Papers by Fabio Durastante
Journal of Maritime Archaeology, Jan 8, 2024
The practice of nocturnal navigation in the Mediterranean Sea could be inferred from both archaeo... more The practice of nocturnal navigation in the Mediterranean Sea could be inferred from both archaeological and written records. While there is sufficient proof that the ships and their crew were quite familiar with nighttime sailing, current scholarship has not satisfactorily investigated how the reduction of visibility could have affected the nautical practice. For this reason, the aim of this contribution is twofold: (1) to evaluate to what extent visibility was reduced at night, and ( ) to understand what kind of strategies (if any) could be put in place to overcome the difficulties of a low level of visibility. Amongst the strategies, we will also assess the impact on visibility of fixed and portable lighting devices, such as torches and pierced amphoras, as documented by the archaeological and literary evidence.
arXiv (Cornell University), Mar 23, 2021
We describe a complete theory for walk-based centrality indices in complex networks defined in te... more We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag-Leffler functions. This overarching theory includes as special cases wellknown centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag-Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modelling and computational issues, and provide guidelines on parameter selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided.
arXiv (Cornell University), Dec 19, 2023
Given a stochastic matrix P partitioned in four blocks P ij , i, j = 1, 2, Kemeny's constant κ(P ... more Given a stochastic matrix P partitioned in four blocks P ij , i, j = 1, 2, Kemeny's constant κ(P ) is expressed in terms of Kemeny's constants of the stochastic complements P 1 = P 11 + P 12 (I -P 22 ) -1 P 21 , and P 2 = P 22 + P 21 (I -P 11 ) -1 P 12 . Specific cases concerning periodic Markov chains and Kronecker products of stochastic matrices are investigated. Bounds to Kemeny's constant of perturbed matrices are given. Relying on these theoretical results, a divide-and-conquer algorithm for the efficient computation of Kemeny's constant of graphs is designed. Numerical experiments performed on real world problems show the high efficiency and reliability of this algorithm.
Journal of Computational Physics, 2019
In this paper we propose a new choice of poles to define reliable rational Krylov methods. These ... more In this paper we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising.
International Journal of Computer Mathematics: Computer Systems Theory
Some aspects of nonlocal dynamics on directed and undirected networks for an initial value proble... more Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. This is a new extension to non-stationary behavior of a class of non-local phenomena on complex networks for which both directed and undirected graphs are considered. Under appropriate assumptions, the existence, uniqueness, and uniform asymptotic stability of the solutions of the underlying initial value problem are proved. Some examples giving a sample of the behavior of the dynamics are also included.
SIAM Journal on Matrix Analysis and Applications, 2021
We describe a complete theory for walk-based centrality indices in complex networks defined in te... more We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag-Leffler functions. This overarching theory includes as special cases wellknown centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag-Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modelling and computational issues, and provide guidelines on parameter selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided.
arXiv (Cornell University), Jul 26, 2023
We propose two approaches, based on Riemannian optimization, for computing a stochastic approxima... more We propose two approaches, based on Riemannian optimization, for computing a stochastic approximation of the pth root of a stochastic matrix A. In the first approach, the approximation is found in the Riemannian manifold of positive stochastic matrices. In the second approach, we introduce the Riemannian manifold of positive stochastic matrices sharing with A the Perron eigenvector and we compute the approximation of the pth root of A in such a manifold. This way, differently from the available methods based on constrained optimization, A and its pth root approximation share the Perron eigenvector. Such a property is relevant, from a modeling point of view, in the embedding problem for Markov chains. The extended numerical experimentation shows that, in the first approach, the Riemannian optimization methods are generally faster and more accurate than the available methods based on constrained optimization. In the second approach, even though the stochastic approximation of the pth root is found in a smaller set, the approximation is generally more accurate than the one obtained by standard constrained optimization.
arXiv (Cornell University), Apr 19, 2023
This work deals with the numerical solution of systems of oscillatory second-order differential e... more This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation.
arXiv (Cornell University), Dec 9, 2021
We consider here a cell-centered finite difference approximation of the Richards equation in thre... more We consider here a cell-centered finite difference approximation of the Richards equation in three dimensions, averaging for interface values the hydraulic conductivity = (), a highly nonlinear function, by arithmetic, upstream and harmonic means. The nonlinearities in the equation can lead to changes in soil conductivity over several orders of magnitude and discretizations with respect to space variables often produce stiff systems of differential equations. A fully implicit time discretization is provided by backward Euler one-step formula; the resulting nonlinear algebraic system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the solution of a sequence of linear systems involving Jacobian matrices. We prove some new results concerning the distribution of the Jacobians eigenvalues and the explicit expression of their entries. Moreover, we explore some connections between the saturation of the soil and the ill conditioning of the Jacobians. The information on eigenvalues justifies the effectiveness of some preconditioner approaches which are widely used in the solution of Richards equation. We also propose a new software framework to experiment with scalable and robust preconditioners suitable for efficient parallel simulations at very large scales. Performance results on a literature test case show that our framework is very promising in the advance towards realistic simulations at extreme scale.
