A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization

TL;DR

This paper introduces a dynamic generalized conditional gradient method for inverse problems with optimal transport regularization, enabling efficient reconstruction of undersampled, noisy dynamic data by iteratively adding measures on curves.

Contribution

It extends the conditional gradient method to a dynamic setting with optimal transport regularization, providing convergence guarantees and practical heuristics.

Findings

Method effectively reconstructs heavily undersampled dynamic data.

Algorithm demonstrates convergence with sublinear rate.

Numerical examples show robustness to noise and undersampling.

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., arXiv:1907.11589, 2019) 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 in iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimize 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 at reconstructing heavily undersampled dynamic data, together with the presence of noise.