A variational mollification approach to circular deconvolution
Abstract
We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.
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BibTeX
@misc{karamehmedovic2026variational,
title = {{A variational mollification approach to circular deconvolution}},
author = {Mirza Karamehmedović and Pierre Maréchal and Mykhailo Petrov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17849},
eprint = {2609.17849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17849},
}
RIS
TY - UNPB
TI - A variational mollification approach to circular deconvolution
AU - Mirza Karamehmedović
AU - Pierre Maréchal
AU - Mykhailo Petrov
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17849
UR - https://arxiv.org/abs/2609.17849
AB - We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.
ER -