In this paper we characterize sparse solutions for variational problems of the form min u∈X φ(u) + F (Au), where X is a locally convex space, A is a linear continuous operator that maps into a finite dimensional Hilbert space and φ is a seminorm. More precisely, we prove that there exists a minimizer that is "sparse" in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer φ (possibly translated by an element in the null space of φ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in [31] under weaker hypotheses.

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, 54:2351M2AN, 54: -2382M2AN, 54: , 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.

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In this paper, we characterize the extremal points of the unit ball of the Benamou-Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite-dimensional data and optimal-transport based regularization.

We propose an accelerated generalized conditional gradient method (AGCG) for the minimization of the sum of a smooth, convex loss function and a convex one-homogeneous regularizer over a Banach space. The algorithm relies on the mutual update of a finite set A k of extreme points of the unit ball of the regularizer and an iterate u k ∈ cone(A k ). Each iteration requires the solution of one linear problem to update A k and of one finite dimensional convex minimization problem to update the iterate. Under standard hypotheses on the minimization problem we show that the algorithm converges sublinearly to a solution. Subsequently, imposing additional assumptions on the associated dual variables, this is improved to a linear rate of convergence. The proof of both results relies on two key observations: First, we prove the equivalence of the considered problem to the minimization of a lifted functional over a particular space of Radon measures using Choquet's theorem. Second, the AGCG algorithm is connected to a Primal-Dual-Active-point Method (PDAP) on the lifted problem for which we finally derive the desired convergence rates.

We study measure-valued solutions of the inhomogeneous continuity equation ∂ t ρ t + div(vρ t ) = gρ t where the coefficients v and g are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger-Kantorovich energy is finite. This principle gives a decomposition of the solution into curves t → h(t)δ γ(t) that satisfy the characteristic system γ(t) = v(t, γ(t)), ḣ(t) = g(t, γ(t))h(t) in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of g where characteristics are not unique with respect to h. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger-Kantorovichtype regularizers are characterized. Such regularizers arise, e.g., in the context of dynamic inverse problems and dynamic optimal transport.

In this paper we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

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