arXiv (Cornell University), Jan 27, 2020
In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG) method for general symm... more In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG) method for general symmetric positive-definite matrices. The method relies on an aggregation algorithm, named coarsening based on compatible weighted matching, which exploits the interplay between the principle of compatible relaxation and the maximum product matching in undirected weighted graphs. The results are based on a general convergence analysis theory applied to the class of AMG methods employing unsmoothed aggregation and identifying a quality measure for the coarsening; similar quality measures were originally introduced and applied to other methods as tools to obtain good quality aggregates leading to optimal convergence for M-matrices. The analysis, as well as the coarsening
Research Square (Research Square), Jun 16, 2023
In this paper we describe some work aimed at upgrading the Alya code with up-to-date parallel lin... more In this paper we describe some work aimed at upgrading the Alya code with up-to-date parallel linear solvers capable of achieving reliability, efficiency and scalability in the computation of the pressure field at each time step of the numerical procedure for solving a LES formulation of the incompressible Navier-Stokes equations. We developed a software module in the Alya's kernel to interface the libraries included in the current version of PSCToolkit, a framework for the iterative solution of sparse linear systems on parallel distributed-memory computers by Krylov methods coupled to Algebraic MultiGrid preconditioners. The Toolkit has undergone some extensions within the EoCoE-II project with the primary goal to face the exascale challenge. Results on a realistic benchmark for airflow simulations in wind farm applications show that the PSCToolkit solvers significantly outperform the original versions of the Conjugate Gradient method available in the Alya kernel in terms of scalability and parallel efficiency and represent a very promising software layer to move the Alya code towards exascale.
arXiv (Cornell University), Jun 29, 2020
Linear solvers for large and sparse systems are a key element of scientific applications, and the... more Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work where we examine some recent developments in a package of AMG preconditioners to improve efficiency, scalability and robustness on extreme scale problems. The main novelty is the design and implementation of a parallel coarsening algorithm based on aggregation of unknowns employing weighted graph matching techniques; this is a completely automated procedure, requiring no information from the user, and applicable to general symmetric positive definite (s.p.d.) matrices. The new coarsening algorithm improves in terms of numerical scalability at low operator complexity over decoupled aggregation algorithms available in previous releases of the package. The preconditioners package is built on the parallel software framework PSBLAS, which has also been updated to progress towards exascale. We present weak scalability results on one of the most powerful supercomputer in Europe for linear systems with sizes up to O(10 10) unknowns.
Software impacts, Mar 1, 2023
arXiv (Cornell University), Sep 19, 2017
Several applicative problems require the evaluation of functions of large and sparse matrices. We... more Several applicative problems require the evaluation of functions of large and sparse matrices. We propose some accelerating techniques for computing $\Psi(A)$ or $\Psi(A)b$ where $\Psi$ is approximated by a rational expansion. $A$ can be not Hermitian or nonsymmetric with real entries and $b$ is a vector. All strategies have good parallel potentialities. The underlying proposals are inspired by an updating framework for incomplete factorizations. Our claims will be confirmed by several numerical tests showing the effectiveness of the approach in a variety of applications.
International Journal of Computer Mathematics: Computer Systems Theory, Jul 3, 2022
Some aspects of nonlocal dynamics on directed and undirected networks for an initial value proble... more Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. This is a new extension to non-stationary behavior of a class of non-local phenomena on complex networks for which both directed and undirected graphs are considered. Under appropriate assumptions, the existence, uniqueness, and uniform asymptotic stability of the solutions of the underlying initial value problem are proved. Some examples giving a sample of the behavior of the dynamics are also included.
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Papers by Fabio Durastante
Relative to the paper:
Bertaccini, Daniele, and Fabio Durastante. "Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses." Numerical Algorithms (2016): 1-22.
$$
\left\lbrace\begin{array}{l}
\displaystyle \min J(y,u) = \frac{1}{2}\|y - z_d\|_2^2 + \frac{\lambda}{2}\|u\|_2^2, \\
\;\;\text{subject to }e(y,u) = 0.
\end{array}\right. \mathbb{R} \ni \lambda > 0
$$
where $J$ and $e$ are two continuously Fr\'{e}chet derivable functionals such that,
$
J\,:\;Y\times U \rightarrow \mathbb{R}, \;\; e:\;Y\times U \rightarrow W,
$
with $Y,U$ and $W$ {reflexive} Banach spaces and $z_d \in U$ given. For $e(y,u)=0$ we select a Fraction Partial Differential Equation, either the \emph{Fractional Advection Dispersion Equation} or the two--dimensional \emph{Riesz Space Fractional Diffusion equation}.
\par\noindent
Indeed, many problems that exhibit \emph{non--local properties} have been modeled using fractional calculus, e.g., anomalous diffusion (i.e diffusion not accurately modeled by the usual advection--dispersion equation), the dynamics of viscoelastic and polymeric materials and many others \cite{podlubny}. We focus on extending the existing strategies for classical PDE constrained optimization to the fractional case. We will present both a theoretical and experimental analysis of the problem in an algorithmic framework based on the L--BFGS method coupled with a Krylov subspace solver.