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 certain digital image segmentation
problems and brittle fracture models.
In the first part of this talk we show new preliminary results on
the existence of minimizers for inverse free-discontinuity problems, by
restricting the solutions to a class of functions with piecewise Lipschitz
discontinuity set.
If we discretize such situations for numerical purposes, the inverse
free-discontinuity problem in the discrete setting can be re-formulated as
that of finding a derivative vector with small components at all but a few
entries that exceed a certain threshold. This problem is similar to those
encountered in the field of "sparse recovery", where vectors
with a small number of dominating components in absolute value are
recovered from a few given linear measurements via the minimization of
related energy functionals.
As a second result, we show that the computation of global minimizers in
the discrete setting is an NP-hard problem.
With the aim of formulating efficient computational approaches in such
a complicated situation, we address iterative thresholding algorithms that
intertwine gradient-type iterations with thresholding steps which were
designed to recover sparse solutions.
It is natural to wonder how such algorithms can be used towards solving
discrete free-discontinuity problems. This talk explores also this
connection, and, by establishing an iterative thresholding algorithm for
discrete inverse free-discontinuity problems, provides new insights on
properties of minimizing solutions thereof.