Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint
Abstract
This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of ℝd, d≥ 1. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.
Related
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
- Well--posedness for the continuum Calogero--Moser equations by parabolic regularization
- On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
- Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint
- Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
BibTeX
@misc{morris2026wellposedness,
title = {{Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint}},
author = {Jason R. Morris and Sedar Ngoma},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28171},
eprint = {2608.28171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28171},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint
AU - Jason R. Morris
AU - Sedar Ngoma
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28171
UR - https://arxiv.org/abs/2608.28171
AB - This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of ℝ^d, d≥ 1. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.
ER -