Two natural ways of modelling Formula 1 race outcomes are a probabilistic approach, based on the exponential distribution, and statistical regression modelling of the ranks. Both approaches lead to exactly soluble race-winning probabilities. Equating race-winning probabilities leads to a set of equivalent parametrisations. This time-rank duality is attractive theoretically and leads to new ways of dis-entangling driver and car level effects as well and a simplified Monte Carlo simulation algorithm. Results are illustrated by applications to the 2022 and 2023 Formula 1 seasons.

We propose a fully-corrective generalized conditional gradient method (FC-GCG) 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 extremal points of the unit ball of the regularizer and of 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 FC-GCG 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 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.

We study measure-valued solutions of the inhomogeneous continuity equation
\( \partial_t \rho_t + div (v \rho_t) = g \rho_t \,, \)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 Hellinge–Kantorovich energy is finite. This principle gives a decomposition of the solution into curves \(t \mapsto h(t) δ\)_{\(γ(t) \)} that satisfy the characteristic system \( γ’ (t)=v(t,γ(t)) \), \(h’ (t)=g(t,γ(t))h(t) \)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–Kantorovich-type regularizers are characterized. Such regularizers arise, for example, in the context of dynamic inverse problems and dynamic optimal transport.

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.

In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy \(Γ\)-converges to a limit functional comprised of two contributions: one is given by a constant \(c_∞> 0 \,\)gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that \(c_∞\,\)is reached when dislocations are evenly-spaced on the one dimensional circle.

In this paper we propose and study a novel optimal transport based regularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that (i) the measured data takes values in a time dependent family of Hilbert spaces, and (ii) the forward operators are time dependent and map, for each time, Radon measures into the corresponding data space. The variational regularization we propose is based on dynamic (un-)balanced optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation, i.e., the Radon measure at time \(t \)is advected by a velocity field \(v \)and varies with a growth rate \(g \), and (ii) are penalized with the kinetic energy induced by \(v \)and a growth energy induced by \(g \). We establish a functional-analytic framework for these regularized inverse problems, prove that minimizers exist and are unique in some cases, and study regularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modelling relevant examples of time varying acquisition strategies, as well as patient motion and presence of contrast agents.

In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimization. For this purpose, we consider a well-known variational model for two-dimensional systems of edge dislocations, within the so-called core radius approach, and we derive the \(Γ\)-limit of the elastic energy functional as the lattice space tends to zero. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.

We study the higher gradient integrability of distributional solutions \(u \)to the equation \(div (σ∇u)=0 \)in dimension two, in the case when the essential range of \(σ\,\)consists of
only two elliptic matrices, i.e., \(σ∈\) { \( \sigma_1, \sigma_2 \) } a.e. in \(Ω\). In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices \(\sigma_1 \)and \(\sigma_2 \),
exponents
\(p = p\)_{\(\sigma_1, \sigma_2 \)} \( ∈(0,+∞) \)and
\(q = q\)_{\(\sigma_1, \sigma_2 \)} \(∈(1,2) \)have been found so that if
\(u ∈W\)^{\(1,q\)} \( (Ω) \)is a solution to the elliptic equation then
\(∇u ∈L^p(Ω,weak) \)and the optimality of the upper exponent \(p \)has been proved. In this paper we complement the above result by proving the optimality of the lower exponent \(q \). Precisely, we show that for every arbitrarily small \(δ\), one can find a particular microgeometry, i.e., an arrangement of the sets
\(σ\)^{\(-1 \)} \( (\sigma_1) \)and
\(σ\)^{\(-1 \)} \( (\sigma_2) \), for which there exists a solution \(u \)to the corresponding elliptic equation such that
\( ∇u ∈L\)^{\(q-δ\)}
but \( ∇u ∉L^q \). The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.

We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases.

The main focus of this PhD thesis is the study of microstructures and geometric patterns in materials, in the framework of the Calculus of Variations. My PhD research, carried out in collaboration with my supervisor Mariapia Palombaro and Marcello Ponsiglione, led to the production of three papers [21, 22, 23]. Papers [21, 22] have already been published, while [23] is currently in preparation.
This thesis is divided into two main parts. In the first part we present the results obtained in [22, 23]. In these two works geometric patterns have to be understood as patterns of dislocations in crystals. The second part is devoted to [21], where suitable microgeometries are needed as a mean to produce gradients that display critical integrability properties.

In this study we present an approach that combines sub-sampled encoding reconstruction and simultaneous object motion computation. For that purpose Optimal Transport is used as convex regularization for motion-afflicted measurements. It reconstructs explicit pixel-wise motion fields simultaneously to the image series. Results based on simulated data show that 8-frame image series can be reconstructed in great detail from 4-fold undersampled k-space series data from a single coil. The high potential of the presented method could be shown for the reconstruction of undersampled image series. For the recovery of the motion fields, further improvements are still necessary.