The modelling of physical processes gives rise to mathematical systems of increasing complexity. Good mathematical models have to reproduce the physical process as precisely as possible while the computing time and the storage resources needed to simulate the mathematical model are limited. As a consequence, there must be a tradeoff between accuracy and computational constraints. At the present time, one is often faced with systems that have an unacceptably high level of complexity. It is then desirable to approximate such systems by systems of lower complexity. This is the Model Reduction Problem. This thesis focuses on the study of new model reduction techniques for linear systems. Our objective is twofold. First, there is a need for a better understanding of Krylov techniques. With such techniques, one can construct a reduced order transfer function that satisfies a set of interpolation conditions with respect to the original transfer function. A study of the generality of such techniques and their extension for MIMO systems via the concept of tangential interpolation constitutes the first part of this thesis. This also led us to study the generality of the projection technique for model reduction. Most large scale systems have a particular structure. They can be modelled as a set of subsystems that interconnect to each other. It then makes sense to develop model reduction techniques that preserve the structure of the original system. Both interpolation-based and gramian-based structure preserving model reduction techniques are developed in a unified way. Second order systems that appear in many branches of engineering deserve a special attention. This constitutes the second part of this thesis.

We consider low-rank solution methods for certain classes of large-scale linear matrix equations. Our aim is to adapt existing low-rank solution methods based on standard, extended and rational Krylov subspaces to solve equations which may viewed as extensions of the classical Lyapunov and Sylvester equations. The first class of matrix equations that we consider are constrained Sylvester equations, which essentially consist of Sylvester's equation along with a constraint on the solution matrix. These therefore constitute a system of matrix equations. The second are generalized Lyapunov equations, which are Lyapunov equations with additional terms. Such equations arise as computational bottlenecks in model order reduction. / Mathematics

Analysis of large networks in biology, science, technology and social systems have become very popular recently. These networks are mathematically represented as graphs. The task is then to extract relevant qualitative information about the empirical networks from the analysis of these graphs. It was found that a graph can be conveniently represented by the spectrum of a suitable difference operator, the normalized graph Laplacian, which underlies diffusions and random walks on graphs. When applied to large networks, this requires computation of the spectrum of large matrices. The normalized Laplacian matrices representing large networks are usually sparse and unstructured. The thesis consists in a systematic evaluation of the available eigenvalue solvers for nonsymmetric large normalized Laplacian matrices describing directed graphs of empirical networks. The methods include several Krylov subspace algorithms like implicitly restarted Arnoldi method, Krylov-Schur method and Jacobi-Davidson methods which are freely available as standard packages written in MATLAB or SLEPc, in the library written C++. The normalized graph Laplacian as employed here is normalized such that its spectrum is confined to the range [0, 2]. The eigenvalue distribution plays an important role in network analysis. The numerical task is then to determine the whole spectrum with appropriate eigenvalue solvers. A comparison of the existing eigenvalue solvers is done with Paley digraphs with known eigenvalues and for citation networks in sizes 400, 1100 and 4500 by computing the residuals.

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.

