A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Flow-induced torsion.

missin

Linearizations of rational matrices

Mención Internacional en el título de doctorThis PhD thesis belongs to the area of Numerical Linear Algebra. Specifically, to the numerical solution of the Rational Eigenvalue Problem (REP). This is a type of eigenvalue problem associated with rational matrices, which are matrices whose entries are rational functions. REPs appear directly from applications or as approx imations to arbitrary Nonlinear Eigenvalue Problems (NLEPs). Rational matrices also appear in linear systems and control theory, among other applications. Nowa days, a competitive method for solving REPs is via linearization. This is due to the fact that there exist backward stable and efficient algorithms to solve the linearized problem, which allows to recover the information of the original rational problem. In particular, linearizations transform the REP into a generalized eigenvalue pro blem in such a way that the pole and zero information of the corresponding rational matrix is preserved. To recover the pole and zero information of rational matrices, it is fundamental the notion of polynomial system matrix, introduced by Rosenbrock in 1970, and the fact that rational matrices can always be seen as transfer functions of polynomial system matrices. This thesis addresses different topics regarding the problem of linearizing REPs. On the one hand, one of the main objectives has been to develop a theory of li nearizations of rational matrices to study the properties of the linearizations that have appeared so far in the literature in a general framework. For this purpose, a definition of local linearization of rational matrix is introduced, by developing as starting point the extension of Rosenbrock’s minimal polynomial system matrices to a local scenario. This new theory of local linearizations captures and explains rigor ously the properties of all the different linearizations that have been used from the 1970’s for computing zeros, poles and eigenvalues of rational matrices. In particu lar, this theory has been applied to a number of pencils that have appeared in some influential papers on solving numerically NLEPs through rational approximation. On the other hand, the work has focused on the construction of linearizations of rational matrices taking into account different aspects. In some cases, we focus on preserving particular structures of the corresponding rational matrix in the li nearization. The structures considered are symmetric (Hermitian), skew-symmetric (skew-Hermitian), among others. In other cases, we focus on the direct construc tion of the linearizations from the original representation of the rational matrix. The representations considered are rational matrices expressed as the sum of their polynomial and strictly proper parts, rational matrices written as general trans fer function matrices, and rational matrices expressed by their Laurent expansion around the point at infinity. In addition, we describe the recovery rules of the information of the original rational matrix from the information of the new lineari zations, including in some cases not just the zero and pole information but also the information about the minimal indices. Finally, in this dissertation we tackle one of the most important open problems related to linearizations of rational matrices. That is the analysis of the backward stability for solving REPs by running a backward stable algorithm on a linearization. On this subject, a global backward error analysis has been developed by considering the linearizations in the family of “block Kronecker linearizations”. An analysis of this type had not been developed before in the literature.Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Froilán Martínez Dopico y codirección de la profesora Silvia Marcaida Bengoechea. Se contó durante cuatro años con un contrato predoctoral FPI, referencia BES-2016-076744, asociado al proyecto ALGEBRA LINEAL NUMERICA ESTRUCTURADA PARA MATRICES CONSTANTES, POLINOMIALES Y RACIONALES, referencia MTM2015-65798-P, del Ministerio de Economía y Competitividad, y cuyo investigador principal fue Froilán Martínez Dopico. Asociado a este contrato, se contó con una ayuda para realizar parte de este trabajo durante dos es tancias internacionales de investigación. La primera estancia de investigación se realizó del 30 de enero de 2019 hasta el 1 de marzo de 2019 en el Department of Mathematical En gineering, Université catholique de Louvain (Bélgica), bajo la supervisión del profesor Paul Van Dooren. La segunda estancia de investigación se realizó del 15 de septiembre de 2019 hasta el 19 de noviembre de 2019 en el Department of Mathematical Sciences, University of Montana (EEUU), bajo la supervisión del profesor Javier Pérez Alvaro. Dado que la entidad beneficiaria del contrato predoctoral es la UC3M mientras que el otro codirector de tesis, la profesora Silvia Marcaida Bengoechea, pertenece al Departamento de Matemáticas de la Universidad del País Vasco (UPV/EHU), el trabajo con la profesora Silvia Marcaida se reforzó mediante visitas a la UPV/EHU, financiadas por ayudas de la RED temática de Excelencia ALAMA (Algebra Lineal, Análisis Matricial y Aplicaciones) asociadas al los proyectos MTM2015-68805-REDT y MTM2017-90682-REDT.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Ion Zaballa Tejada.- Secretario: Fernando de Terán Vergara.- Vocal: Vanni Noferin

