Francesca Pitolli

Laura Pezza, Francesca Pitolli

The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.

Francesca Pitolli

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basis

Chiara Sorgentone, Enza Pellegrino, Francesca Pitolli

Fractional boundary value problems are often used to model complex systems and processes characterized by memory effects and anomalous diffusion. In this paper, we consider fractional boundary value problems involving the Riesz-Caputo operator, which is particularly suited for modeling physical phenomena exhibiting symmetric diffusive effects. We provide an integral representation of the solution to prove existence and uniqueness of the fractional differential problem. We introduce a B-spline collocation method to approximate the solution of the problem and provide a convergence analysis, with both theoretical insights and numerical experiments.

Meriam Zribi, Paolo Pagliuca, Francesca Pitolli

The rapid growth of the Industry 4.0 paradigm is increasing the pressure to develop effective automated monitoring systems. Artificial Intelligence (AI) is a convenient tool to improve the efficiency of industrial processes while reducing errors and waste. In fact, it allows the use of real-time data to increase the effectiveness of monitoring systems, minimize errors, make the production process more sustainable, and save costs. In this paper, a novel automatic monitoring system is proposed in the context of production process of plastic consumables used in analysis laboratories, with the aim to increase the effectiveness of the control process currently performed by a human operator. In particular, we considered the problem of classifying the presence or absence of a transparent anticoagulant substance inside test tubes. Specifically, a hand-designed deep network model is used and compared with some state-of-the-art models for its ability to categorize different images of vials that can be either filled with the anticoagulant or empty. Collected results indicate that the proposed approach is competitive with state-of-the-art models in terms of accuracy. Furthermore, we increased the complexity of the task by training the models on the ability to discriminate not only the presence or absence of the anticoagulant inside the vial, but also the size of the test tube. The analysis performed in the latter scenario confirms the competitiveness of our approach. Moreover, our model is remarkably superior in terms of its generalization ability and requires significantly fewer resources. These results suggest the possibility of successfully implementing such a model in the production process of a plastic consumables company.

Daniela Calvetti, Annalisa Pascarella, Francesca Pitolli et al.

A recently proposed IAS MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its sensitivity and specificity as a function of the activity location is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on their sensitivity and specificity in identifying active brain regions. We use these protocols for a systematic study of the sensitivity and specificity of the IAS MEG inverse solver, comparing the performance with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The sensitivity is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The specificity analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, vs. the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming the knowledge of the number of active regions. The testing protocols suggest that the IAS solver performs well in terms of sensitivity and specificity both with cortical and subcortical activity estimation.

Daniela Calvetti, Francesca Pitolli, Erkki Somersalo et al.

The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the solution of the associated linear system is computed by the Conjugate Gradient for Least Squares (CGLS) method, the additional information can be encoded in the form of a right preconditioner. In this paper we study how the right preconditioned changes the Krylov subspaces where the CGLS iterates live, and draw a tighter connection between Bayesian inference and Krylov subspace methods. The advantages of a Krylov-meet-Bayes approach to the solution of underdetermined linear inverse problems is illustrated with computed examples.