Beaucoup de phénomènes physiques sont modélisés par des équations aux dérivées partielles, la discrétisation de ces équations conduit souvent à des systèmes dynamiques (continus ou discrets) dépendant d'un vecteur de contrôle dont le choix permet de stabiliser le système dynamique. Comme ces problèmes sont, dans la pratique, de grandes tailles, il est intéressant de les étudier via un autre problème dérivé réduit et plus proche du modèle initial. Dans cette thèse, on introduit et on étudie de nouvelles méthodes basées sur les processus de type Krylov rationnel afin d'extraire un modèle réduit proche du modèle original. Des applications numériques seront faites à partir de problèmes pratiques. Après un premier chapitre consacré au rappel de quelques outils mathématiques, on s'intéresse aux méthodes basées sur le processus d'Arnoldi rationnel par blocs pour réduire la taille d'un système dynamique de type Multi-Input/Multi-Output (MIMO). On propose une sélection adaptative de choix de certains paramètres qui sont cruciaux pour l'efficacité de la méthode. On introduit aussi un nouvel algorithme adaptatif de type Arnoldi rationnel par blocs afin de fournir une nouvelle relation de type Arnoldi. Dans la deuxième partie de ce travail, on introduit la méthode d'Arnoldi rationnelle globale, comme alternative de la méthode d'Arnoldi rationnel par blocs. On définit la projection au sens global, et on applique cette méthode pour approcher les fonctions de transfert. Dans la troisième partie, on s'intéresse à la méthode d'Arnoldi étendue (qui est un cas particulier de la méthode d'Arnoldi rationnelle) dans les deux cas (global et par blocs), on donnera quelques nouvelles propriétés algébriques qui sont appliquées aux problèmes des moments. On consièdère dans la quatrième partie la méthode de troncature balancée pour la réduction de modèle. Ce procédé consiste à résoudre deux grandes équations algébriques de Lyapunov lorsque le système est stable ou à résoudre deux équations de Riccati lorsque le système est instable. Comme ces équations sont de grandes tailles, on va appliquer la méthode de Krylov rationnel par blocs pour approcher la solution de ces équations. Le travail de cette thèse sera cloturé par une nouvelle idée, dans laquelle on définit un nouvel espace sous le nom de sous-espace de Krylov rationnelle étendue qui sera utilisée pour la réduction du modèle. / Many physical phenomena are modeled by PDEs. The discretization of these equations often leads to dynamical systems (continuous or discrete) depending on a control vector whose choice can stabilize the dynamical system. As these problems are, in practice, of a large size, it is interesting to study the problem through another one which is reduced and close to the original model. In this thesis, we develop and study new methods based on rational Krylov-based processes for model reduction techniques in large-scale Multi-Input Multi-Output (MIMO) linear time invariant dynamical systems. In chapter 2 the methods are based on the rational block Arnoldi process to reduce the size of a dynamical system through its transfer function. We provide an adaptive selection choice of shifts that are crucial for the effectiveness of the method. We also introduce a new adaptive Arnoldi-like rational block algorithm to provide a new type of Arnoldi's relationship. In Chapter 3, we develop the new rational global Arnoldi method which is considered as an alternative to the rational block Arnoldi process. We define the projection in the global sense, and apply this method to extract reduced order models that are close to the large original ones. Some new properties and applications are also presented. In chapter 4 of this thesis, we consider the extended block and global Arnoldi methods. We give some new algebraic properties and use them for approaching the firt moments and Markov parameters in moment matching methods for model reduction techniques. In chapter 5, we consider the method of balanced truncation for model reduction. This process is based on the soluytions of two major algebraic equations : Lyapunov equations when the system is stable or Riccati equations when the system is unstable. Since these equations are of large sizes, we will apply the rational block Arnoldi method for solving these equations. In chapter 6, we introduce a new method based on a new subspace called the extended-rational Krylov subspace. We introduce the extended-rational Krylov method which will be used for model reduction in large-scale dynamical systems.

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Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov decompositions; matrix relations which, under certain conditions, are associated with these spaces. We investigate the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. We derive standard and harmonic Ritz extraction strategies for approximating the eigenpairs of a matrix and for approximating the action of a matrix function onto a vector. While these topics have been considered previously, our approach does not require the last pole to be infinite, which makes the extraction procedure computationally more efficient. Typically, the computationally most expensive component of the rational Arnoldi algorithm for computing a rational Krylov basis is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become poorly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this non orthogonal basis. As a consequence we obtain a more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using our high performance C++ implementation. We develop an iterative algorithm for solving nonlinear rational least squares problems. The difficulty is in finding the poles of a rational function. For this purpose, at each iteration a rational Krylov decomposition is constructed and a modified linear problem is solved in order to relocate the poles to new ones. Our numerical results indicate that the algorithm, called RKFIT, is well suited for model order reduction of linear time invariant dynamical systems and for optimisation problems related to exponential integration. Furthermore, we derive a strategy for the degree reduction of the approximant obtained by RKFIT. The rational function obtained by RKFIT is represented with the aid of a scalar rational Krylov decomposition and an additional coefficient vector. A function represented in this form is called an RKFUN. We develop efficient methods for the evaluation, pole and root finding, and for performing basic arithmetic operations with RKFUNs. Lastly, we discuss RKToolbox, a rational Krylov toolbox for MATLAB, which implements all our algorithms and is freely available from http://rktoolbox.org. RKToolbox also features an extensive guide and a growing number of examples. In particular, most of our numerical experiments are easily reproducible by downloading the toolbox and running the corresponding example files in MATLAB.

Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and eigenvalue computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie. A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method. / QC 20101013

<p>Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and eigenvalue computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie.</p><p>A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method.</p>

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.

Die vorliegende Arbeit befasst sich mit der Fragestellung, inwiefern sich gegebene Verfahren zur Approximation von rationalen Krylow-Räumen zur Berechnung von Matrixfunktionen eignen. Als Modellproblem wird dazu eine 2.5D-Formulierung eines Problems aus der Gleichstrom-Geoelektrik mit finiten Elementen formuliert und dann mittels Matrixfunktionen auf rationalen Krylow-Unterräumen gelöst. Ein weiterer Teil beschäftigt sich mit dem Vergleich zweier Verfahren zur Transformation bestehender rationaler Krylow-Räume. Bei beiden Varianten werden die zugrunde liegenden Pole getauscht ohne dass ein explizites Invertieren von Matrizen notwendig ist. In dieser Arbeit werden die über mehrere Publikationen verteilten Grundlagen einheitlich zusammengetragen und fehlende Zusammenhänge ergänzt. Beide Verfahren eignen sich prinzipiell um rationale Krylow-Räume zu approximieren. Dies wird anhand mehrerer Beispiele gezeigt. Anhand des Modellproblems werden Beschränkungen der Methoden verdeutlicht.