Iterative refinement methods for eigenproblems

The subject of this thesis is the numerical solution of eigenproblems from the point of view of iterative refinement. On the whole, we will be concerned with linear, symmetric problems, but occasionally we will make forays into non-linearity and non-symmetry. The initial goal was to develop a better understanding of Rayleigh quotient iteration (RQI) and its numerical performance. Along the way it was necessary to look at a variety of methods proposed for the iterative refinement of eigenelements to see what relationships, if any, they have with RQI. As a consequence we identified a natural progression from algebraic (discrete) methods to continuous methods, some of which have direct discrete counterparts. Chapter 1 provides an overview of eigenproblems and some of the main methods for their numerical solution. Particular emphasis is given to two of the key players which will be found throughout the thesis; namely, inverse iteration and the Rayleigh quotient. In Chapter 2, these are combined to form the Rayleigh quotient iteration; a method with remarkable convergence properties (at least for normal, compact operators). The first part of the chapter, Sections 1 to 4, examine RQI, what its properties are, the way it works, and what it does in terms of minimizing naturally occuring functionals. Section 5 completes the chapter by using Taylor’s series to show why RQI is such a special process. Not many numerical procedures are cubically convergent, and the obvious ploy of using the first three terms of the Taylor’s series to get such fast convergence only results in very inelegant iterations when applied to the eigenproblem. Although it must be said that while the evaluation of the second differential of an arbitrary (vector valued) function is in general quite daunting, and the rewards are probably outweighed by the costs, the functions one would expect in the eigenproblem yield second differentials which are quite simple. Chapter 3 is a bridge between inverse iteration in the first two chapters, and continuous methods in Chapter 4. The link is established through the Rayleigh-Schrödinger series which is the motivation behind Rayleigh-Schrödinger iteration and its several variants. Essentially these are inverse iterations, but using generalized inverses which come in as reduced resolvents. For the self-adjoint case, the iterations follow a particularly nice pattern that is reminiscent of the error squaring (superconvergence) property of the Rayleigh quotient. As with RQI, the iterations have a natural interpretation in terms of minimizing functionals. In this chapter, Section 2 is an inset giving a novel way of arriving at the iteration based on matrix calculus. The derivation of the Rayleigh-Schrödinger series itself, however, is as a homotopy method for getting from a known eigenpair of a perturbed operator to an eigenpair of the unperturbed operator. One way of tackling homotopies is via differential equations, and so in Chapter 4 we turn our attention to these matters. The discussion in Chapter 4 is based on continuous analogues of discrete processes which have their genesis in the discovery that the QR algorithm is closely related to the Toda flow. Many discrete methods follow the solution trajectory of a differential equation, either exactly or approximately. For example, Newton’s iteration can be thought of as Euler’s method applied to a particular initial value problem. Other methods though, like the QR algorithm, produce iterates that are exactly on the solution curve, so that one can think of the continuous method as an interpolation of the discrete iteration. Finally Chapter 5 stands apart in the sense that it does not directly continue on from continuous methods; however, inverse iteration does plays the central role. The main idea is to build up information from the traces of a matrix, its powers, and its inverse powers, which can then be used to approximate eigenvalues. Here, Laguerre’s method for finding the roots of a polynomial is shown to be connected with the (standard) method of traces applied to matrices (or integral operators)

Flow-induced torsion.

missin

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth