R. Cavoretto

Reliable approximation of separatrix manifolds in competition models with safety niches

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 Wendlands functions as local approximants.

Scattered and track data interpolation using an efficient strip searching procedure

Abstract A new local algorithm for bivariate interpolation of large sets of scattered and track data is presented. The method, which changes partially depending on the kind of data, is based on the partition of the interpolation domain in a suitable number of parallel strips, and, starting from these, on the construction for any data point of a square neighbourhood containing a convenient number of data points. Then, the well-known modified Shepard’s formula for surface interpolation is applied with some effective improvements. The proposed algorithm is very fast, owing to the optimal nearest neighbour searching, and achieves good accuracy. Computational cost and storage requirements are analyzed. Moreover, the efficiency and reliability of the algorithm are shown by several numerical tests, also performed by Renka’s algorithm for a comparison.

Efficient computation of partition of unity interpolants through a block-based searching technique

In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two- and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in Matlab is here discussed and made available to the scientific community.

Partition of unity interpolation using stable kernel-based techniques

Abstract In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient.

Partition of unity interpolation on multivariate convex domains

In this paper, we present an algorithm for multivariate interpolation of scattered data sets lying in convex domains Ω ⊆ ℝN, for any N ≥ 2. To organize the points in a multidimensional space, we build a kd-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function (RBF) approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in Ω, where Ω can be any convex domain, like a 2D polygon or a 3D polyhedron. Finally, an application to topographical data contained in a pentagonal domain is presented.

Spectral analysis and preconditioning techniques for radial basis function collocation matrices

SUMMARY Meshless collocation methods based on radial basis functions lead to structured linear systems, which, for equispaced grid points, have almost a multilevel Toeplitz structure. In particular, if we consider partial differential equations (PDEs) in two dimensions, then we find almost (up to a ‘low-rank’ correction given by the boundary conditions) two-level Toeplitz matrices, i.e. block Toeplitz with Toeplitz blocks structures, where both the number of blocks and the block-size grow with the number of collocation points. In Bini et al. (Linear Algebra Appl. 2008; 428:508–519), upper bounds for the condition number of the Toeplitz matrices approximating a one-dimensional model problem were proved. Here, we refine the one-dimensional results, by explaining some numerics reported in the previous paper, and we show a preliminary analysis concerning conditioning, extremal spectral behavior, and global spectral results in the two-dimensional case for the structured part. By exploiting the recent tools in the literature, a global distribution theorem in the sense of Weyl is proved also for the complete matrix-sequence, where the low-rank correction due to the boundary conditions is taken into consideration. The provided spectral analysis is then applied to design effective preconditioning techniques in order to overcome the ill-conditioning of the matrices. A wide numerical experimentation, both in the one- and two-dimensional cases, confirms our analysis and the robustness of the proposed preconditioners. Copyright

Visualization Aspects of Motion Tracking and Analysis of the Outer Surface of the Left Ventricle

The quantitative assessment of the motion and deformation of the heart is instrumental to diagnosis. We developed an accurate method for tracking and analysing the regional motion and deformation of the heart. To be of clinical value, the results must be visualized, and we paid much attention to all relevant visualization aspects.

Approximating basins of attraction for dynamical systems via stable radial bases

In applied sciences it is often required to model and supervise temporal evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models for each stable equilibrium point. We propose to reconstruct the basins via an implicit interpolant using stable radial bases, obtaining the surfaces by partitioning the phase space into disjoint regions. An application to a competition model presenting jointly three stable equilibria is considered.

Lung assist devices influence cardio-energetic parameters: Numerical simulation study

We aim at an analysis of the effects mechanical ventilators (MVs) and thoracic artificial lungs (TALs) will have on the cardiovascular system, especially on important quantities, such as left and right ventricular external work (EW), pressure-volume area (PVA) and cardiac mechanical efficiency (CME). Our analyses are based on simulation studies which were carried out by using our CARDIOSIM© software simulator. At first, we carried out simulation studies of patients undergoing mechanical ventilation (MV) without a thoracic artificial lung (TAL). Subsequently, we conducted simulation studies of patients who had been provided with a TAL, but did not undergo MV. We aimed at describing the patients physiological characteristics and their variations with time, such as EW, PVA, CME, cardiac output (CO) and mean pulmonary arterial/venous pressure (PAP/PVP). We were starting with a simulation run under well-defined initial conditions which was followed by simulation runs for a wide range of mean intrathoracic pressure settings. Our simulations of MV without TAL showed that for mean intrathoracic pressure settings from negative (-4 mmHg) to positive (+5 mmHg) values, the left and right ventricular EW and PVA, right ventricular CME and CO decreased, whereas left ventricular CME and the PAP increased. The simulation studies of patients with a TAL, comprised all the usual TAL arrangements, viz. configurations “in series” and in parallel with the natural lung and, moreover, hybrid configurations. The main objective of the simulation studies was, as before, the assessment of the hemodynamic response to the application of a TAL. We could for instance show that, in case of an “in series” configuration, a reduction (an increase) in left (right) ventricular EW and PVA values occurred, whereas the best performance in terms of CO can be achieved in the case of an in parallel configuration.

Spectral Analysis for Radial Basis Function Collocation Matrices

The aim of this paper is to provide tools and results for the analysis of the linear systems arising from radial basis function (RBF) approximations of partial differential equations (PDEs), see e.g., [1,9]. Informally, a radial function \(\phi (x) : \mathbb{R}^n \rightarrow \mathbb{R} \) is a function of the Euclidean norm \(\|x\|\) of x, i.e., \(\phi (x) = \eta (\| x\|) \), for \( \eta (t) : \mathbb{R}^n \rightarrow \mathbb{R}\)