On Convex Envelopes and Regularization of Non-Convex Functionals without moving Global Minima

We provide theory for the computation of convex envelopes of non-convex functionals including an l2-term, and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low rank recovery problems but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the l2-term contains a singular matrix we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional we necessarily have a direction in which the convex envelope is affine.Comment: arXiv admin note: text overlap with arXiv:1609.0937

Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single-Crystal Plasticity

This paper is concerned with the effective modeling of deformation microstructures within a concurrent multiscale computing framework. We present a rigorous formulation of concurrent multiscale computing based on relaxation; we establish the connection between concurrent multiscale computing and enhanced-strain elements; and we illustrate the approach in an important area of application, namely, single-crystal plasticity, for which the explicit relaxation of the problem is derived analytically. This example demonstrates the vast effect of microstructure formation on the macroscopic behavior of the sample, e.g., on the force/travel curve of a rigid indentor. Thus, whereas the unrelaxed model results in an overly stiff response, the relaxed model exhibits a proper limit load, as expected. Our numerical examples additionally illustrate that ad hoc element enhancements, e.g., based on polynomial, trigonometric, or similar representations, are unlikely to result in any significant relaxation in general

Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.Comment: 16 pages, 14 figures; Milan J. Math., (2012

A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection

We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian distribution. The highly flexible variational structure of the corresponding regulariser encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM). A suitable preliminary variable splitting together with a novel result in multivariate non-convex proximal calculus yield a very efficient minimisation algorithm. Several numerical results showing significant quality-improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular in terms of texture and detail preservation

Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature

Gradient flows as a selection procedure for equilibria of nonconvex energies

For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard–Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions $u(x)$ which are nonconvex in the gradient $\nabla u$ and possibly also in $u$. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.Comment: 34 pages, 10 figure

Existência de minimizantes para integrais n-dimensionais não-convexos

Primeiro demonstra-se a existência de minimizantes para o integral múltiplo ∫ Ω ℓ∗∗ ( u (x) , ρ1 (x, u(x))∇u (x) ) ρ2 (x, u(x)) d x on W 1;1 u@ (Ω) , onde Ω ⊂ Rd é aberto e limitado, u : Ω → R pertence ao espaço de Sobolev u@ (·) + W1;1 0 (Ω), u@ (·) ∈ W1;1 (Ω) ∩ C0 ( Ω ) ; ℓ : R×Rd → [0,∞] é superlinear L⊗B−mensurável, ρ1(·, ·), ρ2(·, ·) ∈ C0 (Ω×R) ambos > 0 e ℓ∗∗(·, ·) é apenas sci em (·, 0). Também se considera o caso ∫ Ω L∗∗ (x, u(x),∇u(x) ), embora com hipóteses mais complexas, mas é igualmente possível ter L(x, ·, ξ) não-sci para ξ ̸= 0; Por último demonstra-se a existência de minimizantes radialmente simétricos, i.e. uA(x) = UA ( |x| ), uniformemente contínous para o integral múltiplo ∫ BR L∗∗ ( u(x), |Du(x) | ) d x na bola BR ⊂ Rd, u : Ω → Rm pertence ao espaço de Sobolev A + W1;1 0 (BR, Rm ), L∗∗ : Rm×R → [0,∞] é convexa, sci e superlinear, L∗∗ ( S, · ) é par; note-se também que enquanto no caso escalar, m = 1, apenas necessitamos de mais uma hipótese : ∃ min L∗∗ (R, 0 ), no caso vectorial, m > 1, L∗∗ (·, ·) também tem de satisfazer uma restrição geométrica, a qual chamamos quasi − escalar; sendo o exemplo mais simples de uma função quasi − escalar o caso biradial L∗∗ ( | u(x) | , |Du(x) | ); ABSTRACT: First it is proved the existence of minimizers for the multiple integral ∫ Ω ℓ∗∗ ( u (x) , ρ1 (x, u(x))∇u (x) ) ρ2 (x, u(x)) d x on W 1;1 u@ (Ω) , where Ω ⊂ Rd is open bounded, u : Ω → R is in the Sobolev space u@ (·) + W1;1 0 (Ω), with boundary data u@ (·) ∈ W1;1 (Ω) ∩ C0 ( Ω ) ; and ℓ : R×Rd → [0,∞] is superlinear L⊗B − measurable with ρ1(·, ·), ρ2(·, ·) ∈ C0 (Ω×R) both > 0 and ℓ∗∗(∫ ·, ·) only has to be lsc at (·, 0). The case Ω L∗∗ (x, u(x),∇u(x) ) is also treated, though with less natural hypotheses, but still allowing L(x, ·, ξ) non − lsc for ξ ̸= 0; Lastly it is proved the existence of uniformly continuous radially symmetric minimizers uA(x) = UA ( |x| ) for the multiple integral ∫ BR L∗∗ ( u(x), |Du(x) | ) d x on a ball BR ⊂ Rd, among the vector Sobolev functions u(·) in A + W1;1 0 (BR, Rm ), using a convex lsc L∗∗ : Rm×R → [0,∞] with L∗∗ ( S, · ) even and superlinear; but while in the scalar m = 1 case we only need one more hypothesis : ∃ min L∗∗ (R, 0 ), in the vectorial m > 1 case L∗∗ (·, ·) also has to satisfy a geometric constraint, which we call quasi − scalar; the simplest example being the biradial case L∗∗ ( | u(x) | , |Du(x) | )

On global solvability of a class of possibly nonconvex QCQP problems in Hilbert spaces

We provide conditions ensuring that the KKT-type conditions characterizes the global optimality for quadratically constrained (possibly nonconvex) quadratic programming QCQP problems in Hilbert spaces. The key property is the convexity of a image-type set related to the functions appearing in the formulation of the problem. The proof of the main result relies on a generalized version of the (Jakubovich) S-Lemma in Hilbert spaces. As an application, we consider the class of QCQP problems with a special form of the quadratic terms of the constraints.Comment: arXiv admin note: text overlap with arXiv:2206.0061