Emma Perracchione

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.

R. Cavoretto, A. De Rossi, E. Perracchione

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 \textsc{Matlab} is here discussed and made available to the scientific community.

Roberto Cavoretto, Alessandra De Rossi, Emma Perracchione

The Partition of Unity (PU) method, performed with local Radial Basis Function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.

Stefano De Marchi, Wolfgang Erb, Francesco Marchetti et al.

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising.

R. Cavoretto, A. De Rossi, G. E. Fasshauer et al.

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has already been proved to be an effective tool for solving interpolation or collocation problems when large data sets are considered. It decomposes the original domain into several subdomains or patches so that only linear systems of relatively small size need to be solved. In research on such partition of unity methods, such subdomains usually consist of spherical patches of a fixed radius. However, for particular data sets, such as track data, ellipsoidal patches seem to be more suitable. Therefore, in this paper, we propose a scheme based on a priori error estimates for selecting the sizes of such variable ellipsoidal subdomains. We jointly solve for both these domain decomposition parameters and the anisotropic RBF shape parameters on each subdomain to achieve superior accuracy in comparison to the standard partition of unity method.

Alessandra De Rossi, Emma Perracchione

In this paper, we discuss the problem of constructing Radial Basis In this paper, we discuss the problem of constructing Radial Basis Function (RBF)-based Partition of Unity (PU) interpolants that are positive if data values are positive. More specifically, we compute positive local approximants by adding up several constraints to the interpolation conditions. This approach, considering a global approximation problem and Compactly Supported RBFs (CSRBFs), has been previously proposed. Here, the use of the PU technique enables us to intervene only locally and as a consequence to reach a better accuracy. This is also due to the fact that we select the optimal number of positive constraints by means of an a priori error estimate and we do not restrict to the use of CSRBFs. Numerical experiments and applications to population dynamics are provided to illustrate the effectiveness of the method in applied sciences.

Emma Perracchione

We consider the problem of reconstructing 3D objects via meshfree interpolation methods. In this framework, we usually deal with large data sets and thus we develop an efficient local scheme via the well-known Partition of Unity (PU) method. The main contribution in this paper consists in constructing the local interpolants for the implicit interpolation by means of Rational Radial Basis Functions (RRBFs). Numerical evidence confirms that the proposed method is particularly performing when 3D objects, or more in general implicit functions defined by scattered data, need to be approximated.

Alessandra De Rossi, Roberto Cavoretto, Emma Perracchione

In this paper the Partition of Unity Method (PUM) is efficiently performed using Radial Basis Functions (RBFs) as local approximants. In particular, we present a new space-partitioning data structure extremely useful in applications because of its independence from the problem geometry. Moreover, we study, in the context of wild herbivores in forests, an application of such algorithm. This investigation shows that the ecosystem of the considered natural park is in a very delicate situation, for which the animal population could become extinguished. The determination of the so-called sensitivity surfaces, obtained with the new fast and flexible interpolation tool, indicates some possible preventive measures to the park administrators.

Giacomo Elefante, Wolfgang Erb, Francesco Marchetti et al.

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the context of approximation theory. We will first prove necessary and sufficient conditions on point sets which guarantee the existence and uniqueness of an interpolant. We will then study the Reproducing Kernel Hilbert Spaces (or native spaces) of these kernels and their norms, and provide inclusion relations between spaces corresponding to different kernel parameters. With these spaces at hand, it will be further possible to derive generic error estimates which apply to sufficiently smooth functions, thus escaping the native space. Finally, we will show how to employ an efficient stable algorithm to these kernels to obtain accurate interpolants, and we will test them in some numerical experiment. After this analysis several computational and theoretical aspects remain open, and we will outline possible further research directions in a concluding section. This work builds some bridges between kernel and polynomial interpolation, two topics to which the authors, to different extents, have been introduced under the supervision or through the work of Stefano De Marchi. For this reason, they wish to dedicate this work to him in the occasion of his 60th birthday.

Gianluca Audone, Francesco Marchetti, Emma Perracchione et al.

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Fabiana Camattari, Sabrina Guastavino, Francesco Marchetti et al.

The purpose of this study is to introduce a new approach to feature ranking for classification tasks, called in what follows greedy feature selection. In statistical learning, feature selection is usually realized by means of methods that are independent of the classifier applied to perform the prediction using that reduced number of features. Instead, greedy feature selection identifies the most important feature at each step and according to the selected classifier. In the paper, the benefits of such scheme are investigated theoretically in terms of model capacity indicators, such as the Vapnik-Chervonenkis (VC) dimension or the kernel alignment, and tested numerically by considering its application to the problem of predicting geo-effective manifestations of the active Sun.

Sabrina Guastavino, Katsiaryna Bahamazava, Emma Perracchione et al.

This study addresses the prediction of geomagnetic disturbances by exploiting machine learning techniques. Specifically, the Long-Short Term Memory recurrent neural network, which is particularly suited for application over long time series, is employed in the analysis of in-situ measurements of solar wind plasma and magnetic field acquired over more than one solar cycle, from $2005$ to $2019$, at the Lagrangian point L$1$. The problem is approached as a binary classification aiming to predict one hour in advance a decrease in the SYM-H geomagnetic activity index below the threshold of $-50$ nT, which is generally regarded as indicative of magnetospheric perturbations. The strong class imbalance issue is tackled by using an appropriate loss function tailored to optimize appropriate skill scores in the training phase of the neural network. Beside classical skill scores, value-weighted skill scores are then employed to evaluate predictions, suitable in the study of problems, such as the one faced here, characterized by strong temporal variability. For the first time, the content of magnetic helicity and energy carried by solar transients, associated with their detection and likelihood of geo-effectiveness, were considered as input features of the network architecture. Their predictive capabilities are demonstrated through a correlation-driven feature selection method to rank the most relevant characteristics involved in the neural network prediction model. The optimal performance of the adopted neural network in properly forecasting the onset of geomagnetic storms, which is a crucial point for giving real warnings in an operational setting, is finally showed.

Michele Piana, Federico Benvenuto, Anna Maria Massone et al.

AI-FLARES (Artificial Intelligence for the Analysis of Solar Flares Data) is a research project funded by the Agenzia Spaziale Italiana and by the Istituto Nazionale di Astrofisica within the framework of the ``Attività di Studio per la Comunità Scientifica Nazionale Sole, Sistema Solare ed Esopianeti'' program. The topic addressed by this project was the development and use of computational methods for the analysis of remote sensing space data associated to solar flare emission. This paper overviews the main results obtained by the project, with specific focus on solar flare forecasting, reconstruction of morphologies of the flaring sources, and interpretation of acceleration mechanisms triggered by solar flares.

Sourav Dutta, Peter Rivera-Casillas, Orie M. Cecil et al.

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. In a previous work [arXiv:2104.13962], we explored the use of Neural Ordinary Differential Equations (NODE) as a non-intrusive method for propagating the latent-space dynamics in reduced order models. Here, we investigate employing deep autoencoders for discovering the reduced basis representation, the dynamics of which are then approximated by NODE. The ability of deep autoencoders to represent the latent-space is compared to the traditional proper orthogonal decomposition (POD) approach, again in conjunction with NODE for capturing the dynamics. Additionally, we compare their behavior with two classical non-intrusive methods based on POD and radial basis function interpolation as well as dynamic mode decomposition. The test problems we consider include incompressible flow around a cylinder as well as a real-world application of shallow water hydrodynamics in an estuarine system. Our findings indicate that deep autoencoders can leverage nonlinear manifold learning to achieve a highly efficient compression of spatial information and define a latent-space that appears to be more suitable for capturing the temporal dynamics through the NODE framework.

Emma Perracchione, Paolo Massa, Anna Maria Massone et al.

Space telescopes for solar hard X-ray imaging provide observations made of sampled Fourier components of the incoming photon flux. The aim of this study is to design an image reconstruction method relying on enhanced visibility interpolation in the Fourier domain. % methods heading (mandatory) The interpolation-based method is applied on synthetic visibilities generated by means of the simulation software implemented within the framework of the Spectrometer/Telescope for Imaging X-rays (STIX) mission on board Solar Orbiter. An application to experimental visibilities observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is also considered. In order to interpolate these visibility data we have utilized an approach based on Variably Scaled Kernels (VSKs), which are able to realize feature augmentation by exploiting prior information on the flaring source and which are used here, for the first time, for image reconstruction purposes.} % results heading (mandatory) When compared to an interpolation-based reconstruction algorithm previously introduced for RHESSI, VSKs offer significantly better performances, particularly in the case of STIX imaging, which is characterized by a notably sparse sampling of the Fourier domain. In the case of RHESSI data, this novel approach is particularly reliable when either the flaring sources are characterized by narrow, ribbon-like shapes or high-resolution detectors are utilized for observations. % conclusions heading (optional), leave it empty if necessary The use of VSKs for interpolating hard X-ray visibilities allows a notable image reconstruction accuracy when the information on the flaring source is encoded by a small set of scattered Fourier data and when the visibility surface is affected by significant oscillations in the frequency domain.

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.