Regularized Sequential Reconstructions of Dynamic Nonsmooth Sources from Sparse Boundary Flux Data

1 Introduction

• The problem combines distinct unknown source regions on successive intervals with finitely many fixed boundary sensors, beyond prior results for narrower source or observation settings.Earlier finite-point uniqueness results concern time-independent or separable sources; another treatment of semi-discrete sources requires flux on an open boundary subset [40] [34].Source: §1 · p. 2
• The block lower-triangular forward map lets each interval’s source be reconstructed from its current data and estimates of earlier sources, without later measurements.Forward substitution reduces the coupled problem to sequential inverse problems governed by the same type of current-interval operator.Source: §1 · p. 3
• Because Fτ is unbounded and not closable on L2(Ω), the paper uses a Hilbert-scale penalty that makes the regularized problem one for a bounded operator.The change of variables produces an equivalent standard Tikhonov problem for Aτ,s, which is Hilbert–Schmidt and injective.Source: §1 · p. 4
• Injectivity of the compactified operator, together with forward substitution, yields uniqueness for the full dynamic problem with admissible sensors.The paper states uniqueness on each finite observation window.Source: §1 · p. 4
• The Galerkin–Tikhonov reconstructions converge as the noise level decreases, and thresholding recovers the unknown regions in measure.Convergence holds when the regularization parameters and Galerkin dimensions are chosen consistently with the effective data error.Source: §1 · p. 4
• Earlier reconstruction errors enter later data through history operators with exponential decay, while adding nested sensors improves finite-dimensional inverse stability.For nested sensor sets, the restricted operator’s singular values are nondecreasing as sensors are added.Source: §1 · p. 4

2 Notation and preliminaries

• The paper works on the open unit disk with sources and data in L2(Ω), and uses daggers for exact quantities and superscript δ for noisy data.Source: §2 · p. 5

2.1 The unknown regions

• Each unknown Dk is a Lipschitz region active on its prescribed interval, with source p†k = χDk; equal characteristic functions in L2 imply equal Lipschitz regions.Thus recovering the source in L2 identifies the region, and thresholding returns the unique Lipschitz region up to a null set.Source: §2.1 · p. 5

2.2 Function spaces

• The preliminaries specify Sobolev and Besov spaces, including Hs(D) as restrictions from Hs(R2) for bounded Lipschitz domains.Source: §2.2 · p. 5

2.3 Compact and Hilbert–Schmidt operators

• Compact operators are operator-norm limits of finite-rank operators, Hilbert–Schmidt operators are compact, and a densely defined operator is closable exactly when its adjoint domain is dense.Source: §2.3 · pp. 5–6

2.4 The Dirichlet Laplacian on the disk

• The Dirichlet Laplacian on the disk has an orthonormal eigenbasis with radial and cosine/sine angular modes; for distinct sensors, nonzero sensor vectors yield coefficients that are not square-summable.The partial coefficient sums grow as RM log N + O(1), a logarithmic divergence identified as the key reason the forward operator is unbounded.Source: §2.4 · p. 6

2.5 The Hilbert scale

• For 0 < s < 1/2, D(Ls) = Hs(Ω) and contains characteristic functions of Lipschitz regions, while the inverse L^-s is compact.This makes the Hilbert-scale penalty finite on the sources while providing a compactifying space for the analysis.Source: §2.5 · p. 7

3.1 The forward operator and its representation on each time interval

• On interval k, the flux is the current-source response plus history terms from earlier sources, yielding a block lower triangular system solved by forward substitution.Source: §3.1 · pp. 9–10

3.2 Compactness of history operators and nonclosability of Fτ

• For pairwise-distinct sensors, Fτ is densely defined, unbounded, and nonclosable in X, while it is bounded under the stronger norm ∥Ls·∥X.The history operators are Hilbert–Schmidt; changing the source norm enables Hilbert-scale Tikhonov regularization.Source: §3.2 · pp. 10–12

