- Optimization for imaging:
- Together with Thomas Pock (TU. Graz) we have written recently a short review entitled "An introduction to continuous optimization for imaging", published in Acta Numerica, vol 25.
- With Luca Calatroni (CMAP), we have investigated backtracking strategies for first order forward-backward descent, in strongly convex cases (with the possibility to increase the time steps.) Backtracking strategies for accelerated descent methods with smooth composite objectives.
- With Matthias Ehrhardt (DAMTP/CIA/Cambridge), Peter Richtarik (KAUST), Carola-Bibiane Schönlieb (DAMTP/CIA) we have investigated the efficiency of a "Stochastic Primal-Dual Hybrid Gradient Algorithm With Arbitrary Sampling And Imaging Applications" which yields a very good convergence rate.

- Other imaging applications and variational problems:
- with Thomas Pock, we have just finished a preprint on a "Total Roto-translational variation" which extends to arbitrary images curvature dependent functionals of level sets such as the Elastica, initially proposed for "inpainting" applications.
- With Benoit Merlet (U. Lille) and Luca Ferrari (CMAP),
we are studying the phase-field approximation of Steiner
type (or mass transportation)
problems. A first result in 2D is found in
A phase-field approximation of the Steiner problem in dimension two
. The general case is treated in Variational approximation of size-mass energies for
*k*-dimensional currents. - With Vincent Duval (Telecom), Gabriel Peyré (ENS) and Clarice Poon (U. Cambridge), we have studied some geometric properties of the total variation denoising, for simple functions with small noise, in Total Variation Denoising and Support Localization of the Gradient
- In Regularity for the Optimal Compliance Problem with Length Penalization we have described, with Jimmy Lamboley, Antoine Lemenant and Eugene Stepanov, the structure of 1D compact connected sets minimizing some "compliance" energy.

- Analysis of Brittle Fracture.
- With Sergio Conti (U. Bonn) and Gilles Francfort (U. Paris-Nord),
we have proved a Korn-Poincaré inequality for
*SBD*(also,*GSBD*functions) which show that if their jump set is small enough, then they satisfy a classical Korn-Poincaré inequality, up to a small set, see "Korn-Poincare inequalities for functions with a small jump set" (Indiana U.~Math Journal). As an application, we are able to show the Gamma-convergence of fracture models with non-interpenetration, in 2D, cf Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration. - With Vito Crismale (CMAP), we have studied extensively
the "Griffith problem" for the modeling of fracture growth and
have shown existence of solutions, including strong solutions with
a Dirichlet boundary conditions, in two papers
Compactness and lower-semicontinuity in
*GSBD*(J. Eur. Math. Soc, to appear) and Existence of strong solutions to the Dirichlet problem for the Griffith energy (Cal. Var PDE). - Together with Flaviana Iurlano (U. Paris 6) and Sergio Conti, we have succeeded in proving the "GSBD" (brittle fracture / Griffith's energy) variant of De Giorgi, Carriero and Leaci's result on the Mumford-Shah functional, see Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy (JMPA).

- With Sergio Conti (U. Bonn) and Gilles Francfort (U. Paris-Nord),
we have proved a Korn-Poincaré inequality for
- Nonlocal and Crystalline curvature flows:
- We could prove with Massimiliano Morini (Parma) and
Marcello Ponsiglione (Pisa) the existence of crystalline
curvature flows in any dimension. The paper
is Existence and Uniqueness for a Crystalline Mean Curvature Flow.
With Matteo Novaga, we have considerably extended this
result, showing that one can consider quite arbitrary
mobilities, and showing the convergence for
*any*anisotropy of the approximation scheme of Almgren, Taylor, Wang (1993). The paper is entitled Existence and uniqueness for anisotropic and crystalline mean curvature flows and will appear in the Journal of the American Math. Society. - Before this, still with Massimiliano Morini and Marcello Ponsiglione, we had studied in a quite general setting (nonlocal) curvature flows, in particular which arise from quite general (but translational invariant) nonlocal perimeters.

- We could prove with Massimiliano Morini (Parma) and
Marcello Ponsiglione (Pisa) the existence of crystalline
curvature flows in any dimension. The paper
is Existence and Uniqueness for a Crystalline Mean Curvature Flow.
With Matteo Novaga, we have considerably extended this
result, showing that one can consider quite arbitrary
mobilities, and showing the convergence for

Quite