The group of Numerical Analysis covers research topics from Numerical Linear Algebra and Partial Differential Equations. Details on these topics can be found below.
The group works in close collaboration with the Numerical Analysis group at the Department of Computer Science of the University of Pisa (G. Del Corso, L. Gemignani, F. Poloni), and at Scuola Normale Superiore (M. Benzi).
The group runs a Seminar Series: further information can be found at this link.
Research Topics
Numerical Linear Algebra
The group focuses on numerical methods for largescale linear systems, linear and nonlinear eigenvalue problems, matrix functions and matrix equations, and taking advantage of matrix structures.
Members
Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.
Quantum Walks and Block Encodings
The research in this area focuses on Quantum walks and Quantum Block Encodings. Quantum walks are the quantum counterpart of random walks on a graph and, like classical random walks, they can be defined in a continuoustime or in a discretetime framework. Continuoustime quantum walks evolve according to the Schroedinger equation on a suitable Hilbert space. For discretetime quantum walks, on the other hand, several different definitions of the evolution operator are available, in most cases according to the coined or the Szegedy formalisms. Both continuoustime and discretetime quantum walks have been proved to be universal for quantum computation.
Quantum walks are often used for algorithm design. In particular, thanks to their favorable diffusion properties, they are often employed in the context of unstructured search problems, resulting in a quadratic speedup with respect to classical approaches. Further applications include the formulation of quantum centrality and communicability measures in network analysis.
Research activity in this field focuses on the study of hitting time for various kinds of discretetime quantum walks on directed graphs, and on the applications of continuous and discretetime quantum walks in network analysis.
Quantum Block Encoding (QBE), instead, is a fundamental ingredient in quantum numerical linear algebra. Its importance is motivated by the fact that quantum evolution is necessarily unitary: in particular, all matrices need to be represented in unitary form, for instance as a suitable quantum circuit. Roughly speaking, a blockencoding of a matrix $M$ is a unitary matrix $U$ that contains $M/\alpha$ as its top left block, where $\alpha\geq\M\_2$ is a scaling factor. Clearly, the presence of structure in $M$ facilitates the development of effective blockencoding strategies, which are crucial to ensure significant speedup in quantum algorithms — for instance, solving an $N\times N$ linear system with runtime polylogarithmic in $N$.
Members
Collaborators
High Performance Computing
Research in numerical linear algebra for highperformance computing focuses on developing efficient algorithms and techniques for solving largescale linear algebra problems that arise in various computational tasks, such as solution of partial differential equations, data analysis, optimization, and machine learning.
This field addresses challenges related to the scalability, accuracy, and performance of numerical methods when dealing with massive datasets or complex mathematical models. We investigate algorithms that can exploit the parallelism and vectorization capabilities of modern highperformance computing architectures, such as multicore CPUs, GPUs, and specialized accelerators like FPGAs or TPUs. A particular emphasis is given to opensource software development and the construction and implementation of innovative algorithms into the PSCToolkit suite of libraries (https://psctoolkit.github.io/), and into the development of preconditioners to improve the convergence rate and stability of iterative methods for solving large sparse linear systems, particularly important for solving systems arising from partial differential equations.
Members
Collaborators
Markov Chains and Complex Networks
Many problems from the applications are modeled by Markov chains. Very often the set of the states is huge or even infinite. In these cases customary techniques are not suited to solve this kind of problems.
The goal of this research area is the design and analysis of effective solution algorithms for infinite Markov chains, with special attention to the ones coming from queuing models. This goal is reached by developing theoretical tools relying on on complex analysis, numerical analysis, and structured matrix computations.
A close connection exists between Markov chains and complex networks, which are graphs with particular structures. On these graphs, it is of great interest in applications to define centrality measures, that rank the nodes by importance. The computation of such measures can be challenging for large networks, and techniques based on iterative methods for functions of matrices and largescale linear systems can be adopted in this context.
Members
Collaborators
Matrix Equations and Matrix Functions
Many problems from the real world and from Scientific Computing are modeled by matrix equations or by matrix functions. For instance, the celebrated algebraic Riccati equation which in the continuous time models takes the form $XBX+AX+XD+C=0$, is related with the analysis of stability of dynamical systems. Here, $A,B,C,D$ are given matrices of compatible sizes and $X$ is the unknown matrix. Quadratic equations like $AX^2+BX+C=0$ model damped vibration problems as well as stochastic models encountered in the analysis of queues. In the case of queues of the M/G/1 type the equations take the form $\sum_{i=1}^{\infty} A_{i}X^{i+1}=0$ where a matrix analytic function over a suitable domain is involved.
The goal of the research in this area is to develop tools for designing fast and effective algorithms to solve this kind of equations. These equations, as well as similar ones, can be recast as generalized eigenvalue problems; for instance, the latter is related to finding $\lambda\in\mathbb{C}$ and $x\in\mathbb{C}^{n}$ such that $(\lambda^2 P +\lambda Q + R)x=0$ (quadratic eigenvalue problem). The techniques needed combine the ones used for general nonlinear equations (multivariate Newton methods, fixedpoint iterations) and eigenvalue problems (Schur decompositions, orthogonal reductions, rational approximations).
