PhD Course: Insights and Algorithms for Ill-Posed Problems

Set 262025

Insights and Algorithms for Ill-Posed Problems

Prof. Lothar Reichel and Prof. Laura Dykes
Kent State University, USA

Abstract

The aim of this course is to introduce Master’s and Ph.D. students to linear ill-posed problems. Their properties and applications will be discussed. The focus of the lectures will be on solution methods for linear discrete ill-posed problems and on the numerical linear algebra required for the solution of these problems.

While there are no prerequisites, a basic knowledge of numerical linear algebra, least squares, LU, QR and SVD factorizations, and Matlab programming will be helpful for successfully following the lessons.

Outline

Linear discrete ill-posed problems: Definition, properties, applications
Solution methods for small to moderately sized problems: Regularization, Tikhonov regularization, the singular value decomposition, the generalized singular value decomposition, choice of regularization matrix.
Solution methods for large problems: Iterative methods based on the Lanczos process, the Arnoldi process, and Golub-Kahan bidiagonalization. Regularization by Tikhonov’s method and truncated iteration. Iterative methods for general regularization matrices.
lp-lq minimization for image restoration.

Schedule

Monday September 29, 15-18, room B
Thursday October 2, 15-18, room 2
Friday October 3, 15-18, room 2
Monday October 6, 15-18, room B

The first four lectures will be broadcast on Microsoft Teams for students who cannot attend in person.

The final two lectures, each lasting two hours, will be delivered on Teams after the instructors return to their offices. The schedule will be released during the lectures.

Anyone interested in participating in the course should contact the organizers, Alessandro Buccini and Giuseppe Rodriguez.

Exam

TBA

Acknowledgements

The course is partially supported by the INdAM Visiting Professors Program.

PhD Course: Qualitative Properties of Solutions of Uniformly Parabolic Equations

Mag 232025

Qualitative Properties of Solutions of Uniformly Parabolic Equations

Prof. Daniele Castorina
Università di Napoli Federico II

Dott. Simone Ciani
Università di Bologna

Abstract

The course is divided in two sections:

Section 1 – Second Order Parabolic Equations (Ciani)
In the first lectures we will follow Ch. VII of [1], by setting up a definition of solution of parabolic uniformly elliptic equations. In the sequel we will prove existence and uniqueness for the Cauchy-Dirichlet problem, thanks to the method of Galerkin approximations and a priori estimates. Then we will approach regularity theory: first we will prove that the unique solution to the boundary value problem proposed improves its regularity as much as the initial value datum allows, until we reach the smoothness C-infinity by a bootstrap argument. In the last lecture, time permitting, we will comment on the lack of regularity when the coefficients and the data are rough; and give a glimpse of the possible minimal regularity properties affordable, following Chap XI-XII of [2].

Section 2 – The Alexandrov-Bakelman-Pucci method and its applications (Castorina)
The classical Alexandrov-Bakelman-Pucci (or ABP) estimate is a uniform bound for strong solutions of second order uniformly parabolic operators with bounded measurable coefficients written in nondivergence form. Its main feature is being a basic tool in the regularity theory for fully nonlinear parabolic equations. However, the ABP method is fairly general and it can be adapted to a wide variety of different issues such as obtaining a maximum principle in domains of small measure, as well as simplifying the proofs of several isoperimetric and Sobolev inequalitiees. The aim of this second part course is to introduce the ABP method in detail, discuss some of its generalizations and refinements and to give a detailed and complete overview of its applications, explicitly highlighting the improvements of using this technique with respect to previous and more classical tools.

Outline

Existence and Uniqueness
Regularity Theory I – Improvement of Regularity
Minimal Regularity II – Hölder Continuity, Harnack inequality and Applications
The Alexandrov-Bakelman-Pucci estimate
The Maximum Principle in small domains
The Gidas-Ni-Nirenberg Theorem and Isoperimetric and Sobolev inequalities

Schedule

8/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
9/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
10/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A

Exam

The final assessment consists of one of the two choices: a list of exercises to solve (during the course) and a seminar; or a written elaborate on selected topics of the course.

References

L. Evans, Partial Differential Equations, Second Edition, AMS, 1998.
E. DiBenedetto, U. Gianazza, Partial Differential Equations, Third Edition, Birkhäuser, 2023.
Berestycki, H., Nirenberg, L. On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22, 1991, 1–37.
Berestycki, H., Nirenberg, L., Varadhan, S. R. S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47,1994, 47–92.
Cabré, X. On the Alexandrov-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48, 1995, 539–570.
Cabré, X., Ros-Oton, X. Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations 255, 2013, 4312–4336.
Cabré, X., Ros-Oton, X., Serra, J. Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. 18, 2016, 2971–2998.
Gilbarg, D., Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin-New York, 1983.

PhD Course: Introduction to Compositional Data Analysis and Modelling

Mar 232025

Introduction to Compositional Data Analysis and Modelling

Prof. Fabio Divino
Università del Molise & University of Jyväskylä

Abstract

This course introduces the fundamental concepts of compositional data analysis, including the algebraic structure of the simplex and its main properties. It will then cover the basic tools for analysis before presenting the main regression models:
(a) compositional data as a predictor;
(b) compositional data as a response variable;
(c) compositional data as both predictor and response.

All topics will be explored through hands-on lab sessions in R using real data and simulations.

Outline

Lecture 1: Introduction to Compositional Data Analysis. ALR, ILR, and CLR Transformations
Lecture 2: Descriptive Analysis in the Simplex. Introduction to Regression Models
Lecture 3: Regression Models with Compositional Data

Schedule

The course consists of a total of 6 hours, scheduled as follows:

March 17, 15:00-17:00 – Aula A
March 19, 11:00-13:00 – Aula A
March 21, 15:00-17:00 – Aula A

Exam

The final exam consists of a presentation on a specific topic covered in the course.

References

K. Gerald van den Boogaart & Raimon Tolosana-Delgado, Analyzing Compositional Data with R, Springer, 2013.

PhD Course: Introduction to Algorithmic Fairness

Mar 122025

Introduction to Algorithmic Fairness: Principles, Methods and Regulatory Perspectives

Dr. Erasmo Purificato
European Commission, Joint Research Centre (JRC), Italy

Abstract

The course provides a comprehensive introduction to algorithmic fairness, exploring key concepts such as definitions, bias characterisation and the potential sources of unfairness in machine learning models. Initially, we will thoroughly examine fairness criteria, bias detection metrics, and the limitation of fairness evaluation in binary scenarios. Then, we will analyse the emerging multiclass and multigroup approaches, and cover bias mitigation techniques and their practical trade-offs. Finally, the course will examine legal and ethical frameworks governing algorithmic fairness, with a focus on EU regulations such as the GDPR, DSA, and AI Act, as well as global policies.

Outline

Lecture 1: Foundation of Algorithmic Fairness
- Why fairness matters in AI and ML
- Defining fairness and bias
- Potential causes of unfairness in ML
- Fairness criteria
- Conflicts between fairness goals
Lecture 2 – Measuring Bias and Fairness
- Bias detection metrics
- Challenges in binary scenarios
- Extending fairness metrics to multiclass and multigroup scenarios
Lecture 3 – Mitigating Bias
- Bias mitigation strategies
- Choosing the right fairness intervention
- Trade-offs and practical implementations
Lecture 4 – Legal and Ethical Frameworks for Fairness in AI
- Overview of EU Regulations affecting AI and ML (i.e., GDPR, DSA and AI Act)
- Fairness principles in EU Regulations
- Fairness principles in global regulations
- The future of algorithmic fairness and open research challenges

Schedule

The course will have a total duration of 10 hours, scheduled as follows:

May 21, 14:00-18:00 Aula II
May 22, 10:30-12:30 Aula F
May 22, 14:00-16:00 Aula F
May 23, 10:00-12:00 Aula F

Exam

The final exam consists either in a seminar presentation focusing on a specific topic studied during the course or in a test held on the last day of the course. The definitive format will be announced when the schedule is finalized. The course will be held in person. Please contact me if you are interested in joining.

References

The content of the course is based (but not limited to) the following articles:

Simon Caton and Christian Haas. Fairness in Machine Learning: A Survey. ACM Comput. Surv. 56, 7, Article 166 (2024). https://dl.acm.org/doi/10.1145/3616865
Corbett-Davies, Sam, Johann D. Gaebler, Hamed Nilforoshan, Ravi Shroff, and Sharad Goel. The measure and mismeasure of fairness. Journal of Machine Learning Research 24, no. 312 (2023). https://jmlr.org/papers/v24/22-1511.html
Dana Pessach and Erez Shmueli. A Review on Fairness in Machine Learning. ACM Comput. Surv. 55, 3, Article 51 (2023). https://doi.org/10.1145/3494672
Sahil Verma and Julia Rubin. Fairness definitions explained. In Proceedings of the International Workshop on Software Fairness (FairWare 2018). https://doi.org/10.1145/3194770.3194776
Purificato, Erasmo, Ludovico Boratto, and Ernesto William De Luca. Toward a responsible fairness analysis: from binary to multiclass and multigroup assessment in graph neural network-based user modeling tasks. Minds and Machines 34, no. 3 (2024). https://doi.org/10.1007/s11023-024-09685-x
Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation),
2016, OJ L119/1. http://data.europa.eu/eli/reg/2016/679/oj
Regulation (EU) 2022/2065 of the European Parliament and of the Council of 19 October 2022 on a Single Market For Digital Services and amending Directive 2000/31/EC (Digital Services Act), 2022, OJ L277/1. http://data.europa.eu/eli/reg/2022/2065/oj
Regulation (EU) 2024/1689 of the European Parliament and of the Council of 13 June 2024 laying down harmonised rules on artificial intelligence and amending Regulations (EC) No 300/2008, (EU) No 167/2013, (EU) No 168/2013, (EU) 2018/858, (EU) 2018/1139 and (EU) 2019/2144 and
Directives 2014/90/EU, (EU) 2016/797 and (EU) 2020/1828 (Artificial Intelligence Act), 2024, http://data.europa.eu/eli/reg/2024/1689/oj

