Metodi per la soluzione veloce di una classe di equazioni di Riccati algebriche

Consideriamo l'equazione matriciale $$XCX+B-AX-XE=0,$$ dove $A \in \R^{m \times m}$, $X,B \in \R^{m\times n}$, $C \in \R^{n \times m}$, $E \in \R^{n \times n}$, nota come \emph{equazione di Riccati algebrica non simmetrica} (NARE). In particolare, siamo interessati a studiare un caso particolare dell'equazione, derivante da un problema fisico nell'ambito della teoria del trasporto di neutroni, nel quale i coefficienti sono nella forma \begin{equation}\begin{split}\label{defs} \begin{aligned} B&=ee^T, & C&=qq^T,\\ A&=\Delta-eq^T, & E& =D-qe^T\\ D&=\diag(d_1,\dotsc,d_n),& \Delta&=\diag(\delta_1,\dotsc,\delta_n),\\ d_i&=\frac{1}{cx_i(1-\alpha)},& \delta_i&=\frac{1}{cx_i(1+\alpha)},\\ e&=\mat{1 & 1 & \dotsb & 1}^T,& q_i&=\frac{w_i}{2x_i}. \end{aligned} \end{split} \end{equation} Nella tesi, esaminiamo diversi metodi iterativi noti in letteratura per il calcolo della soluzione minimale non negativa $X^\ast$. \begin{enumerate} \item[(a)] il metodo di Newton [C.H. Guo--Laub, 2000], \item[(b)] lo \emph{structured doubling algorithm} [X.X. Guo--Lin--Xu, 2006], \item[(c)] la riduzione ciclica [Ramaswami, 1999], \item[(d)] il metodo di Newton applicato all'iterazione di Lu [Lu, 2005]. \end{enumerate} Utilizzando le propriet\`a delle matrici con struttura di rango, sviluppiamo versioni specializzate dei quattro algoritmi citati per il problema \eqref{defs}, che permettono di abbassarne il costo computazionale da $O(n^3)$ operazioni aritmetiche per passo del caso generale a $O(n^2)$. Mostriamo inoltre come sia possibile applicare agli algoritmi sviluppati la \emph{tecnica di shift} [He--Meini--Rhee, 2001] per accelerare la convergenza nei cosiddetti \emph{casi critici} del problema (cio\`e, nel caso \eqref{defs}, quando $c=1, \alpha=0$). Gli esperimenti numerici condotti evidenziano l'efficacia dell'approccio proposto. Come risultato supplementare, riusciamo a dimostrare alcune interessanti relazioni algebriche che legano gli algoritmi analizzati e forniscono nuovi spunti per l'analisi della convergenza e lo sviluppo di nuovi algoritmi. In particolare, proviamo che il metodo (d) calcola la stessa iterazione del metodo (a) lavorando direttamente sui generatori della struttura di rango delle matrici coinvolte, e che il metodo (b) coincide essenzialmente con il metodo (c), a meno dell'applicazione di una trasformazione preliminare dell'equazione. We consider the matrix equation $$XCX+B-AX-XE=0,$$ where $A \in \R^{m \times m}$, $X,B \in \R^{m\times n}$, $C \in \R^{n \times m}$, $E \in \R^{n \times n}$, known as \emph{Nonsymmetric algebraic Riccati equation} (NARE). As a special case, we are interested in a Riccati equation appearing in a problem in neutron transport theory, whose coefficients are \begin{equation}\begin{split}\label{defs} \begin{aligned} B&=ee^T, & C&=qq^T,\\ A&=\Delta-eq^T, & E& =D-qe^T\\ D&=\diag(d_1,\dotsc,d_n),& \Delta&=\diag(\delta_1,\dotsc,\delta_n),\\ d_i&=\frac{1}{cx_i(1-\alpha)},& \delta_i&=\frac{1}{cx_i(1+\alpha)},\\ e&=\mat{1 & 1 & \dotsb & 1}^T,& q_i&=\frac{w_i}{2x_i}. \end{aligned} \end{split} \end{equation} Several iterative methods for the calculation of the minimal nonnegative solution $X^\ast$ exist in literature. Among these, we will focus on: \begin{enumerate} \item[(a)] Newton's method [C.H. Guo--Laub, 2000], \item[(b)] the \emph{structured doubling algorithm} [X.X. Guo--Lin--Xu, 2006], \item[(c)] the cyclic reduction [Ramaswami, 1999], \item[(d)] Newton's method applied to Lu's iteration [Lu, 2005]. \end{enumerate} Using the properties of rank-structured matrices, we develop specialized versions of these four algorithms to deal with the problem \eqref{defs}. Their common computational cost is $O(n^2)$ per step, instead of $O(n^3)$ as in the general algorithms. We also show how to apply the \emph{shift technique} [He--Meini--Rhee, 2001] together with our specialized algorithms; this leads to faster convergence in the so-called \emph{critical cases} of the problem (that is, in \eqref{defs}, when $c=1$, $\alpha=0$). The numerical experiments performed confirm the effectiveness of our approach. As a further result, we prove a couple of interesting algebraic relations between the algorithms, which lead to deeper insight on the existing algorithms and provide ideas for the development of new ones. More precisely, we prove that the method (d) calculates the same iterates as (a) but working directly on the displacement generators of the involved matrices, and that (b) is a special case of algorithm (c), applied after a preliminary transformation of the NARE

Feasible generalized least squares estimation of multivariate GARCH(1, 1) models

We provide a feasible generalized least squares estimator for (unrestricted) multivariate GARCH(1, 1) models. We show that the estimator is consistent and asymptotically normally distributed under mild assumptions. Unlike the (quasi) maximum likelihood method, the feasible GLS is considerably fast to implement and does not require any complex optimization routine. We present numerical experiments on simulated data showing the performance of the GLS estimator, and discuss the limitations of our approach. © 2014 Elsevier Inc

MULTIVARIATE TREND–CYCLE EXTRACTION WITH THE HODRICK–PRESCOTT FILTER

The Hodrick–Prescott filter represents one of the most popular methods for trend–cycle extraction in macroeconomic time series. In this paper we provide a multivariate generalization of the Hodrick–Prescott filter, based on the seemingly unrelated time series approach. We first derive closed-form expressions linking the signal–noise matrix ratio to the parameters of the VARMA representation of the model. We then show that the parameters can be estimated using a recently introduced method, called “Moment Estimation Through Aggregation (META).” This method replaces traditional multivariate likelihood estimation with a procedure that requires estimating univariate processes only. This makes the estimation simpler, faster, and better behaved numerically. We prove that our estimation method is consistent and asymptotically normal distributed for the proposed framework. Finally, we present an empirical application focusing on the industrial production of several European countries

Methods for verified stabilizing solutions to continuous-time algebraic Riccati equations

We describe a procedure based on the Krawczyk method to compute a verified enclosure for the stabilizing solution of a continuous-time algebraic Riccati equation A∗X+XA+Q=XGX building on the work of Hashemi (2012) and adding several modifications to the Krawczyk procedure. We show that after these improvements the Krawczyk method reaches results comparable with the current state-of-the-art algorithm (Miyajima, 2015), and surpasses it in some examples. Moreover, we introduce a new direct method for verification which has a cubic complexity in term of the dimension of X, employing a fixed-point formulation of the equation inspired by the ADI procedure. The resulting methods are tested on a number of standard benchmark examples

An inverse-free ADI algorithm for computing Lagrangian invariant subspaces

Summary: The numerical computation of Lagrangian invariant subspaces of large-scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low-rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse-free representation of the subspace and avoids inverses of ill-conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low-rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyrigh

Componentwise accurate fluid queue computations using doubling algorithms

Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue et al. (Numer Math 120:671–700, 2012) provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms. © 2014, Springer-Verlag Berlin Heidelberg

Mutual visibility by luminous robots without collisions

We consider the Mutual Visibility problem for anonymous dimensionless robots with obstructed visibility moving in a plane: starting from distinct locations, the robots must reach, without colliding, a configuration where no three of them are collinear. We study this problem in the luminous robots model, in which each robot has a visible light that can assume colors from a fixed set. Among other results, we prove that Mutual Visibility can be solved in SSynch with 2 colors and in ASynch with 3 colors. If an adversary can interrupt and stop a robot moving to its computed destination, Mutual Visibility is still solvable in SSynch with 3 colors and, if the robots agree on the direction of one axis, also in ASynch. As a byproduct, we provide the first obstructed-visibility solutions to two classical problems for oblivious robots: collision-less convergence to a point (also known as near-gathering) and circle formation

The UAV-based test source as an end-to-end verification tool for aperture arrays

A UAV-mounted radio-frequency transmitter is proposed as a known reference field source to perform a set of functional tests on aperture arrays. The experimental results obtained on complete prototypes (end-to-end) and sub-assemblies provide good confidence on both amplitude and timing verification

The Digital Signal Processing Platform for the Low Frequency Aperture Array: Preliminary Results on the Data Acquisition Unit

A signal processing hardware platform has been developed for the Low Frequency Aperture Array component of the Square Kilometre Array (SKA). The processing board, called an Analog Digital Unit (ADU), is able to acquire and digitize broadband (up to 500MHz bandwidth) radio-frequency streams from 16 dual polarized antennas, channel the data streams and then combine them flexibly as part of a larger beamforming system. It is envisaged that there will be more than 8000 of these signal processing platforms in the first phase of the SKA, so particular attention has been devoted to ensure the design is low-cost and low-power. This paper describes the main features of the data acquisition unit of such a platform and presents preliminary results characterizing its performance

The Signal Processing Firmware for the Low Frequency Aperture Array

The signal processing firmware that has been developed for the Low Frequency Aperture Array component of the Square Kilometre Array is described. The firmware is implemented on a dual FPGA board, that is capable of processing the streams from 16 dual polarization antennas. Data processing includes channelization of the sampled data for each antenna, correction for instrumental response and for geometric delays and formation of one or more beams by combining the aligned streams. The channelizer uses an oversampling polyphase filterbank architecture, allowing a frequency continuous processing of the input signal without discontinuities between spectral channels. Each board processes the streams from 16 antennas, as part of larger beamforming system, linked by standard Ethernet interconnections. There are envisaged to be 8192 of these signal processing platforms in the first phase of the Square Kilometre array so particular attention has been devoted to ensure the design is low cost and low power