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2022
Journal article
Open Access

A new methodology for effective definition and efficient evaluation of dependability-related properties is proposed. The analysis targets the systems composed of a large number of components, each one modeled implicitly through high-level formalisms, such as stochastic Petri nets. Since the component models are implicit, the reward structure that characterizes the dependability properties has to be implicit as well. Therefore, we present a new formalism to specify those reward structures. The focus here is on component models that can be mapped to stochastic automata with one or several absorbing states so that the system model can be mapped to a stochastic automata network with one or several absorbing states. Correspondingly, the new reward structure defined on each component's model is mapped to a reward vector so that the dependability-related properties of the system are expressed through a newly introduced measure defined starting from those reward vectors. A simple, yet representative, case study is adopted to show the feasibility of the method.

**See at: **
ISTI Repository | ieeexplore.ieee.org | CNR ExploRA

2021
Journal article
Open Access

A standard approach to calculate the roots of a univariate polynomial is to compute the eigenvalues of an associated confederate matrix instead, such as, for instance, the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure-exploiting QR algorithm. In [J. L. Aurentz et al., Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942-973] it was shown that a structure-exploiting QR algorithm for companion matrices leads to a structured backward error in the companion matrix. The proof relied on decomposing the error into two parts: a part related to the recurrence coefficients of the basis (a monomial basis in that case) and a part linked to the coefficients of the original polynomial. In this article we prove that the analysis can be extended to other classes of comrade matrices. We first provide an alternative backward stability proof in the monomial basis using structured QR algorithms; our new point of view shows more explicitly how a structured, decoupled error in the confederate matrix gets mapped to the associated polynomial coefficients. This insight reveals which properties have to be preserved by a structure-exploiting QR algorithm to end up with a backward stable algorithm. We will show that the previously formulated companion analysis fits into this framework, and we analyze in more detail Jacobi polynomials (comrade matrices) and Chebyshev polynomials (colleague matrices).

**See at: **
Aaltodoc Publication Archive | arXiv.org e-Print Archive | Electronic Transactions on Numerical Analysis | Electronic Transactions on Numerical Analysis | epub.oeaw.ac.at | ISTI Repository | CNR ExploRA | doi.org

2020
Journal article
Open Access

Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software packages for such matrices often focus on linear systems, their scope of applications is in fact much wider and includes, for example, matrix functions and eigenvalue problems. In this work, we present a new MATLAB toolbox called hm-toolbox, which encompasses this versatility with a broad set of tools for HODLR and HSS matrices, unmatched by existing software. While mostly based on algorithms that can be found in the literature, our toolbox also contains a few new algorithms as well as novel auxiliary functions. Being entirely based on MATLAB, our implementation does not strive for optimal performance. Nevertheless, it maintains the favorable complexity of hierarchical low-rank matrices and offers, at the same time, a convenient way of prototyping and experimenting with algorithms. A number of applications illustrate the use of the hm-toolbox.

**See at: **
arXiv.org e-Print Archive | SIAM Journal on Scientific Computing | ISTI Repository | SIAM Journal on Scientific Computing | doi.org | epubs.siam.org | Infoscience - EPFL scientific publications | CNR ExploRA

2020
Journal article
Open Access

We discuss the efficient computation of performance, reliability, and availability measures for Markov chains; these metrics - and the ones obtained by combining them, are often called performability measures. We show that this computational problem can be recasted as the evaluation of a bilinear form induced by appropriate matrix functions, and thus solved by leveraging the fast methods available for this task.

**See at: **
arXiv.org e-Print Archive | Journal of Computational and Applied Mathematics | ISTI Repository | Journal of Computational and Applied Mathematics | doi.org | CNR ExploRA | www.sciencedirect.com

2020
Journal article
Open Access

We consider a class of linear matrix equations involving semi-infinite matrices which have a quasi-Toeplitz structure. These equations arise in different settings, mostly connected with PDEs or the study of Markov chains such as random walks on bidimensional lattices. We present the theory justifying the existence of the solution in an appropriate Banach algebra which is computationally treatable, and we propose several methods for computing them. We show how to adapt the ADI iteration to this particular infinite dimensional setting, and how to construct rational Krylov methods. Convergence theory is discussed, and numerical experiments validate the proposed approaches.