PhD Course: Introduction to algebraic logic

Ott 242024

Introduction to algebraic logic

Dr. Nicolò Zamperlin
Università degli Studi di Cagliari

Abstract

The course is an introduction to the theory of algebraizability of Blok and Pigozzi. Through an analytic study of the first chapters of Font’s handbook on abstract algebraic logic we will first introduce the elementary notions of universal algebra needed for linking together logic and algebra (closure operators and their lattices, varieties, quasivarieties and equational consequences), then building upon these notions we will consider the case of implicative logics and their algebraic properties, introducing the technique of completeness through the Lindenbaum-Tarski process. Finally we generalize these notions to the class of algebraizable logics (with a glimpse to the larger Leibniz heirarchy), with the ultimate goal of proving the isomorphism theorem and the transfer for the deduction theorem.

Schedule

The course will have a duration of 20 hours, scheduled as follows:

November 7, aula B, h. 15-17
November 14, aula A, h. 15-17
November 22, aula II, h. 9:30-11:30
November 29, aula II, h. 9:30-11:30
December 2, aula II, h. 9:30-11
December 6, aula II, h. 9:30-11:30
December 12, aula B, h. 15-17
January 15, aula B, h. 10-12
January 20, aula B, h. 15-17
February 3, aula B, h. 10-12

Exam

The final exam consists in a seminar presentation. The course will be held in person. Please contact me if you are interested in joining the course

References

Bergman, C., Universal Algebra: Fundamentals and Selected Topics, Chapman & Hall Pure and Applied Mathematics, Chapman and Hall/CRC, 2011.
Blok, W., and Pigozzi, D., Algebraizable logics, vol. 396 of Memoirs of the American Mathematical Society, A.M.S., 1989.
Burris, S., and Sankappanavar, H.P., A course in Universal Algebra, freely available online: https://www.math.uwaterloo.ca/snburris/htdocs/ualg.html, 2012 update.
Czelakowski, J., Protoalgebraic logics, vol. 10 of Trends in Logic: Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
Font, J.M., Abstract Algebraic Logic: An Introductory Textbook, College Publications, 2016

PhD Course: Conic Programming: Theory and applications

Set 232024

Conic Programming: Theory and applications

Prof. Benedetto Manca
Università degli Studi di Cagliari

Abstract

The course covers the theory of conic programming, starting from the simplest case of linear programming and introducing conic quadratic and semi-definite programming. The first part of the course will introduce the theoretical backgrounds needed to define the concept of conic programming. In the second part the case of conic quadratic and semi-definite programming will be addressed together with some applications.

Outline

From Linear to Conic Programming
Conic Quadratic Programming
The quadratic formulation of the Distance Geometry Problem
Semi-definite Programming
The semi-definite relaxation of the Distance Geometry Problem
Diagonally dominant matrices and positive semi-definite matrices
The ellipsoidal separation problem

Schedule

The course consists in 10 hours, two lectures per week. Details will be specified on the occasion of the first lecture, which will be given on October 3, 2024 at 2:30 p.m. in room B of the Department of Mathematics and Computer Science.

Exam

The final exam consists in a presentation on a specific application of conic programming (conic quadratic or semi-definite).

References

Ben-Tal, Aharon, and Arkadi Nemirovski. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for industrial and applied mathematics, 2001.
Liberti, Leo. Distance geometry and data science. Top 28.2 (2020): 271-339
Astorino, Annabella, et al. Ellipsoidal classification via semidefinite programming. Operations Research Letters 51.2 (2023): 197-203.

PhD Course: Introduction to Kähler Geometry

Set 122024

Introduction to Kähler Geometry

Prof. Roberto Mossa, Prof. Giovanni Placini
Università degli Studi di Cagliari

Abstract

This introductory course covers some of the fundamental concepts of Kähler geometry, with particular attention to almost complex and complex manifolds, the properties of Hermitian metrics, and Kähler metrics. Starting from the basics of differential geometry, we will explore the structure of almost complex and complex manifolds. Subsequently, we will delve into the properties of Hermitian metrics, focusing on the definition and characteristics that define Kähler metrics, which play a key role in integrating the complex structure with the Riemannian one. Through concrete examples and applications, students will gain a deep understanding of these concepts, preparing them for advanced studies in Kähler geometry.

