# 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 exascale-enabled 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 non-zero 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. On the quality of matching-based aggregates for algebraic coarsening of SPD matrices in AMG. (2020). Submitted. arXiv preprint arXiv:2001.09969 D'Ambra, Pasqua; Durastante, Fabio; Filippone, Salvatore. AMG preconditioners for Linear Solvers at Extreme Scale. (2021). SIAM J. Scientific Computing. In Press. arXiv preprint arXiv:2006.16147 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: 978-1-4987-6416-2 MR3793630 Bertaccini, Daniele; Donatelli, Marco; Durastante, Fabio; Serra-Capizzano, Stefano. Optimizing a multigrid Runge-Kutta smoother for variable-coefficient convection-diffusion equations. Linear Algebra Appl. 533 (2017), 507--535. MR3695922 Bertaccini, Daniele; Durastante, Fabio. Interpolating preconditioners for the solution of sequence of linear systems. Comput. Math. Appl. 72 (2016), no. 4, 1118--1130. 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 time-fractional differential equations with a new adaptive multi-term discretization of the generalized Caputo-Dzherbashyan derivative. Calcolo 56 (2019), no. 4, Paper No. 36, 24 pp. MR4015149 Bertaccini, Daniele; Durastante, Fabio. Efficient preconditioner updates for semilinear space-time fractional reaction-diffusion equations. Structured matrices in numerical linear algebra, 285--302, Springer INdAM Ser., 30, Springer, Cham, 2019. MR3931580 Bertaccini, D.; Durastante, F. Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation. Appl. Math. Lett. 95 (2019), 92--97. MR3936774 Bertaccini, Daniele; Durastante, Fabio. Limited memory block preconditioners for fast solution of fractional partial differential equations. J. Sci. Comput. 77 (2018), no. 2, 950--970. MR3860196 Bertaccini, Daniele; Durastante, Fabio. Solving mixed classical and fractional partial differential equations using short-memory principle and approximate inverses. Numer. Algorithms 74 (2017), no. 4, 1061--1082. 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 non-trivial way.
 Numerical optimal control of Partial Differential Equations Durastante, Fabio; Furci, Isabella. Spectral analysis of saddle-point matrices from optimization problems with elliptic PDE constraints. Electronic Journal of Linear Algebra 36 (2020), no. 36, 73-798. Open Access. Durastante, Fabio; Cipolla, Stefano. Fractional PDE constrained optimization: box and sparse constrained problems. Numerical methods for optimal control problems, 111--135, Springer INdAM Ser., 29, Springer, Cham, 2018. MR3889795 Cipolla, Stefano; Durastante, Fabio. Fractional PDE constrained optimization: an optimize-then-discretize approach with L-BFGS and approximate inverse preconditioning. Appl. Numer. Math. 123 (2018), 43--57. 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 Matrix-Function-vector 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. In: Vermolen F.J., Vuik C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. (2021) DOI: 10.1007/978-3-030-55874-1_33 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), 470--482. 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, 1799--1817. MR3960343 Matrix-Functions
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: Complex-Networks Bertaccini, Daniele; Durastante, Fabio. Nonlocal diffusion of variable order on graphs. Submitted. Arrigo, Francesca; Durastante, Fabio. Mittag-Leffler 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 non-local 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, 77--97. MR4216832 Benzi, Michele; Bertaccini, Daniele; Durastante, Fabio; Simunec, Igor. Non-local 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/s11075-021-01076-y. 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), 1-22 pp. Applications, intersections, contaminations

### Projects

• Member of the Work-Package 6 (WP6) "Applications and Use cases" in the European TEXTAROSSA project.
• 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 Work-Package 3 (WP3) "Scalable Solvers" in the European EoCoE-II project.
• Writer for the Deliverable: D3.2 Preliminary results and performance evaluation of Linear Algebra solvers'',
• Contributor for the Deliverable: D3.3 Updated results and new releases of LA solvers.''
• Funding for 1200€ for the 2020 GNCS-INDAM Project "Nonlocal models for the analysis of complex networks" with Principal Investigator F. Durastante
• Funding for 3400€ for the 2019 GNCS-INDAM 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 GNCS-INDAM 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) H-Index: 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.