Ezio Venturino

Roberto Cavoretto, Alessandra De Rossi, Emma Perracchione et al.

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models. Precisely, a particular solution of a dynamical system is completely determined by its initial condition and by the parameters involved in the model. Furthermore, when the omega limit set reduces to a point, the trajectory of the solution evolves towards the steady state. But, in case of multi-stability it is possible that several steady states originate from the same parameter set. Thus, in these cases the importance of accurately reconstruct the attraction basins follows. In this paper we focus on dynamical systems of ordinary differential equations presenting three stable equilibia and we design algorithms for the detection of the points lying on the manifolds determining the basins of attraction and for the reconstruction of such manifolds. The latter are reconstructed by means of the implicit partition of unity method which makes use of radial basis functions (RBFs) as local approximants. Extensive numerical test, carried out with a Matlab package made available to the scientific community, support our findings.

Roberto Cavoretto, Alessandra De Rossi, Roberta Freda et al.

Due to complexity and invisibility of human organs, diagnosticians need to analyze medical images to determine where the lesion region is, and which kind of disease is, in order to make precise diagnoses. For satisfying clinical purposes through analyzing medical images, registration plays an essential role. For instance, in Image-Guided Interventions (IGI) and computer-aided surgeries, patient anatomy is registered to preoperative images to guide surgeons complete procedures. Medical image registration is also very useful in surgical planning, monitoring disease progression and for atlas construction. Due to the significance, the theories, methods, and implementation method of image registration constitute fundamental knowledge in educational training for medical specialists. In this chapter, we focus on image registration of a specific human organ, i.e. the lung, which is prone to be lesioned. For pulmonary image registration, the improvement of the accuracy and how to obtain it in order to achieve clinical purposes represents an important problem which should seriously be addressed. In this chapter, we provide a survey which focuses on the role of image registration in educational training together with the state-of-the-art of pulmonary image registration. In the first part, we describe clinical applications of image registration introducing artificial organs in Simulation-based Education. In the second part, we summarize the common methods used in pulmonary image registration and analyze popular papers to obtain a survey of pulmonary image registration.

Roberto Cavoretto, Alessandra De Rossi, Emma Perracchione et al.

In this paper we consider the problem of reconstructing separatrices in dynamical systems. In particular, here we aim at partitioning the domain approximating the boundaries of the basins of attraction of different stable equilibria. We start from the 2D case sketched in \cite{cavoretto11} and the approximation scheme presented in \cite{cavoretto11,C-D-P-V}, and then we extend the reconstruction scheme of separatrices in the cases of three dimensional models with two and three stable equilibria. For this purpose we construct computational algorithms and procedures for the detection and the refinement of points located on the separatrix manifolds that partition the phase space. The use of the so-called meshfree or meshless methods is used to reconstruct the separatrices.

Roberto Cavoretto, Alessandra De Rossi, Emma Perracchione et al.

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey-predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants.