Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating $x$ when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) and $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, and $v$ obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of $x$. Pole selection strategies with explicit convergence bounds are given also in this case

A low-rank technique for computing the quasi-stationary distribution of subcritical Galton-Watson processes

We present a new algorithm for computing the quasi-stationary distribution of subcritical Galton--Watson branching processes. This algorithm is based on a particular discretization of a well-known functional equation that characterizes the quasi-stationary distribution of these processes. We provide a theoretical analysis of the approximate low-rank structure that stems from this discretization, and we extend the procedure to multitype branching processes. We use numerical examples to demonstrate that our algorithm is both more accurate and more efficient than other approaches

Solving rank structured Sylvester and Lyapunov equations

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for large-scale interconnected systems", Automatica, 2016, and by Palitta and Simoncini in "Numerical methods for large-scale Lyapunov equations with symmetric banded data", SISC, 2018, which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems

Low-rank updates and a divide-and-conquer method for linear matrix equations

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 HODLR, HSS, and banded matrices. Numerical experiments demonstrate the advantages of divide-and-conquer over existing approaches, in terms of computational time and memory consumption

On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices

The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show that such a submatrix can always be chosen to be a principal submatrix if $A$ is symmetric semidefinite or diagonally dominant. Then we analyze the low-rank approximation error returned by a greedy method for volume maximization, cross approximation with complete pivoting. Our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices. In particular, for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error. We also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results

On Functions of quasi Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued continuous function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction

Efficient cyclic reduction for QBDs with rank structured blocks

We provide effective algorithms for solving block tridiagonal block Toeplitz systems with $m\times m$ quasiseparable blocks, as well as quadratic matrix equations with $m\times m$ quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size $m\approx 10^2$

HEALTH TECHNOLOGY ASSESSMENT (HTA): DALLO STATO DELL' ARTE ALL'ESPERIENZA NELL' AZIENDA OSPEDALIERO UNIVERSITARIA PISANA

L’esperienza maturata dal gruppo di lavoro aziendale sulle valutazioni delle tecnologie sanitarie (HTA) corrisponde esattamente ai quattro processi principali che accomunano il macroprocesso dell’ hospital based HTA, indicati da Francesconi nel suo Innovazione organizzativa e tecnologica in sanità. Il ruolo dell’health technology assessment (11): • raccolta dei bisogni e definizione della proposta di valutazione; • istruttoria (technology assessment in senso stretto); • selezione delle modalità di adozione e/o acquisizione; • gestione e monitoraggio delle prestazioni. La formalizzazione delle attività aziendali di HTA con la creazione di una nuova Sezione Dipartimentale, inserita in Staff alla Direzione, dedicata alla valutazione delle tecnologie sanitarie (Sez. Health Tecnology Assessment) è il completamento del percorso che ha portato negli ultimi anni allo sviluppo informale in Azienda di attività di valutazione delle tecnologie sanitarie e alla nascita del gruppo di lavoro per supportare le decisioni del vertice aziendale (decision making). E sicuramente i progetti “work in progress” subiranno un accellerazione grazie anche a questa nuova realtà aziendale

Modelli di organizzazione, gestione e controllo alla luce del Decreto Legislativo 8 Giugno 2001 n.231. Un caso applicativo

Con il presente elaborato si intende trattare della responsabilità amministrativa degli Enti dipendente da reato, introdotta dal Decreto Legislativo 8 giugno 2001 n. 231, soffermandosi, in particolar modo, sul ruolo dei Modelli organizzativi e gestionali idonei a prevenire reati della specie di quelli che possono fondare la responsabilità dell’Ente. Nella prima parte, a seguito di alcune considerazioni generali ed inerenti il contesto di riferimento, verranno approfonditi gli aspetti più significativi, soprattutto in termini di responsabilità e di esonero dalla stessa, introdotti dalla nuova disciplina. Il tutto con particolare riguardo verso i soggetti destinatari del decreto, le fattispecie di reato contemplate e le conseguenze sanzionatorie previste. Nella seconda parte l’attenzione si concentrerà sul ruolo dei Modelli di Organizzazione, Gestione e Controllo cui il decreto fa riferimento. Degli stessi verranno descritte: struttura, componenti essenziali e modalità operative di realizzazione. A completamento del lavoro si riporterà il caso di Alfa S.p.A., azienda industriale di medie dimensioni, operante nel settore cartario ed avente sede legale in lucchesia; oggetto dell’approfondimento sarà l’adozione, da parte della stessa società, del modello gestionale di prevenzione dei rischi, secondo le modalità suggerite dallo stesso Decreto 231 e da Confindustria. Il lavoro si concentrerà sulla descrizione di alcuni degli step operativi seguiti nella costruzione del Modello nonché su alcuni degli strumenti operativi impiegati per il medesimo scopo