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A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

Bredies, Kristian; Carioni, Marcello; Fanzon, Silvio; Romero, Francisco

Authors

Kristian Bredies

Marcello Carioni

Francisco Romero



Abstract

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.

Citation

Bredies, K., Carioni, M., Fanzon, S., & Romero, F. (2022). A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Foundations of Computational Mathematics, https://doi.org/10.1007/s10208-022-09561-z

Journal Article Type Article
Acceptance Date Feb 9, 2022
Online Publication Date Mar 30, 2022
Publication Date 2022
Deposit Date May 9, 2023
Publicly Available Date May 12, 2023
Journal Foundations of Computational Mathematics
Print ISSN 1615-3375
Electronic ISSN 1615-3383
Publisher Springer
Peer Reviewed Peer Reviewed
DOI https://doi.org/10.1007/s10208-022-09561-z
Keywords Conditional gradient method; Dynamic inverse problems; Benamou–Brenier energy; Optimal transport regularization; Continuity equation
Public URL https://hull-repository.worktribe.com/output/4270995

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Publisher Licence URL
http://creativecommons.org/licenses/by/4.0

Copyright Statement
© The Author(s) 2022.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.




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