Iterative Thresholding Meets Free-Discontinuity Problems

Iterative Thresholding Meets Free-Discontinuity Problems

Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function. Hence, the derivative of the solution is assumed to be a ‘small’ function almost everywhere except on sets where it concentrates as a singular measure. This is the case, for instance, in crack detection from fractu...

PArameter-free Generic Iterative Thresholding Algorithm (PAGITA)

The Calibration Method for Free Discontinuity Problems

The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.

Regular Minimizers of Some Free Discontinuity Problems

We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine several examples, including elastic-plastic beam and plate with free yield lines and deformable body with free damage. In all cases there is a gap between the condition for solvability (safe load...

Finite Difference Approximation of Free Discontinuity Problems

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers. AMS (MOS) subject classifications: 49J45, 65K10.