Prossimi seminari del Dipartimento di Matematica

The latest years, machine learning has been one of the main directions in the numerical solution of inverse problems, aiming to face the ill-posed nature of these problems. In this talk, we delve into the solution of inverse problems and specifically inverse eigenvalue and inverse source problems, from a machine learning perspective. In the first part, we focus on the inverse Sturm-Liouville eigenvalue problem for sym- metric potentials and the inverse transmission eigenvalue problem for spherically sym- metric refractive indices. We present the main ideas behind supervised machine learning regression and briefly discuss the basic properties of the algorithms we implement, which are k-Nearest Neighbours (kNN), Random Forests (RF) and Neural Networks (MLP). Afterwards, we numerically solve the direct problems using well known methods, in order to produce the spectral data which in turn are used for training the machine learning models. We consider examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. In the second part, we pose the inverse source problem, to identify the number, posi- tions, and strengths of hidden line sources inside a dielectric cylinder. Using classification Neural Networks, we show that we can predict the unknown number of sources with high accuracy. We complete this talk with a discussion on an ongoing work for the inverse source problem to recover the positions and strengths of the sources. Our experiments validate the efficiency of these machine learning models for numerically tackling such inverse problems, providing a proof-of-concept for their applicability in this field. 1. N. Pallikarakis and A. Ntargaras, Application of machine learning regression models to inverse eigenvalue problems, Computers & Mathematics with Applications, 154 (2024). 2. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Predicting the number of line sources inside a cylinder using classification neural networks, (2024), (to appear in: 2024 IEEE Int. Symp. Antennas Propag. and ITNC-USNC-URSI Radio Sci. Meet.). 3. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Exploring the inverse line- source scattering problem in dielectric cylinders with deep neural networks, (2024), (submitted - under review).

The aim of the talk is to analyze an evolutionary Φ- Laplacian problem, with singular and convective reactions: u_t -A u= f+g .The differential operator A considered is an elliptic operator, driven by a Young function Φ, while the reaction terms are Carathéodory functions obeying suitable growth conditions. The problem possesses three features of interest: • the operator A can be non-homogeneous, and with unbalanced growth; • the reaction term f is singular (i.e., it behaves like u^{−γ} with γ ∈ (0, 1)) and f(x, ·) can be non-monotone); • the reaction term g is convective (i.e., it depends on ∇u). We will first introduce the functional setting of the problem, by recalling the most relevant function spaces involved and their basic properties. Secondly, the main issues concerning both singular and convective terms are highlighted, together with the sub-solution and freezing techniques. Finally, we will briefly sketch the proof of our existence result, based on a priori estimates and a semi-discretization (in time) procedure. The seminar will have an introductory fashion, under the trend proposed by the cycle ASK (Analysis Student Kernel), for young analysis researchers, at the University of Bologna (https://sites.google.com/view/askbologna/home?authuser=1).

L'insegnamento fornisce le nozioni necessarie per la comprensione e l'utilizzo dei principali algoritmi di apprendimento. Verranno introdotte le definizioni fondamentali relative ai problemi di apprendimento supervisionato e non supervisionato. Poi verranno presentati alcuni approcci per l'apprendimento statistico supervisionato, come metodi locali e regularization networks, sia nel caso lineare che nonlineare. Verranno altresì introdotte le reti neurali. Il corso conterrà anche un'introduzione a problemi di apprendimento non supervisionato, come clustering a dimensionality reduction. Gli argomenti trattati dal punto di vista teorico, saranno affrontati anche da un punto di vista numerico durante le lezioni in laboratorio.

This talk will present some case studies on using artificial intelligence (AI) to preserve , study and valorize cultural heritage digital assets. At the heart of the approach is recognizing the need for a human AI frameworks in which data scientists and domain experts collaborate synergistically. Moreover promoting interdisciplinary partnerships is essential touphold ethical principles and serve the public good.

Over the years since 1970, Magnetic Resonance Imaging (MRI) has developed into as one of the preferred choices for many radiological exams today. It relies primarily on its ability to detect water, which constitutes a significant portion of most tissues (around 70-90%). Changes in the water content and properties within tissues due to diseases or injuries can be substantial, rendering MRI highly effective in diagnosis due to its sensitivity. ESAOTE specializes in designing MRI systems with low-field technology (0.25 T to 0.4 T), offering several advantages including improved patient comfort, cost reduction, less demanding installation requirements, and lower energy consumption. However, the trade-off for using low-field MRI is a decrease in signal strength, often requiring longer scan times to achieve high-quality diagnostic images. Hence, there is a need for techniques to accelerate image acquisition. Specifically, ESAOTE has developed the Speed-Up technique, inspired to compressed-sensing techniques. Recent advances in collaboration with the Amsterdam University Medical Center aim to improve the reconstruction algorithm using artificial intelligence (AI), while maintaining the diagnostic accuracy of traditional, longer scans. This was achieved by optimizing the k-space undersampling scheme and reconstructions, using the Cascades of Independently Recurrent Inference Machines (CIRIM). Promising image quality was observed up to an acceleration factor of at least 2.5.

Anomaly detection is a primary need for industrial/manufacturing applications and Datalogic business, but many challenges persist. The collection and annotation of data is expensive and most of the time unpractical as the deployed systems are usually not remotely accessible. The image resolution and the frame rate require computing-intensive algorithms that often do not fit the real-time constraint on embedded devices. In this study, we propose a novel solution that, inspired by recent advancements in this field obtained by normalizing-flow and patch-based feature distribution, combines unsupervised learning with efficient processing to deploy an optimized solution to reach a high classification accuracy while working with few training samples.

Confocal laser-scanning microscopy (CLSM) has long been celebrated in life-science research for its unique blend of spatial and temporal resolution, coupled with its versatile applications. However, recent advancements in detector technology have sparked a transformative shift in CLSM, triggered by the introduction of novel single-photon array detectors. These detectors, poised to supplant single-element detectors (also known as bucket detectors), offer access to previously discarded sample information, reshaping the trajectory of CLSM. In traditional CLSM, images are generated by raster scanning a focused laser beam across the sample, with single-element detectors registering a single-intensity value at each sample position. In contrast, single-photon array detectors capture true temporal images at each scanning position, transitioning CLSM into image scanning microscopy (ISM). Image scanning microscopy transcends traditional CLSM by generating not merely a two-dimensional dataset but a five-dimensional one, incorporating four spatial dimensions and a temporal dimension. This enables the reconstruction of highly informative and super-resolved images of the sample. This seminar will delve into the foundational principles of ISM, starting with the formulation of the forward model underlying the technique. Subsequently, a maximum likelihood approach, considering Poissonian noise, will be presented for reconstructing super-resolved images from the four-dimensional spatial dataset. An extension of this framework will incorporate the temporal dimension, enabling the reconstruction of fluorescence lifetime images that integrate structural and functional sample information. Furthermore, the seminar will explore leveraging the ISM dataset and deep learning techniques to accurately estimate the point-spread function of the optical system. This has the potential to significantly enhance the quality of reconstructed super-resolved images. By elucidating these advancements and future prospects, this seminar aims to inspire researchers to harness the full potential of ISM in pushing the boundaries of biomedical imaging.