**See at: **
arXiv.org e-Print Archive | Linear Algebra and its Applications | ISTI Repository | Linear Algebra and its Applications | doi.org | CNR ExploRA | www.sciencedirect.com

2020
Conference article
Open Access

The goal of the paper is to investigate the role of soil-structure interaction in modeling the dy-namic behavior of masonry towers. The study, conducted on the bell tower of the Basilica of San Frediano in Lucca (Italy), is based on both experimental and numerical results. The former were collected during an experimental campaign carried out on the tower using seismometric stations, while the latter have been obtained via the modal analysis and model updating proce-dures implemented in the finite element code NOSA-ITACA. Combining experimental and nu-merical outcomes made it possible to assess the influence of the soil, modeled as a system of elastic springs, on the natural frequencies of the tower. Finite element models of the tower have been calibrated by taking the presence of the adjacent church into account and choosing differ-ent unknown parameters, including the soil stiffness.

**See at: **
eurodyn2020.org | ISTI Repository | CNR ExploRA

2020
Journal article
Open Access

Finite element model updating of a structure made of linear elastic materials is based on the solution of a minimization problem. The goal is to find some unknown parameters of the finite element model (elastic moduli, mass densities, constraints and boundary conditions) that minimize an objective function which evaluates the discrepancy between experimental and numerical dynamic properties. The objective function depends nonlinearly on the parameters and may have multiple local minimum points. This paper presents a numerical method able to find a global minimum point and assess its reliability. The numerical method has been tested on two simulated examples - a masonry tower and a domed temple - and validated via a generic genetic algorithm and a global sensitivity analysis tool. A real case study monitored under operational conditions has also been addressed, and the structure's experimental modal properties have been used in the model updating procedure to estimate the mechanical properties of its constituent materials.

**See at: **
arXiv.org e-Print Archive | Mechanical Systems and Signal Processing | ISTI Repository | Mechanical Systems and Signal Processing | doi.org | CNR ExploRA | www.sciencedirect.com

2019
Journal article
Open Access

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost. The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of k matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well. Numerical experiments illustrate stability and efficiency of the proposed methods.

**See at: **
arXiv.org e-Print Archive | Mathematics of Computation | Lirias | ISTI Repository | CNR ExploRA | www.ams.org | doi.org

2019
Journal article
Open Access

Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$ and $\lVert E \rVert_2$ are finite. These properties allow to approximate any QT-matrix, within any given precision, by means of a finite number of parameters. QT-matrices, equipped with the norm $\lVert A \rVert_{\mathcal QT}=\alpha\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, for $\alpha = (1+\sqrt 5)/2$, are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT-matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

**See at: **
arXiv.org e-Print Archive | Numerical Algorithms | Numerical Algorithms | doi.org | link.springer.com | CNR ExploRA

2019
Contribution to book
Open Access

When Fiedler published his "A note on Companion matrices" in 2003 in Linear Algebra and its Applications, he could not have foreseen the significance of this elegant factorization of a companion matrix into essentially two-by-two Gaussian transformations, which we will name \emph{(scalar) elementary Fiedler factors}. Since then, researchers extended these results and studied the various resulting linearizations, the stability of Fiedler companion matrices, factorizations of block companion matrices, Fiedler pencils, and even looked at extensions to non-monomial bases. In this chapter, we introduce a new way to factor block Fiedler companion matrices into the product of scalar elementary Fiedler factors. We use this theory to prove that, e.g., a block (Fiedler) companion matrix can always be written as the product of several scalar (Fiedler) companion matrices. We demonstrate that this factorization in terms of elementary Fiedler factors can be used to construct new linearizations. Some linearizations have notable properties, such as low band-width, or allow for factoring the coefficient matrices into unitary-plus-low-rank matrices. Moreover, we will provide bounds on the low-rank parts of the resulting unitary-plus-low-rank decomposition. To present these results in an easy-to-understand manner we rely on the flow-graph representation for Fiedler matrices recently proposed by Del Corso and Poloni in Linear Algebra and its Applications, 2017.

**See at: **
ISTI Repository | doi.org | link-springer-com-443.webvpn.fjmu.edu.cn | CNR ExploRA

2019
Report
Unknown

MOSCARDO - ICT technologies for structural monitoring of age-old constructions based on wireless sensor networks and drones. Description of the activities conducted by the Mechanics of Materials and Structures Laboratory of ISTI-CNR. Project Report n. 2.

**See at: **
CNR ExploRA

2019
Report
Open Access

We are concerned with the computation of the mean-time-to-failure(MTTF) for a large system of loosely interconnected components, mod-eled as continuous time Markov chains. In particular, we show that split-ting the local and synchronization transitions of the smaller subsystemsallows to formulate an algorithm for the computation of the MTTF whichis proven to be linearly convergent. Then, we show how to modify themethod to make it quadratically convergent, thus overcoming the difficul-ties for problems with convergent rate close to1.In addition, it is shown that this decoupling of local and synchroniza-tion transitions allows to easily represent all the matrices and vectors in-volved in the method in the tensor-train (TT) format -- and we providenumerical evidence showing that this allows to treat large problems withup to billions of states -- which would otherwise be unfeasible.

**See at: **
dcl.isti.cnr.it | ISTI Repository | CNR ExploRA

2019
Journal article
Open Access

Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of partial differential equations. In this work, we present and analyze a new algorithm, based on tensorized Krylov subspaces, for quickly updating the solution of such a matrix equation when its coefficients undergo low-rank changes. We demonstrate how our algorithm can be utilized to accelerate the Newton method for solving continuous-time algebraic Riccati equations. Our algorithm also forms the basis of a new divide-and-conquer approach for linear matrix equations with coefficients that feature hierarchical low-rank structure, such as hierarchically off-diagonal low-rank structures, hierarchically semiseparable, and banded matrices. Numerical experiments demonstrate the advantages of divide-and-conquer over existing approaches, in terms of computational time and memory consumption.

**See at: **
arXiv.org e-Print Archive | SIAM Journal on Scientific Computing | ISTI Repository | SIAM Journal on Scientific Computing | doi.org | epubs.siam.org | Infoscience - EPFL scientific publications | CNR ExploRA

2019
Journal article
Open Access

This paper describes a model updating procedure implemented in NOSA-ITACA, a finite-element (FE) code for the structural analysis of masonry constructions of historical interest. The procedure, aimed at matching experimental frequencies and mode shapes, allows for fine-tuning the calculations of the free parameters in the model. The numerical method is briefly described, and some issues related to its robustness are addressed. The procedure is then applied to a simple case study and two historical structures in Tuscany, the Clock Tower in Lucca and the Maddalena Bridge in Borgo a Mozzano.

**See at: **
Journal of Performance of Constructed Facilities | ISTI Repository | ascelibrary.org | Journal of Performance of Constructed Facilities | CNR ExploRA

2019
Report
Open Access

The ISTI Young Researcher Award (YRA) selects yearly the best young staff members working at Institute of Information Science and Technologies (ISTI). This award focuses on quality and quantity of the scientific production. In particular, the award is granted to the best young staff members (less than 35 years old) by assessing their scientific production in the year preceding the award. This report documents the selection procedure and the results of the 2019 YRA edition. From the 2019 edition on the award is named as "Matteo Dellepiane", being dedicated to a bright ISTI researcher who prematurely left us and who contributed a lot to the YRA initiative from its early start.

