Research
My research activities are divided into different strands and projects.
The publications are reported by subject area and project in descending
temporal order.
The same information can be found on either my
Google Scholar Profile, my
ResearchGate, or my
Orcid profile.
Solving Linear Algebra problems is a core task in pretty much every Scientific
Challenges, from the numerical solution of partial differential equations, to the
new problems arising in the analysis of larger and larger set of data. Moreover,
we are fastly moving towards the use of exascaleenabled HPC platforms, thus
we need to be capable of pushing our algorithms (and codes) to work efficiently
in this environment. This first research line deals with this type of issue, and
more specifically with the task of solving linear systems of equation,
$$ A \mathbf{x} = \mathbf{b}, \qquad A \in \mathbb{R}^{N \times N}, \; \mathbf{x},\mathbf{b} \in \mathbb{R}^{N}, $$
in which $A$ is a
large, $N \geq 10^9$, and
sparse,
number on nonzero entries proportional to $N$, matrix. The key instruments are
here
Krylov Subspace Methods,
Geometric and
Algebraic Multigrid
preconditioners, and the application of update strategies to deal with more than
one linear system.
Preconditioning for large and sparse linear systems 

D'Ambra, Pasqua; Durastante, Fabio;
Filippone, Salvatore. AMG preconditioners for
Linear Solvers at Extreme Scale. (2020). Submitted.
arXiv preprint arXiv:2006.16147

D'Ambra, Pasqua; Durastante, Fabio;
Filippone, Salvatore. On the quality of matchingbased
aggregates for algebraic coarsening of SPD matrices in AMG. (2020). Submitted.
arXiv preprint arXiv:2001.09969

Bertaccini, Daniele; Durastante, Fabio.
Iterative methods and preconditioning for large and sparse linear systems
with applications. Monographs and Research Notes in Mathematics.
CRC Press, Boca Raton, FL, 2018. xviii+353 pp.
ISBN: 9781498764162
MR3793630

Bertaccini, Daniele; Donatelli, Marco;
Durastante, Fabio; SerraCapizzano, Stefano.
Optimizing a multigrid RungeKutta smoother for variablecoefficient
convectiondiffusion equations. Linear Algebra Appl. 533 (2017),
507535. MR3695922

Bertaccini, Daniele; Durastante, Fabio.
Interpolating preconditioners for the solution of sequence of linear systems.
Comput. Math. Appl. 72 (2016), no. 4, 11181130.
MR3529065


Fractional calculus is that subset of Analysis that studies the possibilities of
defining differential operator of real or complex order. After the definition of
such objects, the needs of numerically approximating the solution of differential
equations defined in terms of them immediately arise. The following papers deal
with the efficient numerical solution of different formulations of such problems.

Durastante, Fabio. Efficient solution of timefractional
differential equations with a new adaptive multiterm discretization of
the generalized CaputoDzherbashyan derivative.
Calcolo 56 (2019), no. 4, Paper No. 36, 24 pp.
MR4015149

Bertaccini, Daniele; Durastante, Fabio.
Efficient preconditioner updates for semilinear spacetime fractional
reactiondiffusion equations. Structured matrices in numerical linear algebra,
285302, Springer INdAM Ser., 30, Springer, Cham, 2019.
MR3931580

Bertaccini, D.; Durastante, F.
Block structured preconditioners in tensor form for the allatonce
solution of a finite volume fractional diffusion equation.
Appl. Math. Lett. 95 (2019), 9297.
MR3936774

Bertaccini, Daniele; Durastante, Fabio.
Limited memory block preconditioners for fast solution of fractional partial
differential equations. J. Sci. Comput. 77
(2018), no. 2, 950970.
MR3860196

Bertaccini, Daniele; Durastante, Fabio.
Solving mixed classical and fractional partial differential equations
using shortmemory principle and approximate inverses.
Numer. Algorithms 74 (2017), no. 4, 10611082.
MR3626328

Numerical treatment of Fractional Differential Equations 
Numerical optimal control theory is a part of numerical optimization that deals
with the problem of finding a
control for either a boundary value problem
or a dynamical system over a period of time in such a way that an objective
function is minimized/maximized. The prototypical problem of this type has the form
$$ (y^*,u^*) = \arg\min_{u \in \mathcal{U}} J(y,u) = \frac{1}{2}\y  y_z\_2^2 + \frac{\lambda}{2} \ u \_2^2, \text{ s.t. } e(y,u) = 0,$$
where $\mathcal{U}$ is a function space containing the admissible controls, $y_z$
a target state, and $e(y,u)=0$ a boundary value problem that the quantity $y$ and
$u$ have to satisfy. The numerical solution of such problem is challenging in many
ways, it mixes together optimization, differential and linear algebra problems in
a nontrivial way.
Numerical optimal control of Partial Differential Equations 

Durastante, Fabio; Furci, Isabella.
Spectral analysis of saddlepoint matrices from optimization problems with
elliptic PDE constraints. Electronic Journal of Linear Algebra
36 (2020), no. 36, 73798.
Open Access.

Durastante, Fabio; Cipolla, Stefano.
Fractional PDE constrained optimization: box and sparse constrained problems.
Numerical methods for optimal control problems,
111135, Springer INdAM Ser., 29, Springer, Cham, 2018.
MR3889795

Cipolla, Stefano; Durastante, Fabio.
Fractional PDE constrained optimization: an optimizethendiscretize
approach with LBFGS and approximate inverse preconditioning.
Appl. Numer. Math. 123 (2018), 4357.
MR3711990

A matrix function is simply a function which maps a matrix to another matrix,
indeed to maintain interesting properties one have to extend the concept of
scalar function to a matrix in a clever way. If, e.g., $f$ is a function
admitting a Taylor expansion with radius of convergence $r$, and $A$ is a matrix
such $\A\ < r$ for any compatible norm $\\cdot\$, then $f(A)$ can be
defined as
$$ f(A) = \sum_{j=0}^{+\infty} c_k A^k, $$
for $c_k$ the Taylor coefficients of $f$. Equivalently, if $A$ is diagonalizable,
and $f$ is defined on the eigenvalues of $A$, then
$$ f(A) = X f(\Lambda) X^{1}, \qquad A = X \Lambda X^{1}, \; f(\Lambda)_{i,i} = f(\lambda_i(A)), $$
and in several other ways for more general $A$ and $f$. In any case, one of the
principal numerical task that one can face is the computation of MatrixFunctionvector
products, i.e., $\mathbf{y} = f(A)\mathbf{x}$, for $A$ a
large and sparse
matrix. The following line of research deals with these kind of problems by
different techniques, and for different purposes (solution of Fractional
differential equations, network analysis,
etc.).

Bertaccini, D.; Durastante, F.
Computing function of large matrices by a preconditioned rational
Krylov method. Numerical Mathematics and Advanced Applications
ENUMATH 2019. Lecture Notes in Computational
Science and Engineering. Springer International Publishing, 2020.
In press

Bertaccini, D.; Durastante, F.
Computing functions of very large matrices with small TT/QTT ranks
by quadrature formulas. J. Comput. Appl. Math. 370 (2020),
112663, 15 pp.
MR4046619

Aceto, L.; Bertaccini, D.;
Durastante, F.; Novati, P.
Rational Krylov methods for functions of matrices with applications
to fractional partial differential equations.
J. Comput. Phys. 396 (2019), 470482.
MR3989621

Bertaccini, D.; Popolizio, M.;
Durastante, F. Efficient approximation of functions
of some large matrices by partial fraction expansions.
Int. J. Comput. Math. 96 (2019), no. 9,
17991817.
MR3960343

MatrixFunctions 
Network science is the studies of complex networks that are nothing more than
graphs that are both not random and not structured, e.g., one usually do not
consider Cayley graphs encoding the structure of an abstract group. Examples
are telecommunication networks, computer networks, biological networks,
social networks, and many others. In each case every actor is represented by
a vertex of the graph and their connections as an edge. My interest in this
field lies both in the modelling and the computational aspects, specifically
in the task of extracting information on the networked phenomena by looking at
the topology of the graph.
Graphs and Network Science: ComplexNetworks 

Bertaccini, Daniele; Durastante, Fabio.
Nonlocal diffusion of variable order on graphs. Submitted.

Arrigo, Francesca; Durastante, Fabio.
MittagLeffler functions and their applications in network science.
Submitted.
arXiv preprint arXiv:2103.12559

Bianchi, Davide; Donatelli, Marco;
Durastante, Fabio; Mazza, Mariarosa.
Compatibility, embedding and regularization of nonlocal random walks
on graphs. Submitted. arXiv preprint arXiv:2101.00425.

Cipolla, Stefano; Durastante, Fabio;
Tudisco, Francesco. Nonlocal pagerank.
ESAIM Math. Model. Numer. Anal. 55 (2021),
no. 1, 7797.
MR4216832

Benzi, Michele; Bertaccini, Daniele;
Durastante, Fabio; Simunec, Igor.
Nonlocal network dynamics via fractional graph Laplacians.
J. Complex Netw. 8 (2020), no. 3, cnaa017,
29 pp. MR4130854

Inverse problems are involved in the task of inferring the parameters of model
from a set of (possibly noisy) measurement requires. These type of problems
can be difficult to solve for several reason, e.g., one could be dealing with a
lack of unicity, that is: different values of parameters are consistent with the
measurement, and discovering the parameters may be challenging from the computational
point of view. I have done some work in using linear algebra techniques for
a particular type of inverse problem that is
image denoise and deblurring,
that is the process of removing blurring artifacts and noise from images (e.g.,
somebody shaked the camera while we were taking the photo).

Cipolla, Stefano; Donatelli, Marco;
Durastante, Fabio. Regularization of Inverse
Problems by an Approximate Matrix–Function Technique.
Numer. Algorithms. DOI. 10.1007/s1107502101076y. In Press.

Cipolla, Stefano; Di Fiore, Carmine;
Durastante, Fabio; Zellini, Paolo.
Regularizing properties of a class of matrices including the optimal and
the superoptimal preconditioners. Numer. Linear Algebra Appl.
26 (2019), no. 2, e2225, 17 pp.
MR3911025

Regularization and Inverse Problems

Every so often a mathematician with a solution meets a person with that
problem, this is what happens.

Mauro, Chiara; Durastante, Fabio.
Evaluating visibility at sea: Instrumental data and historical nautical
records. Mount Etna from the Calabrian Ionian coast (Italy).
The Journal of Island and Coastal Archaeology (2020),
122 pp.

Applications, intersections, contaminations 
Projects
 Funding for
15000€ for the organization of the INdAM workshop on "Fractional Differential Equations:
Modeling, Discretization, and Numerical Solvers" (
https://fractionalworkshop.github.io/). Organizing committee:
Angelamaria Cardone, Marco Donatelli, Fabio Durastante, Roberto Garrappa,
Mariarosa Mazza, Marina Popolizio.
 Member of
the WorkPackage 3 (WP3) "Scalable Solvers" in the European
EoCoEII project.

Funding for 1200€ for the 2020 GNCSINDAM Project "Nonlocal models for
the analysis of complex networks" with Principal Investigator F. Durastante

Funding for 3400€ for the 2019 GNCSINDAM Project "Tecniche innovative e
parallele per sistemi lineari e non lineari di grandi dimensioni, funzioni ed equazioni
matriciali ed applicazioni" with Principal Investigator D. Bertaccini together
with L. Bergamaschi, P. D’Ambra, M. Ferronato, C. Janna, F. Marcuzzi, A.
Martinez Calomardo, V. Simoncini, M. Porcelli, D. Brandoni

Funding for 4500€ for the 2018 GNCSINDAM project "Tecniche innovative
per problemi di algebra lineare" with Principal Investigator D. Bertaccini together
with D. Bini, G. Del Corso, D. Fasino, L. Gemignani, B. Iannazzo, N.
Mastronardi, B. Meini, F. Poloni, A. Fayyaz, P. Boito, M. Fasi, I. Furci, S.
Massei and L. Robol.
Collaboration Network
Most of the work described here has been done with several collaborators. I am
always working on expanding my network of collaborators by looking for people to
do interesting things with.
Bibliometric indexes
Not everything that counts can be counted, and not everything that can be counted counts.


(Scopus) HIndex: 6 
(Scopus) Subject Areas 
Editorial Work
I serve as editor for the
Journal of Mathematical Modeling (J. Math. Model.)
I have served as reviewer for manuscripts submitted to the following journals.