arXiv subjects

Francesco Marchetti

Publications and source records attributed to Francesco Marchetti.

At least 37 records · Page 2Linked to original sources

arXiv2023

Physics-driven machine learning for the prediction of coronal mass ejections' travel times

Coronal Mass Ejections (CMEs) correspond to dramatic expulsions of plasma and magnetic field from the solar corona into the heliosphere. CMEs are scientifically relevant because they are involved in the physical mechanisms characterizing the active Sun. However, more recently CMEs have attracted attention for their impact on space weather, as they are correlated to geomagnetic storms and may induce the generation of Solar Energetic Particles streams. In this space weather framework, the present paper introduces a physics-driven artificial intelligence (AI) approach to the prediction of CMEs travel time, in which the deterministic drag-based model is exploited to improve the training phase of a cascade of two neural networks fed with both remote sensing and in-situ data. This study shows that the use of physical information in the AI architecture significantly improves both the accuracy and the robustness of the travel time prediction.

astro-ph.SR↗

arXiv2023

Mapped Variably Scaled Kernels: Applications to Solar Imaging

Variably scaled kernels and mapped bases constructed via the so-called fake nodes approach are two different strategies to provide adaptive bases for function interpolation. In this paper, we focus on kernel-based interpolation and we present what we call mapped variably scaled kernels, which take advantage of both strategies. We present some theoretical analysis and then we show their efficacy via numerical experiments. Moreover, we test such a new basis for image reconstruction tasks in the framework of hard X-ray astronomical imaging.

math.NA↗

arXiv2023

Moving Least Squares Approximation using Variably Scaled Discontinuous Weight Function

Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable {\it piecewise} Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.

math.NA↗

arXiv2023

Data-driven kernel designs for optimized greedy schemes: A machine learning perspective

Thanks to their easy implementation via Radial Basis Functions (RBFs), meshfree kernel methods have been proved to be an effective tool for e.g. scattered data interpolation, PDE collocation, classification and regression tasks. Their accuracy might depend on a length scale hyperparameter, which is often tuned via cross validation schemes. Here we leverage approaches and tools from the machine learning community to introduce two-layered kernel machines, which generalize the classical RBF approaches that rely on a single hyperparameter. Indeed, the proposed learning strategy returns a kernel that is optimized not only in the Euclidean directions, but that further incorporates kernel rotations. The kernel optimization is shown to be robust by using recently improved calculations of cross validation scores. Finally, the use of greedy approaches, and specifically of the Vectorial Kernel Orthogonal Greedy Algorithm (VKOGA), allows us to construct an optimized basis that adapts to the data. Beyond a rigorous analysis on the convergence of the so-constructed two-Layered (2L)-VKOGA, its benefits are highlighted on both synthesized and real benchmark data sets.

math.NA↗

arXiv2022

Interpolation with the polynomial kernels

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.

math.NA↗

arXiv2022

On Kosloff Tal-Ezer Least-Squares Quadrature Formulas

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map's parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way.

math.NA↗

arXiv2022

Operational solar flare forecasting via video-based deep learning

Operational flare forecasting aims at providing predictions that can be used to make decisions, typically at a daily scale, about the space weather impacts of flare occurrence. This study shows that video-based deep learning can be used for operational purposes when the training and validation sets used for the network optimization are generated while accounting for the periodicity of the solar cycle. Specifically, the paper describes an algorithm that can be applied to build up sets of active regions that are balanced according to the flare class rates associated to a specific cycle phase. These sets are used to train and validate a Long-term Recurrent Convolutional Network made of a combination of a convolutional neural network and a Long-Short Memory network. The reliability of this approach is assessed in the case of two prediction windows containing the solar storm of March 2015 and September 2017, respectively.

astro-ph.SR↗

arXiv2022

Polynomial mapped bases: theory and applications

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects.

math.NA↗

arXiv2022

Convergence analysis for image interpolation in terms of the cSSIM

Assessing the similarity of two images is a complex task that attracts significant efforts in the image processing community. The widely used Structural Similarity Index Measure (SSIM) addresses this problem by quantifying a perceptual structural similarity. In this paper we consider a recently introduced continuous SSIM (cSSIM), which allows one to analyze sequences of images of increasingly fine resolutions, and further extend the definition of the index to encompass the locally weighted version that is used in practice. For both the local and the global versions, we prove that the continuous index includes the classical SSIM as a special case, and we provide a precise connection between image similarity measured by the cSSIM and by the $L_2$ norm. Using this connection, we derive bounds on the cSSIM by means of bounds on the $L_2$ error, and we even prove that the two error measures are equivalent in certain circumstances. We exploit these results to obtain precise rates of convergence with respect to the cSSIM for several concrete image interpolation methods, and we further validate these findings by different numerical experiments. This newly established connection paves the way to obtain novel insights into the features and limitations of the SSIM, including on the effect of the local weighted window on the index performances.

math.NA↗

arXiv2022

Variably Scaled Persistence Kernels (VSPKs) for persistent homology applications

In recent years, various kernels have been proposed in the context of persistent homology to deal with persistence diagrams in supervised learning approaches. In this paper, we consider the idea of variably scaled kernels, for approximating functions and data, and we interpret it in the framework of persistent homology. We call them Variably Scaled Persistence Kernels (VSPKs). These new kernels are then tested in different classification experiments. The obtained results show that they can improve the performance and the efficiency of existing standard kernels.

math.NA↗

arXiv2021

A stochastic extended Rippa's algorithm for LpOCV

In kernel-based approximation, the tuning of the so-called shape parameter is a fundamental step for achieving an accurate reconstruction. Recently, the popular Rippa's algorithm [14] has been extended to a more general cross validation setting. In this work, we propose a modification of such extension with the aim of further reducing the computational costs. The resulting Stochastic Extended Rippa's Algorithm (SERA) is first detailed and then tested by means of various numerical experiments, which show its efficacy and effectiveness in different approximation settings.

math.NA↗

arXiv2021

Microwave Quantum Radar using a Josephson Traveling Wave Parametric Amplifier

Detection of low-reflectivity objects can be improved by the so-called quantum illumination procedure. However, quantum detection probability exponentially decays with the source bandwidth. The Josephson Parametric Amplifiers (JPAs) technology utilized as a source, generating a pair of entangled signals called two-mode squeezed vacuum states, shows a very narrow bandwidth limiting the operation of the microwave quantum radar (MQR). In this paper, for the first time, a microwave quantum radar setup based on quantum illumination protocol and using a Josephson Traveling Wave Parametric Amplifier (JTWPA) is proposed. Measurement results of the developed JTWPA, pumped at 12 GHz, show an ultrawide bandwidth equal to 10 GHz at X-band making our MQR a promising candidate for the detection of stealth objects.

quant-ph↗

arXiv2021

Implementation paradigm for supervised flare forecasting studies: a deep learning application with video data

Solar flare forecasting can be realized by means of the analysis of magnetic data through artificial intelligence techniques. The aim is to predict whether a magnetic active region (AR) will originate solar flares above a certain class within a certain amount of time. A crucial issue is concerned with the way the adopted machine learning method is implemented, since forecasting results strongly depend on the criterion with which training, validation, and test sets are populated. In this paper we propose a general paradigm to generate these sets in such a way that they are independent from each other and internally well-balanced in terms of AR flaring effectiveness. This set generation process provides a ground for comparison for the performance assessment of machine learning algorithms. Finally, we use this implementation paradigm in the case of a deep neural network, which takes as input videos of magnetograms recorded by the Helioseismic and Magnetic Imager on-board the Solar Dynamics Observatory (SDO/HMI). To our knowledge, this is the first time that the solar flare forecasting problem is addressed by means of a deep neural network for video classification, which does not require any a priori extraction of features from the HMI magnetograms.

astro-ph.SR↗

arXiv2021

Efficient Reduced Basis Algorithm (ERBA) for kernel-based approximation

The main purpose of this work is the one of providing an efficient scheme for constructing reduced interpolation models for kernel bases. In literature such problem is mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely Efficient Reduced Basis Algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa's algorithm, our ERBA takes advantage of a fast implementation.

math.NA↗

arXiv2021

MANTRA: Memory Augmented Networks for Multiple Trajectory Prediction

Autonomous vehicles are expected to drive in complex scenarios with several independent non cooperating agents. Path planning for safely navigating in such environments can not just rely on perceiving present location and motion of other agents. It requires instead to predict such variables in a far enough future. In this paper we address the problem of multimodal trajectory prediction exploiting a Memory Augmented Neural Network. Our method learns past and future trajectory embeddings using recurrent neural networks and exploits an associative external memory to store and retrieve such embeddings. Trajectory prediction is then performed by decoding in-memory future encodings conditioned with the observed past. We incorporate scene knowledge in the decoding state by learning a CNN on top of semantic scene maps. Memory growth is limited by learning a writing controller based on the predictive capability of existing embeddings. We show that our method is able to natively perform multi-modal trajectory prediction obtaining state-of-the art results on three datasets. Moreover, thanks to the non-parametric nature of the memory module, we show how once trained our system can continuously improve by ingesting novel patterns.

cs.CV↗

arXiv2021

Stable discontinuous mapped bases: the Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA) method

The mapped bases or Fake Nodes Approach (FNA), introduced in [10], allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied in preventing the appearance of the Gibbs phenomenon when interpolating discontinuous functions. However, the originally proposed S-Gibbs map suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge's phenomenon. Here, we propose a novel approach, termed Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated. After providing a theoretical analysis of the Lebesgue constant associated to the mapped nodes, we test the new approach by performing different numerical experiments which confirm the theoretical findings.

math.NA↗

arXiv2021

Score-oriented loss (SOL) functions

Loss functions engineering and the assessment of forecasting performances are two crucial and intertwined aspects of supervised machine learning. This paper focuses on binary classification to introduce a class of loss functions that are defined on probabilistic confusion matrices and that allow an automatic and a priori maximization of the skill scores. The performances of these loss functions are validated during the training phase of two experimental forecasting problems, thus showing that the probability distribution function associated with the confusion matrices significantly impacts the outcome of the score maximization process.

cs.LG↗

arXiv2021

On $(β,γ)$-Chebyshev functions and points of the interval

In this paper, we introduce the class of $(β,γ)$-Chebyshev functions and corresponding points, which can be seen as a family of {\it generalized} Chebyshev polynomials and points. For the $(β,γ)$-Chebyshev functions, we prove that they are orthogonal in certain subintervals of $[-1,1]$ with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of $(β,γ)$-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters $β$ and $γ$.

math.NA↗