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We define a family of functionals, called p-oscillation functionals, that can be interpreted as discrete versions of the classical total variation functional for p=1 and of the p-Dirichlet functionals for p>1. We introduce the notion of…
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…
This paper is on $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the…
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: first, we prove that…
We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, \Gamma)=\int_{B_1\backslash\Gamma} |\nabla u|^2 dx + \lambda^2 \frac{\pi}{2} \mathcal{H}^1(\Gamma). $$ It is known that if the minimizer has a crack-tip in…
We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and…
The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may…
The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions.…
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce…
We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincar\'e…
We propose an alternative to $k$-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies…
In this paper we introduce two conceptual algorithms for minimising abstract convex functions. Both algorithms rely on solving a proximal-type subproblem with an abstract Bregman distance based proximal term. We prove their convergence when…
Discrete approximations to the equation \begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A'(x)+H(x)) u^{(1)} + B(x) u = f, \; x\in[0,1] \end{equation*} are considered. This is an extension of the Sturm-Liouville case…
We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected.…
This paper can be seen as an attempt of rethinking the {\em Extra-Gradient Philosophy} for solving Variational Inequality Problems. We show that the properly defined {\em Reduced Gradients} can be used instead for finding approximate…
We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.
A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is…
The article considers the discrete analogue of the method of quickest descent for an inverse Acoustics problem in case of a smooth source. The authors derived the gradient of functional in differential and discrete cases, described the…
We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form \[ \mathcal{F}[u] := \int_{\Omega} f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : \Omega…
This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review…