This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first formula employs a block generalization of the residual polynomial, while the second leverages the block collinearity of the residuals. A posteriori error bounds based on the knowledge of spectral information of the argument are derived and tested on a set of examples. Notably, both error formulas and their corresponding upper bounds do not require the use of quadratures for their practical evaluation.

This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form A(1)XB(1)(inverted perpendicular)+A(2)XB(2)(inverted perpendicular)+& ctdot;+A(& ell;)XB(& ell;)(inverted perpendicular)=F, as they arise from discretized partial differential equations and control problems. One often finds that X admits good low-rank approximations, in particular when the right-hand-side matrix F has low rank. For & ell;<= 2 terms, the solution of such equations is well studied, and effective low-rank solvers have been proposed, including alternating direction implicit (ADI) methods for Lyapunov and Sylvester equations. For & ell;>2, several existing methods try to approach X through combining a classical iterative method, such as the conjugate gradient (CG) method, with low-rank truncation. In this work, we consider a more direct approach that approximates X on manifolds of fixed-rank matrices through Riemannian CG. One particular challenge is the incorporation of effective preconditioners into such a first-order Riemannian optimization method. We propose several novel preconditioning strategies, including a change of metric in the ambient space, preconditioning the Riemannian gradient, and a variant of ADI on the tangent space. Combined with a strategy for adapting the rank of the approximation, the resulting method is demonstrated to be competitive for a number of examples representative for typical applications.

This article addresses the presence of noise when solving the inverse problem of determining the material properties of an elastic engineering structure from measured vibration data. After identifying the frequencies, material properties are determined by solving an optimization problem. However, if the experimental frequencies are affected by noise (as often happens in practice), this can lead to inaccuracy due to the ill-conditioning of the problem. A regularization strategy based on Tikhonov regularization is discussed, and an automatic regularization parameter choice is introduced to deal with cases where the noise level is not known beforehand. The algorithm is tested on three examples, including artificial models where the exact solution is known and a case study from the Matilde donjon in Livorno, with experimental data measured by seismic stations during ambient vibration tests. The proposed method consistently performs well across all tests, validating the reliability of the approach.

Given a nonlinear matrix-valued function F(λ) and approximate eigenpairs (λ_i, v_i), we discuss how to determine the smallest perturbation δ F such that [F + δ F](λ_i) v_i = 0; we call the distance between the F and F + δ F the backward error for this set of approximate eigenpairs. We focus on the case where F(λ) is given as a linear combination of scalar functions multiplying matrix coefficients F_i, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the F_i have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the δ F_i are also given.

In the last decade, tensors have shown their potential as valuable tools for various tasks in numerical linear algebra. While most of the research has been focusing on how to compress a given tensor in order to maintain information as well as reducing the storage demand for its allocation, the solution of linear tensor equations is a less explored venue. Even if many of the routines available in the literature are based on alternating minimization schemes (ALS), we pursue a different path and utilize Krylov methods instead. The use of Krylov methods in the tensor realm is not new. However, these routines often turn out to be rather expensive in terms of computational cost and ALS procedures are preferred in practice. We enhance Krylov methods for linear tensor equations with a panel of diverse randomization-based strategies which remarkably increase the efficiency of these solvers making them competitive with state-of-the-art ALS schemes. The up-to-date randomized approaches we employ range from sketched Krylov methods with incomplete orthogonalization and structured sketching transformations to streaming algorithms for tensor rounding. The promising performance of our new solver for linear tensor equations is demonstrated by many numerical results.

The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this paper, we take these advancements a step further by introducing a higher-order variant of Nystr\"om's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nystr\"om technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nystr\"om method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.

The aim of this work is to develop a fast algorithm for approximating the matrix function f(A) of a square matrix A that is symmetric and has hierarchically semiseparable (HSS) structure. Appearing in a wide variety of applications, often in the context of discretized (fractional) differential and integral operators, HSS matrices have a number of attractive properties facilitating the development of fast algorithms. In this work, we use an unconventional telescopic decomposition of A, inspired by recent work of Levitt and Martinsson on approximating an HSS matrix from matrix-vector products with a few random vectors. This telescopic decomposition allows us to approximate f(A) by recursively performing low-rank updates with rational Krylov subspaces while keeping the size of the matrices involved in the rational Krylov subspaces small. In particular, no large-scale linear system needs to be solved, which yields favorable complexity estimates and reduced execution times compared to existing methods, including an existing divide-and-conquer strategy. The advantages of our newly proposed algorithms are demonstrated for a number of examples from the literature, featuring the exponential, the inverse square root, and the sign function of a matrix. Even for matrix inversion, our algorithm exhibits superior performance, even if not specifically designed for this task.

We present an algorithm for the solution of Sylvester equations with right-hand side of low rank. The method is based on projection onto a block rational Krylov subspace, with two key contributions with respect to the state-of-the-art. First, we show how to maintain the last pole equal to infinity throughout the iteration, by means of pole reodering. This allows for a cheap evaluation of the true residual at every step. Second, we extend the convergence analysis in [Beckermann B., An error analysis for rational Galerkin projection applied to the Sylvester equation, SINUM, 2011] to the block case. This extension allows to link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field-of-values of the involved matrices. This is in contrast with the non-block case, where the minimum problem is scalar, instead of matrix-valued. Replacing the norm of the objective function with an easier to evaluate function yields several adaptive pole selection strategies, providing a theoretical analysis for known heuristics, as well as effective novel techniques.

This paper introduces the application of a reduced-order modeling technique for accurate temperature monitoring in Power Electronics modules. The methodology involves coupling the Finite Element Method with the radiosity equation to obtain a high-fidelity representation of the physical behavior of the device. These models account also for surface-to-surface radiation, an aspect that can have a high impact when the operating temperatures increase, and the components are close to each other, which is a common scenario for high power density and compact modules. The Model Order Reduction is performed via the Proper orthogonal decomposition, coupled with the Discrete Empirical Interpolation Method to handle the nonlinearity introduced by the surface-to-surface radiation. This approach allows to reduce the computation time with a limited effect on the accuracy of the prediction. The method is applied to reduce the order of the thermal model of a DC-DC converter. Numerical and experimental results demonstrate the approach’s effectiveness, showcasing high accuracy with minimal computation time and memory cost.

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

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.

The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is computing inverse fractional powers of the standard discretized Laplace operator. In this work, an exponential sum approximation for such fractional powers is derived. It is accurate over all positive real numbers larger than one, and allows to efficiently approximate the action of such operators on tensors stored in a variety of low-rank formats (CP, TT, Tucker). The results are relevant from a practical and theoretical perspective, as they predict the low-rank approximability of the solutions of these linear systems in low-rank tensor formats.

Linear systems with a tensor product structure arise naturally when considering the discretization of Laplace type differential equations or, more generally, multidimensional operators with separable coefficients. In this work, we focus on the numerical solution of linear systems of the form $$ \left(I\otimes \dots\otimes I \otimes A_1+\dots + A_d\otimes I \otimes\dots \otimes I\right)x=b,$$ where the matrices $A_t\in\mathbb R^{n\times n}$ are symmetric positive definite and belong to the class of hierarchically semiseparable matrices. We propose and analyze a nested divide-and-conquer scheme, based on the technology of low-rank updates, that attains the quasi-optimal computational cost $\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa) \log(\epsilon^{-1})))$ where $\kappa$ is the condition number of the linear system, and $\epsilon$ the target accuracy. Our theoretical analysis highlights the role of inexactness in the nested calls of our algorithm and provides worst case estimates for the amplification of the residual norm. The performances are validated on 2D and 3D case studies.

This chapter presents the finite element code NOSA-ITACA for static and modal analyses of masonry structures of architectural interest. NOSA-ITACA adopts the constitutive equation of masonry-like materials, which considers masonry a nonlinear elastic material with zero tensile strength. The capability of modelling restoration and consolidation operations makes the code a helpful tool for maintaining historical buildings. In recent years, long-term vibration monitoring turned out to be an effective non-destructive technique to investigate the dynamic behaviour and check the health status of historical buildings. Changes in their dynamic properties, such as natural frequencies, can represent effective damage indicators. The latest NOSA-ITACA developments are oriented towards structural health monitoring. The availability of the experimental modal properties of a structure makes it possible to calibrate its finite element model via model updating procedures. In particular, the unknown structure’s characteristics, such as materials’ properties and boundary conditions, can be determined by solving a minimum problem whose objective function is expressed as the discrepancy between experimental frequencies and mode shapes and their numerical counterparts. Several case studies are presented to show the main features of NOSA-ITACA and its effectiveness in the conservation of architectural heritage.

New composite materials are always subjected to non-destructive evaluation (NDE) prior to being placed on the market. This is to fully understand the reactions (i.e., development of defects) at the interface between two subsequent layers. Active infrared thermography (aIRT) can help in this regard, especially if anticipated by a simulation of the heat transfer from the exterior (lamp) to the interior (multilayer). Comsol Multiphysics® was used in this work as a tool by developing an innovative approach, which is designed – on the one hand – to minimize the computational cost and – on the other hand – to optimize the radiation to be delivered. The innovation produced by our work also concerns the pre-processing step of the thermal images; in fact, the 2D Fast Iterative Filtering (FIF2) is here introduced, discussing its benefits in comparison to previously developed techniques. Pre-processed data were further analyzed during the post-processing step demonstrating the reliability of FIF2 in enhancing thermal imprints, which leads to an improved detection of subsurface features. In particular, enhanced thermal imprints highlight the shape of the grid of glass fibres present beneath an external coating of hemp fibres (and, in general, added to the whole specimen along the x-y vectors). This grid of glass fibres was recently introduced as an insulation material for buildings. A brief review of the use of the pre-processing step in aIRT allows the reader to better understand the decisive step forward provided by FIF2 combined with a clever numerical simulation in the applied thermal engineering field. Qualitative and quantitative IRT results are shown and discussed thoroughly. Finally, a validation among numerical and experimental (thermographic) data is provided thanks to the Parker (laser flash) method.

