Convex Integrands Research Articles

Bilevel Training Schemes in Imaging for Total Variation--Type Functionals with Convex Integrands

In the context of image processing, given a $k$-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the regularizers are integrals of spatially inhomogeneous integrands with convex dependence on the differential operator applied to the image function. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on $BV$. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularization parameters, thus allowing for good flexibility and preservation of details in the reconstructed image. We establish existence of optima for the scheme and we finally substantiate its feasibility by numerical examples in image denoising. The cases that we treat are Huber versions of the first and second order total variation with both the Huber and the regularization parameter being spatially dependent. Notably the spatially dependent version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the spatially dependent Huber parameter leads to a further enhancement of the image details.

Second-order conditions for non-uniformly convex integrands: quadratic growth in $L^1$

We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Applications to non-smooth optimal control problems are given.

Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD

In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV-ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBDp functions.

Extremal rank-one convex integrands and a conjecture of \u0160ver\xe1k

We show that, in order to decide whether a given probability measure is laminate, it is enough to verify Jensen’s inequality in the class of extremal non-negative rank-one convex integrands. We also identify a subclass of these extremal integrands, consisting of truncated minors, thus proving a conjecture made by Šverák (Arch Ration Mech Anal 119(4):293–300, 1992).

The subdifferential of measurable composite max integrands and smoothing approximation

The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential of the expectation equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about the calculation of Clarke subgradients for the expectation of non-regular integrands. The focus of this contribution is to approximate Clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand. A framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands. This class contains non-regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning.

Homogenization of networks in domains with oscillating boundaries

ABSTRACTWe consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a -limit with respect to an appropriate convergence, is defined in a ‘stratified’ Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for p-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively.

Strictly convex Wulff shapes and 𝐶¹ convex integrands

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C 1 C^1 . Moreover, applications of this result are given.

STABILITY OF <i>C</i><b><sup>∞ </sup></b>CONVEX INTEGRANDS

In this paper, it is shown that the set consisting of stable convex integrands $S^n\to \mathbb{R}_+$ is open and dense in the set consisting of $C^\infty$ convex integrands with respect to Whitney $C^\infty$ topology. Moreover, an application of the proof of this result is also shown.

Two-Scale Convergence of Integral Functionals with Convex, Periodic and Nonstandard Growth Integrands

We study the periodic homogenization for a family of functionals defined on Orlicz-Sobolev spaces. One fundamental in this topic is to extend the classical compactness results of the two-scale convergence method to this type of spaces.

Gamma–closure of some classes of nonstandard convex integrands

Within the framework of the theory of Γ-convergence for convex functionals with nonstandard coercivity and growth conditions, we study the inheritance of some properties (for example, strict convexity, differentiablility, and Δ2-property) of integrands under taking the Γ-closure. We focus on power integrands |ξ| p(x) with variable exponents. The results obtained are also of interest in the case of the Γ-convergence theory for standard functionals. Bibliography: 15 titles.

Automatic convexity of rank-1 convex functions

We announce new structural properties of 1-homogeneous rank-1 convex integrands, and discuss some of their consequences.

Deterministic homogenization of integral functionals with convex integrands

In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence.

Structure of approximate solutions of variational problems with extended-valued convex integrands

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand f : Rn ×Rn R 1 , where Rn is the n-dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

Structure of extremals of autonomous convex variational problems

In this paper we study the structure of approximate solutions of autonomous variational problems with convex integrands. We are interested in a turnpike property of the extremals which is independent of the length of the interval, for all sufficiently large intervals. To have this property means, roughly speaking, that the approximate solutions of the variational problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.

Minimal rearrangements, strict convexity and critical points

An asymmetry estimate for extremal functions in Pólya–Szegö type inequalities for Dirichlet integrals is established in presence of critical points and nonstrictly convex integrands.

Regularization of convex variational problems with applications to generalized Newtonian fluids

We study variational problems with convex integrands of very general structure by introducing certain regularizations leading to particular minimizers. In a second part we apply the method to stationary generalized Newtonian fluids which gives the existence of solutions under weak hypotheses on the dissipative potential. In particular the potential is not required to be a strictly convex function.

FACIAL STRUCTURE OF CONVEX SETS IN BANACH SPACES AND INTEGRAND REPRESENTATION OF CONVEX OPERATORS

Many types of convex operators which take values in some complete lattices can be represented by convex integrands. We consider a certain structure of faces of convex sets, and give a new proof of the representation theorem which is applicable in infinite-dimensional cases. As an application of such representations, we consider the conjugate duality of convex operators.

A Uniqueness Theorem for the Dual Problem Associated to a Variational Problem with Linear Growth

Uniqueness is proved for solutions of the dual problem that is associated with the minimum problem \(\smallint _\Omega f(\nabla u)dx \to {\text{ min}}\) among the mappings \(\mathbb{R}^n \supset \Omega \to \mathbb{R}^N \) with prescribed Dirichlet boundary data and for smooth strictly convex integrands f of linear growth. No further assumptions on f or its conjugate function \(f *\) are imposed, in particular, \(f *\) is not assumed to be strictly convex. A special solution of the dual problem is seen to be a mapping into the image of \(\nabla f\), which immediately implies uniqueness. Bibliography: 13 titles.

Optimal Solutions of Linear Periodic Control Systems with Convex Integrands

In this work we study the existence and asymptotic behavior of overtaking optimal trajectories for linear control systems with convex integrands. We extend the results obtained by Artstein and Leizarowitz for tracking periodic problems with quadratic integrands [2] and establish the existence and uniqueness of optimal trajectories on an infinite horizon. The asymptotic dynamics of finite time optimizers is examined.

Optimal Solutions of Linear Control Systems with Nonperiodic Convex Integrands

In this work we study the existence and asymptotic behaviour of overtaking optimal trajectories for a linear system with a nonperiodic convex cost function. We extend the results obtained by Leizarowitz (Leizarowitz, A. 1986. Tracking nonperiodic trajectories with the overtaking criterion. Appl. Math. Opt. 14 155–171.) for a problem of tracking a prescribed nonperiodic trajectory with a quadratic cost function and establish the existence and uniqueness of optimal trajectories on an infinite horizon. The asymptotic dynamics of finite time optimizers is examined.