Regularized Sequential Reconstructions of Dynamic Nonsmooth Sources from Sparse Boundary Flux Data
Abstract
We consider an inverse source problem for the heat equation on the unit disk Ω⊂ℝ2 with flux measurements at finitely many boundary points. The source is piecewise constant in time on prescribed intervals and, on each interval, is the characteristic function of an unknown region with Lipschitz boundary. The measurements on each interval split into the contribution of the current source and a history term from earlier sources; subtracting the latter leaves a linear inverse problem for a forward operator Fτ that is densely defined and unbounded on L2(Ω). We prove that Fτ is not closable, so the classical regularization theory for closed unbounded operators does not apply. We regularize instead with a Hilbert-scale penalty that compactifies it. The resulting Galerkin--Tikhonov reconstruction is sequential: each source is recovered from the flux observed up to the end of its interval and from the previously reconstructed sources. We prove its convergence and uniqueness, show that earlier reconstruction errors propagate into later data with exponential decay in time, and establish stability estimates with respect to the number of sensors. A numerical experiment on noisy measurements shows the feasibility of the sequential approximations, refined by Fourier approximation and boundary correction by a neural network.
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BibTeX
@misc{bianchi2026regularized,
title = {{Regularized Sequential Reconstructions of Dynamic Nonsmooth Sources from Sparse Boundary Flux Data}},
author = {Davide Bianchi and Tianrui Dai and Hongqin Zhang and Shupeng Zhang and Zhidong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35173},
eprint = {2610.35173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35173},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularized Sequential Reconstructions of Dynamic Nonsmooth Sources from Sparse Boundary Flux Data
AU - Davide Bianchi
AU - Tianrui Dai
AU - Hongqin Zhang
AU - Shupeng Zhang
AU - Zhidong Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35173
UR - https://arxiv.org/abs/2610.35173
AB - We consider an inverse source problem for the heat equation on the unit disk Ω⊂ℝ^2 with flux measurements at finitely many boundary points. The source is piecewise constant in time on prescribed intervals and, on each interval, is the characteristic function of an unknown region with Lipschitz boundary. The measurements on each interval split into the contribution of the current source and a history term from earlier sources; subtracting the latter leaves a linear inverse problem for a forward operator F_τ that is densely defined and unbounded on L^2(Ω). We prove that F_τ is not closable, so the classical regularization theory for closed unbounded operators does not apply. We regularize instead with a Hilbert-scale penalty that compactifies it. The resulting Galerkin--Tikhonov reconstruction is sequential: each source is recovered from the flux observed up to the end of its interval and from the previously reconstructed sources. We prove its convergence and uniqueness, show that earlier reconstruction errors propagate into later data with exponential decay in time, and establish stability estimates with respect to the number of sensors. A numerical experiment on noisy measurements shows the feasibility of the sequential approximations, refined by Fourier approximation and boundary correction by a neural network.
ER -