Matrix structures (such as entrywise nonnegativity, symmetry and symplecticity) play a crucial role in all of this; they are needed for defining the solutions of these equations in the first place, and to ensure feasibility, accuracy and computational efficiency of the numerical algorithms.
Applications of these equations include control and system theory, queuing theory and structured Markov chain modelling in applied probability, and time series estimaton in statistics. It is useful to interface directly with researchers working in these application fields: the different points of view provide useful insight, and the algorithms can be better tailored to the needs of the practitioners.
In several applications, different sets of measurements produce different symmetric positivedefinite matrices as a result; a natural problem is finding the most plausible correct value for the desired matrix. This corresponds to a form of averaging. The plain arithmetic mean $\frac{1}{n}(X_1+X_2+\dots+X_n)$ is not always the best choice from this point of view.
The concept of geometric mean of a set of positive number can be extended to the case of a set of positive definite matrices. However, this extension is not so trivial and, while for the case of two matrices there is a unique definition, in the case of several matrices there exist an infinite number of valid definitions.
The most general setting under which to study this problem is the one of Riemannian geometry: one gives a Riemannian scalar product on the manifold of symmetric positivedefinite matrices, and studies the point which minimizes the sum of squared distances from the given matrices (Cartan mean). This gives rise to a mean which is compatible in some sense with matrix inversions, much like the geometric mean in the scalar case.
In addition to the theoretical aspects, practical computation of these means is an interesting problem, a special case of optimization on manifolds. The classical multivariate methods from optimization need to be adapted to work on a generic manifold; one needs to consider the role of its tangent space and construct suitable maps to and from it.
This kind of mean is of great importance in the applications, especially in engineering and in problems of radar detection.
Our goal is to provide a better understanding of the different concepts involved in this research, new definitions of mean which are better suited for the applicative models, faster and more reliable algorithms for their computation.
Members
Collaborators
Nonlinear eigenvalue problems and polynomial rootfinding
Given a matrixvalued function $F(z)$, the eigenvalues are the points in the complex plane where $F(z)$ has a nontrivial kernel, or its rank drops. Computing the eigenvalues and eigenvectors (the vectors in the kernel) for general nonlinear function is challenging, and can often by achieved by appropriate matrix iterations, or by linearizing the problem and solving a generalized eigenvalue problem.
The latter strategy is particularly appealing when $F(z)$ is either a matrix or a scalar polynomial. For this specific case, particularly structured version of the QR iteration have been developed. For the scalar case, the group has developed a software package capable of computing roots of highdegree polynomial with arbitrary accuracy (MPSolve).
This area is closely related with structured matrices, in particular semiseparable and rankstructured one.
Members
Collaborators
LowRank Approximation and Structured Matrices
This research area deal with the problem of efficiently finding lowrank factorization of matrices $A$, in the form
$$A \approx UV^T,$$
where $U,V$ are tall and skinny matrices. In most cases of interest one assumes to either be able to evaluate $X \mapsto AX$ and $Y \mapsto A^TY$ or to sample selected entries of $A$ at a reduced cost. Several strategies may be adopted for retrieving matrices $U,V$, such as randomized linear algebra methods, or crossapproximation techniques.
A natural evolution of this idea is to consider structured matrices, which are – in general – not lowrank, but may have lowrank blocks, or obtained as lowrank perturbations of particularly structured matrices. The aim is to represent such matrices using $\mathcal O(n)$ or even $\mathcal O(1)$ storage and perform operations with a similar complexity (up to logarithmic factors in the dimension).
In particular, the following structures are considered:
 Rankstructured matrices: Matrices with offdiagonal blocks of lowrank (HODLR), which may be efficiently representable using hierarchically structured bases (HSS).
 Lowrank perturbations of Toeplitz matrices: matrices with this structure form an algebra, which has important applications in the study of queues and Markov chains. See also the section on the numerical solution of Markov chains below. Matrices in this class can often be represented with a storage that is independent of the dimensions, enabling the numerical study of infinite and semiinfinite matrices.
For matrices in this form, efficient arithmetic operations are available. MATLAB toolboxes for handling such matrices are available (see hmtoolbox for rankstructured matrices and cqttoolbox for quasiToeplitz ones).
Members
Collaborators
Partial Differential Equations
The group focuses on Advanced Finite Element methods, FluidStructure Interaction, NonMatching methods, Structurepreservation, Computational Plasma Physics, Model Order Reduction, Data Assimilation, Fractional Differential Equations.
Members
Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.
Computational Plasma Physics
Plasma phenomena are ubiquitous in many physical applications ranging from astrophysics to controlled thermonuclear fusion. Efficient and accurate simulations of plasma models are however challenged by the complex, highly nonlinear and inherent multiscale nature of the physical phenomena involved. Research activities of the group are devoted to the development of numerical methods for fluid and kinetic plasma models based on discontinuous Galerkin, spectral methods and discrete differential forms. Explicit and implicit time integration schemes are investigated. The emphasis is on the study of computationally efficient techniques able to preserve the conservation properties of the problem while accurately coupling the microscopic physics with the macroscopic systemscale dynamics.