**See at: **
ISTI Repository | CNR ExploRA

2019
Conference article
Open Access

This paper describes the automated nonlinear model updating procedure for ma-sonry structures implemented in the NOSA-ITACA code. The algorithm, aimed at matchingnumerical and experimental natural frequencies and mode shapes, combines nonlinear staticanalysis, linear perturbation and modal analysis and allows fine-tuning the free parametersof the model. The numerical method is applied to two simple case studies, which prove itseffectiveness.

**See at: **
2019.compdyn.org | ISTI Repository | CNR ExploRA

2019
Contribution to conference
Open Access

The paper describes a preliminary experimental campaign conducted on The Matilde donjon in Livorno (Italy) aimed to characterize its dynamic behaviour, and the corresponding numerical analysis conducted on a FE model created via the NOSA-ITACA code.

**See at: **
ISTI Repository | CNR ExploRA

2019
Contribution to conference
Open Access

Combining ambient vibration monitoring and finite element (FE) modelling through suitable model updating procedures allows for obtaining an estimate of the boundary conditions and mechanical material properties of engineering structures. Application of FE model updating to historical buildings is relatively recent and involves the solution of a constrained minimum problem, whose objective function is generally expressed as the discrepancy between experimental and numerical quantities, such as natural frequencies and mode shapes. The paper presents an algorithm for FE model updating based on the construction of local parametric reduced-order models embedded in a trust-region scheme and implemented in NOSA-ITACA, a non-commercial FE code developed by the authors. The algorithm exploits the structure of the stiffness and mass matrices and the fact that only a few of the smallest eigenvalues have to be calculated. This new procedure enables to compute eigenvalues and eigenvectors cheaply and thus to solve the minimum problem very efficiently. Besides reducing the overall computation time of the numerical process and enabling the accurate analysis of large scale models with little effort, the proposed algorithm allows for getting information on both the reliability of the solution and its sensitivity to noisy experimental data. Some case studies are presented and discussed and the adoption of regularization techniques to recover meaningful solutions investigated.

**See at: **
ISTI Repository | CNR ExploRA

2019
Journal article
Open Access

A novel method for performing model updating on finite element models is presented. The approach is particularly tailored to modal analyses of buildings, by which the lowest frequencies, obtained by using sensors and system identification approaches, need to be matched to the numerical ones predicted by the model. This is done by optimizing some unknown material parameters (such as mass density and Young's modulus) of the materials and/or the boundary conditions, which are often known only approximately. In particular, this is the case when considering historical buildings. The straightforward application of a general-purpose optimizer can be impractical, given the large size of the model involved. In the paper, we show that, by slightly modifying the projection scheme used to compute the eigenvalues at the lowest end of the spectrum one can obtain local parametric reduced order models that, embedded in a trust-region scheme, form the basis for a reliable and efficient specialized algorithm. We describe an optimization strategy based on this approach, and we provide numerical experiments that confirm its effectiveness and accuracy.

**See at: **
arXiv.org e-Print Archive | Journal of Computational and Applied Mathematics | ISTI Repository | Journal of Computational and Applied Mathematics | doi.org | Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna | CNR ExploRA | www.sciencedirect.com

2019
Conference article
Open Access

The KAES methodology for efficient evaluation of dependability-related properties is proposed. KAES targets systems representable by Stochastic Petri Nets-based models, composed by a large number of submodels where interconnections are managed through synchronization at action level. The core of KAES is a new numerical solution of the underlying CTMC process, based on powerful mathematical techniques, including Kronecker algebra, Tensor Trains and Exponential Sums. Specifically, advancing on existing literature, KAES addresses efficient evaluation of the Mean-Time-To-Absorption in CTMC with absorbing states, exploiting the basic idea to further pursue the symbolic representation of the elements involved in the evaluation process, so to better cope with the problem of state explosion. As a result, computation efficiency is improved, especially when the submodels are loosely interconnected and have small number of states. An instrumental case study is adopted, to show the feasibility of KAES, in particular from memory consumption point of view.

**See at: **
arpi.unipi.it | link.springer.com | ISTI Repository | CNR ExploRA | doi.org