We consider a variational model for image segmentation proposed in Sandberg et al. (2010) [12]. In such a model the image domain is partitioned into a finite collection of subsets denoted as phases. The segmentation is unsupervised, i.e., the model finds automatically an optimal number of phases, which are not required to be connected subsets. Unsupervised segmentation is obtained by minimizing a functional of the Mumford-Shah type (Mumford and Shah, 1989 [1]), but modifying the geometric part of the Mumford-Shah energy with the introduction of a suitable scale term.

It is known that symplectic algorithms do not necessarily conserve energy even for the harmonic oscillator. However, for separable Hamiltonian systems, splitting and composition schemes have the advantage to be explicit and can be constructed to preserve energy. In this paper we describe and test an integrator built on a one-parameter family of symplectic symmetric splitting methods, where the parameter is chosen at each time step so as to minimize the energy error.

We consider the regularization of linear inverse problems by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the L 2 distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator.

In the fracture model presented in this paper, the basic assumption is that the energy is the sum of two terms, one elastic and one cohesive, depending on the elastic and inelastic part of the deformation, respectively. Two variants are examined: a local model, and a nonlocal model obtained by adding a gradient term to the cohesive energy.

We extend the analysis of Chiba et al. [Phys. Rev. D 75, 124014 (2007)] of Solar System constraints on f(R) gravity to a class of nonminimally coupled (NMC) theories of gravity. These generalize f(R) theories by replacing the action functional of general relativity with a more general form involving two functions f1(R) and f2(R) of the Ricci scalar curvature R. While the function f1(R) is a nonlinear term in the action, analogous to f(R) gravity, the function f2(R) yields a NMC between the matter Lagrangian density Lm and the scalar curvature.