3.3 Source-side compactification

• Changing the source norm yields a Hilbert–Schmidt compact operator Aτ,s; it is injective for every τ > 0 when ΓM is admissible.The transformed formulation preserves the sources and fluxes while expressing the Hilbert-scale penalty as a standard Tikhonov penalty.Source: §3.3 · p. 12

3.4 Uniqueness

• For admissible ΓM, agreement of boundary fluxes through time tk uniquely determines the source functions and regions on the first k intervals.Source: §3.4 · p. 13

3.5 Finite-dimensional Galerkin–Tikhonov regularization

• The Galerkin–Tikhonov subsection begins by fixing an interval and treating its diagonal block, then introduces the exact shifted data.Source: §3.5 · p. 13
• For every α > 0, the finite-dimensional functional has a unique minimizer, and under the parameter rule the reconstructions converge strongly in X.The discretization error tends to zero by the Hilbert–Schmidt property.Source: §3.5 · pp. 13–14

3.6 The main convergence theorem

• The method uses forward substitution: it subtracts reconstructed earlier-source contributions and regularizes each diagonal block through a compactified problem.The finite-dimensional minimization is a linear least-squares problem on each Galerkin space.Source: §3.6 · p. 15
• Under the stated parameter conditions, sequential Galerkin–Tikhonov reconstructions converge on every interval as noise tends to zero.The proof proceeds by induction, using bounded history operators to show that errors in earlier reconstructions vanish in the shifted data for each later interval.Source: §3.6 · p. 15
• Thresholding the convergent source reconstructions recovers the unknown regions in measure as the noise vanishes.The thresholded regions differ from the true regions only on sets whose measure vanishes with the noise.Source: §3.6 · p. 15

4.1 Propagation of reconstruction errors

• Earlier reconstruction errors enter later shifted data through the history operators, with an estimate showing exponential decay in time.The estimate depends on the domain and fixed sensor set.Source: §4.1 · p. 16, §4 · p. 16

4.2 Sensor number and stability

• The sensor analysis considers the compactified operator built from a sensor set, with measurements in the unweighted Euclidean norm across sensors.Source: §4.2 · p. 16
• For nested sensor sets, adding a sensor does not decrease the smallest singular value on V_n when the operator is injective there, so the stability constant does not increase.Source: §4.2 · p. 16

• Experiments compare box constraints, Fourier boundary approximation, and pretrained contour-network correction on three-interval sequences with 0.1% or 1% noise per interval.History terms come from classical Galerkin–Tikhonov reconstructions, while refinements act only on the current source.Source: §5 · p. 17

5.1 Discretization, sequential Tikhonov and bounded reconstruction

• The spectral Galerkin space retains 792 eigenfunctions with angular order at most 16 and radial index at most 24.The truncation is by angular order and radial index, not by eigenvalue.Source: §5.1 · p. 17
• Each interval contributes 2,000 measurements from 16 sensors sampled at 125 relative times.The same relative sampling times are used on all three intervals, yielding identical interval forward matrices.Source: §5.1 · p. 17

5.1.1 A bounded version

• The experimental box-constrained variant enforces source values in [0, 1], but its convergence and stability are not analyzed.It uses a different penalty and feasible set from classical Tikhonov and initializes both Fourier fitting and network refinement.Source: §5.1.1 · p. 18

5.1.2 Sources and noisy data

• The test design uses noisy sequences of one-component disks or simple polygons, with 24 disjoint test sequences instances across the stated data partitions.Noise is 0.1% or 1% of the isolated current-source signal; the test sequences comprise 24 source instances across eight sequences.Source: §5.1.2 · p. 18

5.1.3 Reconstruction errors

• Reconstruction error is assessed with Dice overlap and boundary distances, including a modified distance that reduces the influence of isolated large boundary errors.The modified distance uses the 95th percentile; boundary distances are reported in grid-spacing units.Source: §5.1.3 · pp. 18–19

