### Low-rank Structures and Numerical Methods in Matrix and Tensor Computations and their Application

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget: €70.056,00

Principal Investigator: Valeria Simoncini (Università di Bologna)

Local coordinator: Beatrice Meini (Università di Pisa)

##### Participants

Alberto Bucci (Università di Pisa), Luca Gemignani (Università di Pisa)

##### Description

The proposal aims 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, to name a few. 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.
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). Attention will be paid to both theoretical and computational aspects including the implementation of effective algorithms, software design and dissemination.