All my dissemination activity divided in
denotes invited contribution
Oral Presentations
2023

AIP 2023: 11th Applied Inverse Problems Conference
University of Göttingen, Germany, 48 Sep 2023
Sparse optimization Algorithms for Dynamic Imaging
In this talk we introduce a FrankWolfetype algorithm for sparse optimization in Banach spaces. The functional we want to optimize consist of the sum of a smooth fidelity term and of a convex
onehomogeneous regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. We apply this algorithm to the problem of particles tracking from heavily undersampled MRI data. This talk is based on the works cited below.
[1] K. Bredies, M. Carioni, S. Fanzon, D. Walter. Asymptotic linear convergence of FullyCorrective Generalized Conditional Gradient methods. Mathematical Programming, 2023
[2] K. Bredies, S. Fanzon. An optimal transport approach for solving dynamic inverse problems in spaces of measures. ESAIM:M2AN, 54(6): 23512382, 2020
[3] K. Bredies, M. Carioni, S. Fanzon, F. Romero. A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Found Comput Math, 2022
[4] K. Bredies, M. Carioni, S. Fanzon. On the extremal points of the ball of the Benamou–Brenier energy. Bull. London Math. Soc., 53: 14361452, 2021
[5] K. Bredies, M. Carioni, S. Fanzon. A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovichregular coefficients. Communications in Partial Differential Equations, 47(10): 20232069, 2022
2022

Sussex Mathematics Seminar
University of Sussex, UK, 3 Nov 2022
Sparsity and convergence analysis of generalized conditional gradient methods
In this talk we introduce suitable generalized conditional gradient algorithms for solving variational inverse problems
in Banach spaces. Specifically, the functionals considered consist of the sum of a smooth fidelity term and a convex
coercive regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. Finally, we give concrete applications, as for example the solution of dynamic inverse problems regularized with optimal transport energies, which covers the case of dynamic MRI reconstruction.

Seminar, Department of Mathematics
HeriotWatt University, UK, 13 Sep 2022
Sparsity and convergence analysis of generalized conditional gradient methods
In this talk we introduce suitable generalized conditional gradient algorithms for solving variational inverse problems
in Banach spaces. Specifically, the functionals considered consist of the sum of a smooth fidelity term and a convex
coercive regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. Finally, we give concrete applications, as for example the solution of dynamic inverse problems regularized with optimal transport energies, which covers the case of dynamic MRI reconstruction.

Seminar, Department of Mathematics & Scientific Computing
University of Graz, Austria, 18 Feb 2022
Sparsity and convergence analysis of Generalized Conditional Gradient Methods
In this talk we introduce suitable generalized conditional gradient algorithms for solving variational inverse problems consisting of a smooth fidelity term and a convex coercive regularizer. We exploit the sparse structure of the variational problem by designing iterates as suitable linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a linear convergence rate. Finally, we apply our algorithm to solve dynamic inverse problems regularized with optimal transport energies. This will cover the case of dynamic MRI reconstruction.
2021

SIMAI 20202021 Parma
University of Parma, Italy, 30 Aug  3 Sep 2021
Uniform distribution of dislocations at semicoherent interfaces
We will introduce variational models for edge dislocations at semicoherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we prove that, in the large interface limit, the elastic energy \(Γ\)converges to a limit functional comprised of two contributions: one is given by a constant 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 semicoherent to coherent interfaces, we show that the optimal configuration consists in evenlyspaced dislocations on the one dimensional circle. This is joint work with M. Ponsiglione and R. Scala
2019

M.A.G.A. Days (MongeAmpère et Géométrie Algorithmique)
Laboratoire de mathematiques d’Orsay, France, 2021 Nov 2019
Optimal transport regularization for dynamic inverse problems
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 bases on dynamic optimal transport. We apply this abstract framework to variational reconstruction in dynamic undersampled MRI. Further we will present some ideas on conditional gradient methods for sparse reconstruction. This is joint work with Kristian Bredies, Marcello Carioni and Francisco Romero

1st Austrian Calculus of Variations Day
University of Vienna, Austria, 1718 Oct 2019
Optimal transport regularization for dynamic inverse problems
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 bases on dynamic optimal transport. We apply this abstract framework to variational reconstruction in dynamic undersampled MRI. This is joint work with Kristian Bredies, Marcello Carioni and Francisco Romero

