Keywords: image processing computer algebra geometric integration image processing data approximation bio-statistics bio-engineering of tissues

- Parallel direct solvers, for sparse structured linear systems
- Iterative solvers for linear programming
- Parallel iterative solvers, for big dimension, ill conditioned linear systems
- Iterative solvers for big dimension non linear systems, arising
in microwave circuits simulation

Numerical Integration and Approximation and Symbolic Calculus

- Geometric integration of ordinary differential equations

- Elementary Differential Runge-Kutta methods
- Automatic analysis of algorithms stability
- Data approximation and geometric modelling, in Medicine and Astronomy

- In the area of geometric integration, the initial interest has been in the integration of Hamiltonian systems by means of symplectic integrators, derived from the so-called 'generating functions'; more generally, the attention is focused on the solution of ordinary differential equations with Runge-Kutta methods (RK).

The computer algebra environment, in which it is faced solving such systems of equations, is that of Mathematica, within which many functionalities have been built for the analysis and derivation of numerical solvers, also in the case of stiff systems.

The aim has been and still is that of obtaining in an efficient way the automatic generation of high order numerical methods, that can keep certain stability characteristics.

The construction of a uniform framework, modular and hierarchical, facilitates the development of techniques for integrating equations and systems that possess particular properties, such as time reversibility. Such methods are known as 'geometric integrators'.

Orthogonal projector methods, in particular, have been developed for differential systems, the solution of which keep orthonormality, as well as solutors that are employable with separable Hamiltonian systems.

One of the most important and original results, in this area, has been obtained with the introduction of a new class of numerical integrators, which we called 'Elementary--Differential Runge--Kutta' (EDRK): it constitutes a generalization of RK methods and it is built and implemented so that it keeps some specific geometric properties of the flux.

Splitting and composition methods are also important in the field of the Geometric Integration of differential equations, as they allow to augment the order of the base integrator, therefore allowing to increase the accuracy of the obtained approximation, while at the same time they mantain the qualitative features of the solution. By employing the numerical, symbolic, graphical possibilities of Mathematica, and its tools for parallelism, new integrators have been developed, that are quite efficient and unknown in the literature: they namely exploit splitting and composition.

This last research area also involves the search for a global minimum, as the problem faced can be modeled as one of non linear Optimization.

- The problems considered, within the area of experimental data Approximation, are essentially related to medical Imaging.

It is of particular interest the reconstruction, from a modeling - numerical point of view, of a tridimensional mathematical surface, starting from scintigraphic, ultrasound or micro-computerised tomographic images, of the macroscopic profile of a parenchymal organ. Here, the gland of interest is the adult human thyroid, of which the reconstructed surface must represent a good approximation, both visually that in volumetric terms; the final aim is that of recreating the thyroid vascular/stromal skeleton (SSV), internal to the reconstructed object, employing the information somehow included in the shape of the organ itself.

A fractal modeling is considered, following the observation that the thyroid SSV, like many vascular structures in the human body, is more adequately represented via fractal geometry, instead of classical Euclidean geometry; to such an aim, information is employed on the distribution of the arterial vessels which lay on the profile of each lobe, together with the knowledge of their calibers.

Further to the fractal approach, which is of probabilistic kind, the problem of thyroid SSV reconstruction will be faced also by means of numerical methods, of deterministic type, employing spatial information on the vascularisation itself, such as information coming from micro-computerized tomographic images or tridimensional ultrasound images.

A software has been developed, composed of a computational kernel of mathematical and visualization functions, enriched by a graphical interface. The numerical routines, in such a package, are being validated on a wide set of data, in particular, on synthetic matter casts of the arterial thyroid tree.

The most recent results in this research area concern a semi-automatic method for the edge detection of the thyroid lobe border from ultrasound data.

Furthermore, the results obtained in the reconstruction of the thyroid lobe and SSV are being extended to other tissues and organs, such as bone tissues.