Summer School on
Approximation and Quadrature for Applicative Integral Models

The school is organized in the context of the PRIN-PNRR AQuAInt Project whose research team belongs to University of Cagliari (UniCa) and University of Basilicata (UniBas).
It consists of 3 courses that will provide background in Numerical Methods for Integral Equations, Integration formulas and Numerical Linear Algebra problems. Moreover, there will be 3 seminars on some specific applications.

An Overview

The aim of the school is to provide participants with knowledge, methods and tools related to integral equations of different types, with a focus on their use in applied sciences.
Numerical methods to approximate the solutions of Fredholm and Volterra integral equations of the second kind will be presented, as well as examples of stationary partial differential problems reducible to singular and hypersingular integral equations. Moreover, it will be discussed how to deal with first kind Fredholm integral equations, well-known ill-posed problems that frequently arise in applications, by regularization tecniques.
For an adequate presentation of the main themes, it will be necessary to introduce basic mathematical tools related to functional analysis, approximation theory, numerical integration and approximation of singular and hypersingular integral transforms.
Some lectures on the application of integral equations models in image restoring and evolution problems will be also given.
The school is primarily dedicated to doctoral or post-doctoral students working in the field of integral equations, as well as in fields related to this topic, such as approximation theory or numerical analysis. Nonetheless, the school can also be useful for students or young researchers working in other fields but interested in some of the multiple applications involving integral equations.

Admission requirements

Participants are required to have basic knowledge of mathematical analysis, numerical analysis and Linear Algebra. Admission is subject to acceptance.

The number of participants is limited. Applications can be submitted to the e-mail address aquint2025@unibas.it before May 1, 2025 and should include a short curriculum vitae et studiorum (CV). Participation in the summer school is free. Acceptance will be notified by May 7, 2025.
Naples is a very welcoming tourist city and participants are adviced to make flight and hotel reservations as soon as possible, after acceptance. Accommodations close to a stop on Naples Metro Line 1 are recommended.

Courses

Giovanni Monegato (Politecnico di Torino, Italy)
Integral equations' methods for the numerical solution of partial differential equations
The course will focus on the following topics:

• Definitions, properties and applications of integrals defined as Cauchy principal value or Hadamard finite part;
• Examples of stationary partial differential problems which are reducible to weakly singular, singular, or hypersingular integral equations. Numerical methods for their solution;
• Examples of time-dependent problems defined on unbounded domains and their reformulation as integral equations in the space-time domain. Use of the latter also as a non-reflecting condition on an artificial boundary that delimits the physical domain of interest. Numerical methods for their solution.

Gradimir V. Milovanović (Serbian Academy of Sciences and Arts)
Quadrature methods for integral equations
The program will be connected with the following problems:

• Iterative processes and (nonlinear) integral equations of the second kind;
• Numerical treatment of Fredholm integral equations of the second kind;
• Gauss quadrature rules for integrals involving weight functions with variable exponents;
• Software realizations;
• Applications of quadrature rules to weakly singular Volterra integral equations and to some fractional-order boundary value problems.

Giuseppe Rodriguez (University of Cagliari, Italy)
An introduction to the numerical solution of integral equations of the first kind: basic properties, discretization, regularization methods.
First-kind integral equations look formally similar to second-kind equations, but the underlying theory is very different. They appear in mathematical models that characterize many important applicative problems and are a typical example of an ill-posed problem. Indeed, some of them have infinitely many solutions, or no solution at all, and their discretization istypically ill-posed.
The lectures will present basic mathematical tools for their analysis, standard techniques for their discretization, and the ideas that allow regularization methods to select physically meaningful solutions.

Seminars

Giuseppe Mastroianni (University of Basilicata, Italy)

• A numerical method for Fredholm integral equations of the second kind (1 h);
• Convergence of interpolation processes: from Lagrange to high-order Hermite polynomials (1 h).

Angelamaria Cardone (University of Salerno, Italy)
Applications of integral and nonlocal evolutionary models (2 h).

Alessandro Buccini (University of Cagliari, Italy)
Variational and Krylov methods for the solution of discrete inverse problems (2 h).

The venue is at Area Territoriale di Ricerca Napoli 1 (ADR NA1), via Pietro Castellino 111, Naples, Italy, about 50 mt from Montedonzelli Station of Naples Metro Line 1.