5.2 Fourier refinement

• The Fourier fit represents the region with a free center and polar radius parameterized by boundary harmonics through mode 16.The parameter vector has 33 entries.Source: §5.2 · p. 19
• The forward response of a fitted region is computed from boundary integrals rather than by first representing it on the spatial grid.The fit is optimized by Levenberg–Marquardt continuation and retains candidates satisfying the stated radius and disk-interior constraints.Source: §5.2 · p. 19
• The discrepancy threshold is scaled by calibrated effective-error factors of 1.073 and 1.180 at 1% noise, and 1.094 and 1.227 at 0.1% noise for intervals two and three.If no feasible approximation meets the criterion, the bounded reconstruction is retained without neural correction.Source: §5.2 · p. 19

5.3 Pretrained contour-network refinement

• The circular residual U-Net corrects the fitted radius while keeping its center fixed, using eight input channels and a fixed Fourier projection for output corrections.Its inputs omit shape family, absolute position, and the observation vector; the architecture has about 1.3 × 10^6 parameters.Source: §5.3 · p. 20

5.3.1 Pretraining and interval-specific fine-tuning

• Training proceeds from synthetic-contour pretraining to fine-tuning on earlier physical geometries and then interval-specific training on current training inputs.Each interval-specific minibatch pairs source instances with synthetic contours; validation ASSD averaged across shape families selects parameters.Source: §5.3.1 · p. 20

5.4 Overall results

• At 0.1% noise, Fourier plus DNN refinement improves recovery of corners missed by Fourier fitting, especially in stars, while Fourier alone has lower mean Dice than the bounded reconstruction.At 1% noise, large classical ASSD is mainly attributed to empty thresholded supports under the stated convention.Source: §5.4 · p. 20
• At 0.1% and 1% noise, the fully refined reconstruction gives the best overall results, although later intervals and stars remain harder.The reported comparisons cover all 24 test intervals from eight complete test sequences.Source: Table 2 · p. 20, §5.4 · p. 20
• Evaluating the network adds negligible cost relative to the preceding refinement stages.Source: §5.4 · p. 20

A From boundary data to a first contour

• The contour pipeline starts from box-constrained L2 using 16 sensors and 125 history-subtracted times, then fits modes 2–16 before a periodic boundary correction.Source: A From boundary data to a first contour · p. 21

C Preserve the fitted center and area

• The refinement workflow uses synthetic pretraining and physical fine-tuning, with interval-specific fine-tuning on one current interval at a time.Source: C Preserve the fitted center and area · p. 21
• Boundary refinement leaves the classical history unchanged.Source: C Preserve the fitted center and area · p. 21
• The fixed Fourier projection is followed by a radial transformation that preserves the fitted center and Fourier-estimated area.The figure presents the first interval of one test sequence; orange contours show the truth for display only.Source: Figure 1 · p. 21
• Table 3 reports mean Dice and mean ASSD by interval and true shape family at 0.1% and 1% noise.Each interval has eight cases, and each family has six.Source: Table 3 · p. 22
• Figure 2 shows the four reconstruction stages for triangle, square, and star examples at 0.1% and 1% noise.The examples share coordinates and an intensity scale of [0, 1], with Dice and ASSD labels.Source: Figure 2 · p. 22

6 Conclusions

• For 0 < s < 1/2, source-side smoothing compactifies the nonclosable forward operator and yields finite-window uniqueness and Galerkin–Tikhonov convergence under consistent regularization and discretization choices.The forward operator is block lower triangular in time, enabling reconstruction by forward substitution.Source: §6 · p. 23
• Earlier reconstruction errors enter later-interval data with exponential decay in time.Source: §6 · p. 23
• The experiments use Fourier and neural boundary refinement after sequential Galerkin–Tikhonov reconstruction to reduce Hilbert-scale-penalty smoothing.Future work includes theoretical analysis of the box-constrained algorithm and neural-network penalty terms.Source: §6 · p. 23