Members
Collaborators
Advanced Finite Element Methods for the numerical solution of Partial Differential Equations
This research area focuses on the development and application of advanced finite element methods (FEM) to tackle complex problems in computational mechanics and physics. Key topics include the use of discontinuous and discontinuous Galerkin approximations, nonmatching techniques for finite element approximation of interface problems, and adaptive FEM for elliptic problems using regularized forcing data. The motivation behind these studies is to improve the accuracy, efficiency, and applicability of FEM in scenarios where traditional methods face limitations due to geometric complexities, material discontinuities, or the need for adaptive mesh refinement. A particular emphasis is given to opensource software development, and to highperformance computing applications based on the deal.II library.
Members
Collaborators
Computational Techniques for FluidStructure Interaction (FSI)
This area delves into the numerical simulation of fluidstructure interaction problems, employing cuttingedge computational techniques. The research includes the immersed finite element method, arbitrary LagrangianEulerian discretizations, and the development of efficient preconditioners for complex FSI solvers. These works aim to address the challenges in accurately modelling and predicting the behaviour of complex FSI problems, which are critical in many engineering and biomedical applications.
Members
Collaborators
Nonmatching, Interface, and Coupling Methods for mixeddimensional PDEs
Research in this area explores innovative numerical strategies for the effective coupling of mixeddimensional domains and the finite element approximation of problems involving interfaces. Topics include immersed finite element methods, cutfem, and the reduced Lagrange multiplier approach for nonmatching coupling of mixeddimensional domains. The objective is to develop robust and efficient methods for the simulation of multiphysical phenomena that span across different physical dimensions or require the coupling of disparate mathematical models, enhancing the simulation accuracy and computational performance in applications ranging from engineering to applied sciences.
Members
Collaborators
StructurePreserving Numerical Methods
In recent years it has been realized that formulating numerical methods compatible with the physical properties of a model problem – and not just approximating them – has paved the way for numerical discretization with superior accuracy and stability properties. To this aim, structurepreserving or geometric numerical methods seek to achieve physically consistent numerical solutions by preserving the geometric and topological structure of the governing mathematical model. We work at the development of numerical methods and model order reduction for the structurepreserving approximation of differential equations arising in e.g. Hamiltonian mechanics, dissipative dynamics and fluid models.
Members
Collaborators
Model Order Reduction
Numerical simulation of parametrized differential equations is of crucial importance
in the study of realworld phenomena in applied science and engineering. The parameter here has to be understood in a broad sense as it can represent time, boundary condition, a physical parameter, etc. Computational methods for the simulation of such problems often
require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing lowcomplexity highfidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. In this line of research we are interested in theoretical aspects and application of model order reduction techniques.
Members
Collaborators
Filtering and Data Assimiliation
In this line of research we are concerned with the inverse problem of reconstructing an unknown function $u$ or an unknown parameter y from a finite set of measurements of $u(y)$. To counteract illposedness additional a priori information is given by assuming that the studied phenomena can be well described by certain physical models. These models can come in the form of ordinary or partial differential equations or they can be given by a prior probability distribution. The first hypothesis is associated with deterministic reconstruction algorithms (e.g. PBDW), while the second hypothesis is the starting point of Bayesian inversion where the data are used to compute a posterior distribution that represents the uncertainty in the solution. In both research directions, we are interested in the development of efficient filtering algorithms and in the study of related aspects, such as accuracy quantification, computational efficiency of the sampling strategies, dynamical approximation of the state, adaptation of the measurements via sensors’ update, observation noise, modeling errors, etc.
Members
Collaborators
Fractional Differential Equations
Research on numerical methods for Fractional Partial Differential Equations (FPDEs) is an emerging and rapidly growing field at the intersection of fractional calculus and computational mathematics. It focuses on developing efficient and accurate numerical techniques for solving FPDEs, which generalize classical partial differential equations by incorporating fractional derivatives. We focus on designing efficient solvers for the resulting discrete systems, considering the potentially large dense and structured matrices that arise from discretizing FPDEs. This includes developing preconditioning techniques and leveraging parallel computing architectures for improved performance. Our research is also involved in exploring applications of FPDEs in various scientific and engineering fields, including anomalous diffusion, viscoelastic materials, optimal control, and application to the field of complex networks.
Members
Collaborators
People
Faculty
Name  Surname  Personal Card  

