Abstract
Description and objectives We aim at forging new computational approaches to tackle problems involving very large matrices and tensors, many of which cannot be solved with current techniques. Such problems are ubiquitous in modern Data Science and permeate different fields such as uncertainty quantification and dynamical systems monitoring, image and signal processing, finance, network science, and computational chemistry. The overarching theme of the project is the development of dimensionality reduction techniques based on the identification and exploitation of (approximate) low rank structure in the data. Techniques for computing low-rank approximations of matrices and tensors are currently receiving considerable attention and can lead to the efficient solution of problems previously considered intractable due to the curse of dimensionality. Exploitation of low-rank and tensor structures Many problems in network science and in applied probability require the solution of large linear systems, the solution of eigenvalue problems, and the approximation of functions of large matrices. Often the matrices involved have (approximate or perturbed) rank and/or tensor (Kronecker) structure. Techniques for solving such problems, including new preconditioning techniques and specialized rational Krylov subspace methods, will be developed. We will also tackle the important problem of efficiently updating centrality measures and other network quantities after the network undergoes a low-rank change, such as addition, deletion or rewiring of a few links. In another direction, low-rank formulations are needed to approximate the solution of challenging partial differential equations such as in space-time or parameter-dependent problems. There, suitable linear algebra should be considered that adheres to the design of tensor discrete spaces, thus preserving structural properties typical of the continuous solution. Part of the proposal concerns the development of algorithms for solving multiterm matrix and tensor linear equations stemming from problems of this type. The results of the research project will impact several application areas including network analysis, parameter-dependent numerical modeling and applied probability (Markovian modeling). Results Achieved In the following we report the list of topics described in the proposal, that have been explored during the project allotted time, resulting in an extensive number of impactful results. - Low rank approximation of tensor operators and tensor equations - Matrix equations - Large scale linear systems, optimization and preconditioning - Linear inverse problems/Randomized inverse problems - Matrix functions - Structured matrices - Markov chains - Random walks and Complex Networks - Matrix Analysis and Computations The project members have given a large number of talks at workshops and conferences in Italy and abroad, supported by the project funds, to disseminate the results of the project. The participation in special programs, workshops and conferences by team members, and visits to international institutions also provided an opportunity to share the project results, and to interact with international colleagues for mutual benefit. The unit of UNIBO worked on the following topics: i) Development of computational strategies for solving matrix and tensor linear equations. With the contribution of the hired PostDoc Martina Iannacito, funded by this project, we have developed a new class of low-rank methods for solving multiterm matrix equations; we have extended the idea to tensor linear equations. In 2025 we have started a collaboration with prof. Catherine Powell, University of Manchester (UK), who visited UNIBO in September 2025 partially funded by this project; together we attacked the numerical solution of a family of stochastic PDEs by generalizing the methodology to nonsymmetric and heterogeneous problems; this project is still in progress. We have developed matrix-oriented approaches for matrix and tensor least squares problems with application to data science . ii) Low-rank matrix approximations using randomized strategies and applications: we have theoretically analyzed the use of sketching techniques within Krylov subspace methods in matrix function evaluations, matrix and tensor equation solving. We have also devised new computational strategies. Time-dependent matrix equations have also been studied, and the visit of prof. Alessandro Alla, Universita’ di Roma La Sapienza in October 2024 was in this direction of research. iii) Adaptive discretization methods: we have developed new adaptive discretization techniques for Galerkin FEM, and new preconditioning strategies designed for discontinuous Galerkin FEM. In 2024 at ICERM (visit partially funded by this project) Ruggeri and Simoncini have started a collaboration with prof. Ricardo Nochetto, Maryland University (USA), who then visited UNIBO in June 2025, on exploiting grid-related iterative NLA solvers for enhancing the computational cost of the adaptivity step; this project is still in progress. The visit of prof. Marco Verani, Politecnico di Milano in December 2024 was aimed at the same topic. iv) Polynomials and Quadrature formulas: we have deepened our understanding of quadrature formulas, and their use in finite precision arithmetic. We have also addressed the development of fast and reliable algorithms for computing zeros of certain classes of polynomials. During the visit at ICERM in 2024, a collaboration started, leading to [ZDS25]. The research activity of this unit has also been carried out in collaboration with the following additional researchers: Davide Palitta (UNIBO), Yiding Lin (Southwestern University of Finance and Economics, Chengdu, China), Yihong Wang, (Nanjing Normal University China), Marcel Schweitzer (U Wuppertal, Germany), Margherita Porcelli and Stefania Bellavia (U Firenze), Gergor Gantner (U Bonn, Germany), Nick Van Buggenhout, Marc Van Barel, Ralf Vandebril, Paul Van Dooren (U Leuven/U Catholine Louvain, Belgium), Teresa Laudadio (CNR, Bari), Lorenzo Piccinini (UNIBO), Sijing Liu (WPI, Massachusetts, USA)
Project details
Unibo Team Leader: Valeria Simoncini
Unibo involved Department/s:
Dipartimento di Matematica
Coordinator:
ALMA MATER STUDIORUM - Università di Bologna(Italy)
Total Eu Contribution: Euro (EUR) 187.072,00
Total Unibo Contribution: Euro (EUR) 89.221,00
Project Duration in months: 24
Start Date:
28/09/2023
End Date:
28/02/2026
