Related papers: A quantitative variational analysis of the stairca…

We investigate the asymptotic behavior of minimizers for the singularly perturbed Perona-Malik functional in one dimension. In a previous study, we have shown that blow-ups of these minimizers at a suitable scale converge to staircase-like…

We consider the one-dimensional Perona-Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term.…

This paper deals with the so-called staircasing phenomenon, which frequently arises in total variation based denoising models in image analysis. We prove in particular that staircasing always occurs at global extrema of the datum and at all…

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…

We study a functional defined on the class of piecewise constant functions, combining a jump penalization, which discourages discontinuities, with a fidelity term that penalizes deviations from a given linear function, called the forcing…

For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call…

We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in dimension one. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves…

The Gamma-limit of higher-order singular perturbations of the Perona-Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a k-th order regularization designed to balance bulk and…

This paper provides a set of sensitivity analysis and activity identification results for a class of convex functions with a strong geometric structure, that we coined "mirror-stratifiable". These functions are such that there is a…

We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based…

We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$…

Stochastic methods for minimizing a convex integral functional, as initiated by Robbins and Monro in the early 1950s, rely on the evaluation of a gradient (or subgradient if the function is not smooth) and moving in the corresponding…

We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral…

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of…

We study a one dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio-Baradat-Brenier, of the discrete Monge-Amp\`ere gravitational model, which describes the motion of interacting…

We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_\Omega f(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation.

We present an asymptotic description of local minimization problems, and of quasi-static and dynamic evolutions of scaled Perona-Malik functionals. The scaling is chosen such that these energies $\Gamma$-converge to the Mumford-Shah…

We investigate regularity of minimizers in two dimensions for certain classes of non-smooth convex functionals. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of…

This paper is devoted to second-order variational analysis of a rather broad class of extended-real-valued piecewise liner functions and their applications to various issues of optimization and stability. Based on our recent explicit…

This paper establishes a strict mathematical relationship between an arbitrary continuous function on a compact set and its global minima, like the well-known first order optimality condition for convex and differentiable functions. By…