1) Study of numerical regularization methods for the solution of
discrete ill-posed problems, deriving, for example, from the
discretization of a Fredholm integral equation of the first kind,
from linear and non linear control problems.
The discretization of a continuous ill posed problem is a difficult problem to be solved numerically, expecially when te data are affected by errors.
The study of regularization methods for the solution of large size ill-posed problems is a growing research area of numerical analysis. The regularization problem is solved by minimizing a functional (linear or nonlinear, constrained or unconstrained) containing a regularization term penalized by a regularization parameter and by using the semiconvergence property of the descent methods for the minimization of the fitting functional (usually not regularized). Particularly important is the study of and efficient and stable choice of the regularization parameter, that still is an open problem.
2) Study of numerical methods for the minimization of unconstrained
and constrained functionals.
We study methods for the minimization of nonconvex regularization functions, in the Compressive Sampling (CS) theory. These methods are applied in different areas, such as image reconstruction in medicie, astronomy, biology. The methods are implemented and tested on simulated and real data given by researchers working in the specific areas.
3) Reconstruction of blurred and noisy images in medicine,
astronomy, biology and of tomographic images from limited data.