Talks

Oral Presentations

2023

1. AIP

   AIP 2023: 11th Applied Inverse Problems Conference  

   University of Göttingen, Germany, 4-8 Sep 2023

   Sparse optimization Algorithms for Dynamic Imaging

   Event Abs Slides

   In this talk we introduce a Frank-Wolfe-type 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 one-homogeneous 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 Fully-Corrective 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): 2351-2382, 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: 1436-1452, 2021
   [5] K. Bredies, M. Carioni, S. Fanzon. A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients. Communications in Partial Differential Equations, 47(10): 2023-2069, 2022

2022

1. Sussex

   Sussex Mathematics Seminar  

   University of Sussex, UK, 3 Nov 2022

   Sparsity and convergence analysis of generalized conditional gradient methods

   Event Abs Slides

   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.

2. Heriot-Watt

   Seminar, Department of Mathematics  

   Heriot-Watt University, UK, 13 Sep 2022

   Sparsity and convergence analysis of generalized conditional gradient methods

   Venue Abs Slides

   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.

3. Uni Graz

   Seminar, Department of Mathematics & Scientific Computing  

   University of Graz, Austria, 18 Feb 2022

   Sparsity and convergence analysis of Generalized Conditional Gradient Methods

   Venue Abs Slides

   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

1. SIMAI

   SIMAI 2020-2021 Parma  

   University of Parma, Italy, 30 Aug - 3 Sep 2021

   Uniform distribution of dislocations at semi-coherent interfaces

   Event Abs Slides

   We will introduce variational models for edge dislocations at semi-coherent 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 semi-coherent to coherent interfaces, we show that the optimal configuration consists in evenly-spaced dislocations on the one dimensional circle. This is joint work with M. Ponsiglione and R. Scala

2019

1. Orsay

   M.A.G.A. Days (Monge-Ampère et Géométrie Algorithmique)  

   Laboratoire de mathematiques d’Orsay, France, 20-21 Nov 2019

   Optimal transport regularization for dynamic inverse problems

   Event Abs Slides

   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

2. Uni Wien

   1st Austrian Calculus of Variations Day  

   University of Vienna, Austria, 17-18 Oct 2019

   Optimal transport regularization for dynamic inverse problems

   Event Abs Slides

   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

3. ICCOPT

   ICCOPT: 6th International Conference on Continuous Optimization  

   Technical University Berlin, Germany, 3-8 Aug 2019

   Optimal transport regularization for dynamic inverse problems

   Event Abs Slides

   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

1. ULisboa

   Topics in Nonlinear Analysis: Calculus of Variations and PDEs  

   University of Lisbon, Portugal, 10-12 Oct 2018

   Optimal lower exponent of solutions to two-phase elliptic equations in 2D

   Event Abs Slides

   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.

2. Uni Graz

   Seminar, Department of Mathematics & Scientific Computing  

   University of Graz, Austria, 31 Jan 2018

   Linearised polycrystals from a 2D system of edge dislocations

   Venue Abs Slides

   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

1. CIRM

   XXVII National meeting of Calculus of Variations

   Levico Terme, Italy, 6-10 Feb 2017

   A variational model for dislocations at semi-coherent interfaces

   Event Abs Slides

   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.

2016

1. Sapienza

   Working Seminar on Calculus of Variations  

   Sapienza University, Italy, 19 Dec 2016

   Variational models for semi-coherent interfaces

   Venue Abs

   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.

Poster Presentations

2021

1. TraDE-OPT

   ITN TraDe-OPT Winter School  

   Online, 15-19 Feb 2021

   Optimal transport regularization for dynamic inverse problems

   Event Poster Slides

2016

1. Oxford

   Hysteresis, Avalanches and Interfaces in Solid Phase Transformations

   University of Oxford, UK, 19-21 Sep 2016

   A variational model for dislocations at semi-coherent interfaces

   Event Abs Poster

   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 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 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.

2. CMU

   PIRE-CNA. New Frontiers in Nonlinear Analysis for Materials

   Carnegie Mellon University, US, 2-10 Jun 2016

   A variational model for dislocations at semi-coherent interfaces

   Event Abs Poster Slides

   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 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 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.