies assess accuracy and noise sensitivity for the prescribed inverse spaces and fixed regularization parameters.Source: §5.1 · p. 43, Table 1 · p. 43

5.2. Example 5.2. Two-dimensional exterior-to-interior reconstruction.

• The two-dimensional interior mesh study keeps the exterior input and reduced inverse grids fixed, so it does not test the dense-space limit in Theorem 4.19.The source counts are selected using discrete visibility and tail-rank diagnostics, without comparing trial reconstructions to the manufactured solution.Source: §5.2 · p. 45
• Refinement from h = 0.32 to h = 0.04 reduces the exterior L2(W0)-error from 6.2905×10−2 to 8.8340 × 10−3 and the conditional interior L2(Ω)-error from 2.3709 × 10−1 to 3.4056 × 10−2.The errors change little over the three smallest noise levels and increase at larger noise; exterior L∞(W0)-errors are nonmonotone under refinement.Source: §5.2 · p. 45, Figure 3 · p. 46, Table 2 · p. 46

• The complete exterior-to-interior reconstruction converges under the stated approximation and parameter conditions.Source: §6 · p. 46
• Exterior estimates separate concentration, approximation, regularization, and total data-discrepancy errors, including the critical logarithm in the Hölder rate.Source: §6 · p. 46
• For full observation spaces, discrete continuation gives logarithmic state-bias and interior-conductivity estimates.Source: §6 · p. 46
• Under fixed-source conditions, the clipped potential and exterior tail converge without couplings involving discrete visibility constants.Source: §6 · p. 46

• For a finite-dimensional space E and α > 0, the quadratic Tikhonov functional has a unique minimizer satisfying the variational normal equation.Source: Appendix A · p. 48
• The variational matrix is symmetric positive definite for every basis of E.Source: Appendix A · p. 48
• With T, xc, and α fixed, minimizers for data b1 and b2 satisfy a stability estimate controlled by the data difference.Source: Appendix A · pp. 48–49
• The appendix also bounds solutions to residual-load equations and minimizer changes when the operator and data are perturbed.Source: Appendix A · pp. 48–49

• Reference calibration and optional radius extrapolation form the nodal data used in the experiments and connect modified samples to the exterior reconstruction estimates.Source: Appendix B · p. 49
• With multiple retained radii, the extrapolated sample is the intercept of a least-squares fit in {1, r2s}; with one radius, the calibrated value is retained directly.Source: Appendix B · p. 49
• Under the stated conductivity bounds, continuity, and concentration assumptions, calibrated and extrapolated samples satisfy the appendix's consistency estimate.Source: Appendix B · pp. 49–50
• Distinct radii alone do not ensure the required weight bound, and an r2s fit does not generally cancel an r2s log r term under Hölder regularity alone.When additional noise is added after extrapolation, the sample error includes both the pre-noise error and the absolute amplitude of that noise.Source: Appendix B · p. 50