Schedule

The course consists of 32 hours divided into 16 lectures. This is the schedule of the lectures:

Martedì 14 Gennaio 15-17 Aula III (Giovanni Placini)
Giovedì 16 Gennaio 11-13 Aula III (Giovanni Placini)
Martedì 21 Gennaio 11-13 Aula III (Giovanni Placini)
Giovedì 23 Gennaio 11-13 Aula III (Giovanni Placini)
Martedì 28 Gennaio 11-13 Aula III (Giovanni Placini)
Giovedì 30 Gennaio 11-13 Aula III (Roberto Mossa)
Martedì 4 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 6 Febbraio 11-13 Aula III (Roberto Mossa)
Martedì 11 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 13 Febbraio 11-13 Aula III (Roberto Mossa)
Lunedì 17 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 20 marzo 11-13 Aula F (Roberto Mossa)
Giovedì 27 marzo 11-13 Aula F (Roberto Mossa)
Giovedì 3 aprile 11-13 Aula F (Giovanni Placini)
Martedì 8 aprile 11-13 Aula F (Giovanni Placini)
Giovedì 10 aprile 11-13 Aula F (Giovanni Placini)

Exam

The final exam consists in a seminar on a topic building on the content of the course. The topic for the final exam may be proposed by the students themselves or chosen from a list provided at the end of the lectures.

Seminar: Numerical Analysis with Deep Neural Networks

Ago 212024

Numerical Analysis with Deep Neural Networks

Prof. Yuesheng Xu
Old Dominion University and Syracuse University, USA

Abstract

This four-talk lecture sequence aims to introduce numerical analysis with deep neural networks. Traditional function classes used in numerical analysis include polynomials, trigonometric polynomials, splines, finite elements, wavelets, and kernels. Deep neural networks were recently employed in numerical analysis as a class of approximation functions, demonstrating advantages over traditional function classes. These talks will cover the following topics:

Deep neural network representation of a function
Optimization problems that learn a neural network
Adaptive solutions of integral equations with deep neural networks
Adaptive solutions of partial differential equations with deep neural networks.

Schedule

September 12nd, 9.30-11.30 Aula A
September 13nd, 9.30-11.30 Aula A

Seminar: QUBO and Quantum Annealing

Lug 222024

QUBO and Quantum Annealing

Prof. Michele Marchesi
University of Cagliari and NetService spa

Abstract

The seminar, lasting about an hour, presents the problems of unconstrained binary quadratic optimization, where the variables assume binary values (0/1 or -1/1), and the function to be optimized is a quadratic form with real coefficients. It will discuss various real problems that can be represented as QUBO and how to incorporate constraints using penalty coefficients. Exact classical solvers, which can only be used for small problems due to the NP-complete complexity of QUBO problems, and the main heuristic solvers: Tabu Search and Simulated Annealing, will then be presented. Finally, the Quantum Annealing approach for solving this type of problem on specialized quantum computers will be presented.

Schedule

July 25th, 9:30-10:30 (Palazzo delle Scienze, Aula B)

Seminar: Research in Blockchain and Quantum Technologies

Lug 032024

Research in Blockchain and Quantum Technologies

Dr. Ernestas Filatovas
Vilnius University, Lithuania

Abstract

Blockchain and Quantum technologies are among the most groundbreaking advancements, attracting significant attention from industry, government, and academia. This talk highlights the research advances of our “Blockchain and Quantum Technologies Group” in both fields. In the first part of the talk, we introduce Blockchain technology, covering its main concepts such as decentralization, consensus protocols, transaction flow, etc. These key concepts later are summarized within a layered structure. We then present our recent research, including a systematic review and empirical analysis of blockchain simulators, a multi-criteria decision-making (MCDM) framework for selecting consensus protocols, a data-driven classification of consensus protocols using machine learning, and an empirical analysis of wealth decentralization in blockchain networks. This part concludes with an introduction to our novel blockchain-based platform designed to enhance research reproducibility in machine learning. The second part of the talk shifts to Quantum Computing, beginning with an overview of the current state of this technology and its potential applications across various industries. We then highlight our recent achievements, such as the development of more efficient quantum circuits for integer division and the implementation of a quantum blockchain based on hypergraphs. The talk finishes with a presentation of our ongoing research, where we propose an improved quantum annealing method to scale vehicle routing problems.

Schedule

June 20th, 10:00-12:00 (Palazzo delle Scienze, Aula Magna Matematica)

Precedenti