ICCOPT: 6th International Conference on Continuous Optimization
Technical University Berlin, Germany, 38 Aug 2019
Optimal transport regularization for dynamic inverse problems
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 bases on dynamic optimal transport. We apply this abstract framework to variational reconstruction in dynamic undersampled MRI. This is joint work with Kristian Bredies, Marcello Carioni and Francisco Romero
2018

Topics in Nonlinear Analysis: Calculus of Variations and PDEs
University of Lisbon, Portugal, 1012 Oct 2018
Optimal lower exponent of solutions to twophase elliptic equations in 2D
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
\( ∇u ∈L\)^{\(qδ\)}
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.

Seminar, Department of Mathematics & Scientific Computing
University of Graz, Austria, 31 Jan 2018
Linearised polycrystals from a 2D system of edge dislocations
In this talk I will present the results obtained in a recent paper in collaboration with Mariapia Palombaro and Marcello Ponsiglione.
The aim of this paper is to describe polycrystalline structures from the variational point of view. Grain boundaries and the corresponding grain orientations are not introduced as internal variables of the energy, but they spontaneously arise as a result of energy minimisation, under suitable boundary conditions.
We work under the hypothesis of linear planar elasticity, with the reference configuration \(Ω⊂R^2 \)representing a section of an infinite cylindrical crystal. The elastic energy functional \(E_\varepsilon \)depends on the lattice spacing \(\varepsilon \,\)of the crystal and we allow \(N_\varepsilon \)edge dislocations in the reference configuration, with \(N_\varepsilon \to ∞ \)as \(\varepsilon \to 0\). Each dislocation contributes by a factor \(  \log \varepsilon  \)to the elastic energy, so that the natural rescaling for the energy functional is \(N_\varepsilon  \log \varepsilon  \). We work in the energy regime
\[
N_\varepsilon ≫\log \varepsilon .
\]We will see that this energy regime will account for polycrystals containing grains that are mutually rotated by an infinitesimal angle \(θ≈0\).
Specifically, we show that the energy functional \(E_\varepsilon\), rescaled by \(N_\varepsilon \log \varepsilon\), \(Γ\)converges as \(\varepsilon \to 0 \)to a certain functional \(F\), whose dependence on the elastic and plastic parts of the macroscopic strain is decoupled.
Imposing piecewise constant Dirichlet boundary conditions on the plastic part of the limit strain, we then show that \(F \)is minimised by strains that are locally constant and take values into the set of antisymmetric matrices. We call these strains linearised polycrystals.
2017

XXVII National meeting of Calculus of Variations
Levico Terme, Italy, 610 Feb 2017
A variational model for dislocations at semicoherent interfaces
We propose and analyze a simple variational model for dislocations at semicoherent 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 semidiscrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the farfield strain vanishes as the interface size increases.
2016

Working Seminar on Calculus of Variations
Sapienza University, Italy, 19 Dec 2016
Variational models for semicoherent interfaces
We propose and analyze a simple variational model for dislocations at semicoherent 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 semidiscrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the farfield strain vanishes as the interface size increases.
Poster Presentations
2021

ITN TraDeOPT Winter School
Online, 1519 Feb 2021
Optimal transport regularization for dynamic inverse problems
2016

Hysteresis, Avalanches and Interfaces in Solid Phase Transformations
University of Oxford, UK, 1921 Sep 2016
A variational model for dislocations at semicoherent interfaces
We propose and analyze a simple variational model for dislocations at semicoherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations that compensate the lattice misfit at the interface, and a far field elastic energy, spent to decrease the amount of needed dislocations. We prove that the former scales like the surface area of the interface, the latter like its diameter.
The proposed continuum model is deduced from some heuristic derivation from the semidiscrete 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.

PIRECNA. New Frontiers in Nonlinear Analysis for Materials
Carnegie Mellon University, US, 210 Jun 2016
A variational model for dislocations at semicoherent interfaces
We propose and analyze a simple variational model for dislocations at semicoherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations that compensate the lattice misfit at the interface, and a far field elastic energy, spent to decrease the amount of needed dislocations. We prove that the former scales like the surface area of the interface, the latter like its diameter.
The proposed continuum model is deduced from some heuristic derivation from the semidiscrete 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.