Various numerical linear algebra problems can be formulated as evaluating bivariate function of matrices. The most notable examples are the Fr\'echet derivative along a direction, the evaluation of (univariate) functions of Kronecker-sum-structured matrices and the solution of Sylvester matrix equations. In this work, we propose a recursive block diagonalization algorithm for computing bivariate functions of matrices of small to medium size, for which dense liner algebra is appropriate. The algorithm combines a blocking strategy, as in the Schur-Parlett scheme, and an evaluation procedure for the diagonal blocks. We discuss two implementations of the latter. The first is a natural choice based on Taylor expansions, whereas the second is derivative-free and relies on a multiprecision perturb-and-diagonalize approach. In particular, the appropriate use of multiprecision guarantees backward stability without affecting the efficiency in the generic case. This makes the second approach more robust. The whole method has cubic complexity and it is closely related to the well-known Bartels-Stewart algorithm for Sylvester matrix equations when applied to $f(x,y)=\frac{1}{x+y}$. We validate the performances of the proposed numerical method on several problems with different conditioning properties.

TAPAS is a new tool for efficient evaluation of dependability and performability attributes of systems composed of many interconnected components. The tool solves homogeneous continuous time Markov chains described by stochastic automata network models structured in submodels with absorbing states. The measures of interest are defined by a reward structure based on submodels composed through transition-based synchronization. The tool has been conceived in a modular and flexible fashion, to easily accommodate new features. Currently, it implements an array of state-based solvers that addresses the state explosion problem through powerful mathematical techniques, including Kronecker algebra, Tensor Trains and Exponential Sums. A simple, yet representative, case study is adopted, to present the tool and to show the feasibility of the supported methods, in particular frommemory consumption point of view.

Given their crucial role for a society and economy, an essential component of critical infrastructures is the Bad State Estimator (BSE), responsible for detecting malfunctions affecting elements of the physical infrastructure. In the past, the BSE has been conceived to mainly cope with accidental faults, under assumptions characterizing their occurrence. However, evolution of the addressed systems category consisting in pervasiveness of ICT-based control towards increasing smartness, paired with the openness of the operational environment, contributed to expose critical infrastructures to intentional attacks, e.g. exploited through False Data Injection (FDI). In the flow of studies focusing on enhancements of the traditional BSE to account for FDI attacks, this paper proposes a new solution that introduces randomness elements in the diagnosis process, to improve detection abilities and mitigate potentially catastrophic common-mode errors. Differently from existing alternatives, the strength of this new technique is that it does not require any additional components or alternative source of information with respect to the classic BSE. Numerical experiments conducted on two IEEE transmission grid tests, taken as representative use cases, show the applicability and benefits of the new solution.

A novel framework for hierarchical low‐rank matrices is proposed that combines an adaptive hierarchical partitioning of the matrix with low‐rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the flexibility of the format makes it possible to deal with functions that feature singularities in small, localized regions. To deal with time evolution and relocation of singularities, the partitioning can be dynamically adjusted based on features of the underlying data. Our format can be leveraged to efficiently solve linear systems with Kronecker product structure, as they arise from discretized partial differential equations (PDEs). For this purpose, these linear systems are rephrased as linear matrix equations and a recursive solver is derived from low‐rank updates of such equations. We demonstrate the effectiveness of our framework for stationary and time‐dependent, linear and nonlinear PDEs, including the Burgers' and Allen–Cahn equations.

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A=T(a)+E where T(a) is the Toeplitz matrix with entries (T(a))_i,j=a_j-i , for a_j-i∈ℂ , i,j≥ 1 , while E is a matrix representing a compact operator in ℓ ^2 . The matrix A is finitely representable if a_k=0 for k<-m and for k>n , given m,n>0 , and if E has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs (λ ,v) such that Av=λv , with λ∈ℂ , v=(v_j)_j∈ℤ^+ , v 0 , and ∑_j=1^∞ |v_j|^2<∞ . It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind WU(λ )β=0 , where W is a constant matrix and U depends on λ and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton’s method applied to the equation det WU(λ )=0 are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741–769].