  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.
