AQuAIntAQuAInt - Approximation and Quadrature for Applicative Integral Models

Piano Nazionale di Ripresa e Resilienza (PNRR) – Missione 4 “Istruzione e Ricerca” – Componente 2 “Dalla Ricerca all’Impresa” Investimento 1.1 “Fondo per il Programma Nazionale di Ricerca e Progetti di Rilevante Interesse Nazionale (PRIN)”.

The project in brief

In the context of environmental surveying, the geotechnical characterization of the soil,the monitoring of contaminants in land and aquifers, and the estimation of the water content and salinity in agricultural terrain are of particular importance. Aerosols play a fundamental role in many atmospheric processes. Although only a minor constituent of the atmosphere, they influence the earth’s radiation budget, air quality, clouds, and precipitation, as well as the chemistry of the troposphere and stratosphere. In these and other environmental settings, one is interested in determining some physical parameters of the domain under investigation from measurements taken outside the domain, mathematical models lead to identification problems in differential equations and to integral equations.

While differential models have been widely studied in this field, less attention has been paid to integral equations, which often give more direct information on the physical system, as they are more difficult to deal with. In particular, to ensure close adherence to the model, it is essential to adopt an effective discretization, based on a careful analysis of the involved integral operators in suitably weighted function spaces. Moreover, the resulting linear and nonlinear systems of equations have to be solved by efficient algorithms, coupled with regularization techniques in case of ill-posedness and ill-conditioning. Indeed, in some cases, the existence and uniqueness of the solution are missing and there is a strong propagation of experimental error, which implies the need for specific computational approaches. Therefore, new approximation methods, numerical algorithms, and open-access software are needed.

This project aims at gathering researchers from different fields with two strategic objectives:

-in the short term, to push forward the state of the art of the numerical treatment of integral equations, with a specific focus on the integral models arising in applied geophysics and earth sciences, and to make freely available prototypal software tools implementing the developed numerical methods;

-in the medium term, to create a network of collaborations, involving national and international research partners and private companies, to share results, software, and datasets.