Paola  Boito  paola.boito@unipi.it  
Maurizio  Ciampa  maurizio.ciampa@unipi.it  
Fabio  Durastante  fabio.durastante@unipi.it  
Silvia  Gazzola  silvia.gazzola@unipi.it  
Luca  Heltai  luca.heltai@unipi.it  
Stefano  Massei  stefano.massei@unipi.it  
Beatrice  Meini  beatrice.meini@unipi.it  
Cecilia  Pagliantini  cecilia.pagliantini@unipi.it  
Leonardo  Robol  leonardo.robol@unipi.it 
Affiliate Members
Name  Surname  Personal Card  

Dario Andrea  Bini  bini@dm.unipi.it  
Sergio  Steffè  sergio.steffe@unipi.it 
Former Members
there is no data
Postdoctoral Fellows
there is no data
Ph.D. Students at the University of Pisa
Name  Surname  Personal Card  

Alberto  Bucci  alberto.bucci@phd.unipi.it 
Ph.D. Students at other institutions
Name  Surname  Affiliation 

Angelo A.  Casulli  SNS, Pisa 
Marco  Feder  SISSA, Trieste 
Ph.D. Theses supervised by members of the group
awarded by the University of Pisa
Year  Name  Surname  Title of the Thesis  Supervisor(s) 

2012  Vanni  Noferini  Polynomial Eigenproblems: a RootFinding Approach  Dario Andrea Bini and Luca Gemignani 
2007  Bruno  Iannazzo  Numerical Solution of Certain Nonlinear Matrix Equations  Dario Andrea Bini 
2004  Manuela  Bagnasco  Il metodo QR per matrici semiseparabili: aspetti teorici e computazionali  Dario Andrea Bini 
1998  Beatrice  Meini  Fast Algorithms For The Numerical Solution of Structured Markov Chains  Dario Andrea Bini 
1997  Giuseppe  Fiorentino  Tau Matrices and Generating Functions for Solving Toeplitz Systems  Dario Andrea Bini 
1994  Enrico  Bozzo  Matrix Algebras and Discrete Transforms  Dario Andrea Bini 
awarded by another institution
Year  Name  Surname  Title of the Thesis  Institution  Supervisor(s) 

2017  Stefano  Massei  Exploiting rank structures in the numerical solution of Markov chains and matrix functions  SNS, Pisa  Dario Andrea Bini 
2015  Leonardo  Robol  Exploiting rank structures for the numerical treatment of matrix polynomials  SNS, Pisa  Dario Andrea Bini 
2010  Federico Giovanni  Poloni  Algorithms for quadratic matrix and vector equations  SNS, Pisa  Dario Andrea Bini 
2007  Paola  Boito  Structured Matrix Based Methods for Approximate Polynomial GCD  SNS, Pisa  Dario Andrea Bini 
1995  Stefano  Serra  Analisi di proprietà spettrali di matrici di Toeplitz ed applicazioni ai metodi di gradiente coniugato precondizionato per certe classi di sistemi lineari strutturati  Università degli Studi di Milano (La Statale)  Dario Andrea Bini 
Grants
Current

Dipartimento di Eccellenza 20232027 (Progetto Ministeriale Nazionale)
Coordinator of the Research Unit: Matteo Novaga
Project period: Jan 01, 2023 – Dec 31, 2027

Lowrank Structures and Numerical Methods in Matrix and Tensor Computations and their Application (Prin 2022)
Principal Investigator: Valeria Simoncini
Coordinator of the Research Unit: Beatrice Meini
Project period: Sep 28, 2023 – Sep 27, 2025

