Numerical Linear Algebra and Parallel Computing
- 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.