Immersed methods for multiscale and multiphysics problems (IMMEDIATE) (Prin 2022)
Principal Investigator: Paolo Zunino
Coordinator of the Research Unit: Luca Heltai
Project period: Sep 28, 2023 – Sep 27, 2025

Model reduction methods for modal analysis in computational mechanics (KAUST Opportunity Fund Program (OFP))
Principal Investigator: Daniele Boffi
Coordinator of the Research Unit: Luca Heltai
Project period: Jan 01, 2024 – Jun 30, 2025

HighPerformance Mathematics (MUR – FFO Finalizzato)
Principal Investigator: Fabio Durastante
Project period: Jan 01, 2024 – Jul 31, 2024
Past

Analisi di reti complesse: dalla teoria alle applicazioni (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Principal Investigator: Federico Giovanni Poloni
Project period: Jul 07, 2020 – Dec 31, 2022

Metodi lowrank per problemi di algebra lineare con struttura datasparse (Progetto Giovani GNCS)
Principal Investigator: Leonardo Robol
Project period: Mar 09, 2020 – Dec 31, 2021
Visitors
Prospective
First Name  Last Name  Affiliation 

Jie  Meng  Ocean University of China (OUC) 
Current
First Name  Last Name  Affiliation  Building  Floor  Office 

Xu  Li  Lanzhou University of Technology  Building A  First floor  218 
Grouped by year
2024
First Name  Last Name  Affiliation  From  To 

Ivan  Bioli  École Polytechnique Fédérale de Lausanne (EPFL)  Sep 18, 2023  Jan 19, 2024 
Elena  Celledoni  Norwegian University of Science and Technology  Apr 02, 2024  Apr 05, 2024 
Virginie  Ehrlacher  École des Ponts ParisTech  Apr 02, 2024  Apr 05, 2024 
Patrick  Farrell  University of Oxford  Apr 02, 2024  Apr 05, 2024 
Dario  Fasino  Università degli Studi di Udine  Feb 05, 2024  Feb 08, 2024 
Salvatore  Filippone  Università degli Studi di Roma Tor Vergata  May 22, 2024  May 23, 2024 
Martin  Gander  Université de Genève  Apr 02, 2024  Apr 05, 2024 
Miryam  Gnazzo  Gran Sasso Science Institute  Feb 11, 2024  Feb 17, 2024 
Guy  Latouche  Université libre de Bruxelles  Feb 11, 2024  Feb 16, 2024 
Xu  Li  Lanzhou University of Technology  Sep 01, 2023  Aug 31, 2024 
Christian  Lubich  Universität Tübingen  Apr 02, 2024  Apr 05, 2024 
Jie  Meng  Ocean University of China (OUC)  Jul 10, 2024  Jul 25, 2024 
Raf  Vandebril  Katholieke Universiteit Leuven  Apr 08, 2024  Apr 12, 2024 
2023
First Name  Last Name  Affiliation  From  To 

Ivan  Bioli  École Polytechnique Fédérale de Lausanne (EPFL)  Sep 18, 2023  Jan 19, 2024 
Alice  Cortinovis  Stanford University  Jun 08, 2023  Jun 10, 2023 
Martina  Iannacito  Katholieke Universiteit Leuven  Nov 02, 2023  Nov 27, 2023 
Xu  Li  Lanzhou University of Technology  Sep 01, 2023  Aug 31, 2024 
Chiara  Mauro  Universidad Complutense de Madrid  Jul 03, 2023  Jul 14, 2023 
Aleksey Yordanov  Nikolov  Technical University of Sofia  Feb 04, 2023  Mar 04, 2023 
Stoyan  Popov  Technical University of Sofia  Feb 04, 2023  Mar 04, 2023 
Raf  Vandebril  Katholieke Universiteit Leuven  Apr 13, 2023  Apr 13, 2023 
Lu  Xia  Eindhoven University of Technology  Jul 10, 2023  Jul 21, 2023 
2022
First Name  Last Name  Affiliation  From  To 

Gianluca  Ceruti  École Polytechnique Fédérale de Lausanne (EPFL)  Dec 12, 2022  Dec 17, 2022 
Bruno  Iannazzo  Università degli Studi di Perugia  Dec 09, 2022  Dec 12, 2022 
Stefano  Massei  Eindhoven University of Technology  Feb 07, 2022  Mar 06, 2022 
Mariarosa  Mazza  Università degli Studi dell’Insubria  Nov 28, 2022  Dec 01, 2022 
Aaron  Melman  Santa Clara University  Nov 09, 2022  Nov 10, 2022 