Made available in DSpace on 2015-03-03T15:32:44Z (GMT). No. of bitstreams: 1 ElvisNM_DISSERT.pdf: 1325328 bytes, checksum: 26a5738f48a900e63cafc3f1e0b1d776 (MD5) Previous issue date: 2014-04-22 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / Neste trabalho estudamos o problema n?o linear F(X) = 0, onde F ? continuamente diferenci?vel com F : Rn-> Rn. Para solucion?-lo empregamos o m?todo de Newton Inexato obtendo um sistema linearizado J(xk)sk =-F(xk), onde J(xk) representa a matriz Jacobiana no ponto xk e o passo iterativo sk ? calculado por meio do m?todo do Res?duo M?nimo Generalizado (GMRES), que pertence ? fam?lia dos m?todos de proje??o em subespa?os de Krylov. Afim de evitar de evitar o acr?scimo no custo computacional devido ao aumento a cada itera??o na dimens?o do subespa?o de Krylov utilizamos o GMRES com recome?os ou GMRES(m), o qual pode apresentar problemas de estagna??o (duas solu??es consecutivas iguais ou quase iguais). Uma das maneiras de contornar essa estagna??o est? no uso de precondicionadores no sistema inicial Ax = b, passando a um sistema equivalente do tipo M-1Ax = M-1b onde a matriz M ? chamada de precondicionador e tem o papel de facilitar a solu??o do sistema inicial. A escolha de precondicionadores ? uma ?rea de pesquisa que remete ao conhecimento espec?fico a priori do problema a ser resolvido e/ou da estrutura da matriz dos coeficientes A. Neste trabalho buscamos estudar o precondicionamento pela esquerda no m?todo do Newton Inexato - GMRES(m). Apresentamos tamb?m uma estrat?gia que permite a mudan?a entre 3 tipos de precondicionadores (Jacobi, ILU e SSOR) dependendo de informa??es advindas da aplica??o do GMRES(m) a cada itera??o do Newton Inexato, ou seja, a cada vez que se resolve o sistema linearizado precondicionado. Assim fazemos ao final uma compara??o entre nossas estrat?gias e o uso de precondicionadores fixos na resolu??o de problemas teste por meio do NI-GMRES

This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems. For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders. Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework. Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.

Comme la puissance de calcul des systèmes de calcul haute performance continue de croître, en utilisant un grand nombre de cœurs CPU ou d’unités de calcul spécialisées, les applications hautes performances destinées à la résolution des problèmes de très grande échelle sont de plus en plus sujettes à des pannes. En conséquence, la communauté de calcul haute performance a proposé de nombreuses contributions pour concevoir des applications tolérantes aux pannes. Cette étude porte sur une nouvelle classe d’algorithmes numériques de tolérance aux pannes au niveau de l’application qui ne nécessite pas de ressources supplémentaires, à savoir, des unités de calcul ou du temps de calcul additionnel, en l’absence de pannes. En supposant qu’un mécanisme distinct assure la détection des pannes, nous proposons des algorithmes numériques pour extraire des informations pertinentes à partir des données disponibles après une pannes. Après l’extraction de données, les données critiques manquantes sont régénérées grâce à des stratégies d’interpolation pour constituer des informations pertinentes pour redémarrer numériquement l’algorithme. Nous avons conçu ces méthodes appelées techniques d’Interpolation-restart pour des problèmes d’algèbre linéaire numérique tels que la résolution de systèmes linéaires ou des problèmes aux valeurs propres qui sont indispensables dans de nombreux noyaux scientifiques et applications d’ingénierie. La résolution de ces problèmes est souvent la partie dominante; en termes de temps de calcul, des applications scientifiques. Dans le cadre solveurs linéaires du sous-espace de Krylov, les entrées perdues de l’itération sont interpolées en utilisant les entrées disponibles sur les nœuds encore disponibles pour définir une nouvelle estimation de la solution initiale avant de redémarrer la méthode de Krylov. En particulier, nous considérons deux politiques d’interpolation qui préservent les propriétés numériques clés de solveurs linéaires bien connus, à savoir la décroissance monotone de la norme-A de l’erreur du gradient conjugué ou la décroissance monotone de la norme résiduelle de GMRES. Nous avons évalué l’impact du taux de pannes et l’impact de la quantité de données perdues sur la robustesse des stratégies de résilience conçues. Les expériences ont montré que nos stratégies numériques sont robustes même en présence de grandes fréquences de pannes, et de perte de grand volume de données. Dans le but de concevoir des solveurs résilients de résolution de problèmes aux valeurs propres, nous avons modifié les stratégies d’interpolation conçues pour les systèmes linéaires. Nous avons revisité les méthodes itératives de l’état de l’art pour la résolution des problèmes de valeurs propres creux à la lumière des stratégies d’Interpolation-restart. Pour chaque méthode considérée, nous avons adapté les stratégies d’Interpolation-restart pour régénérer autant d’informations spectrale que possible. Afin d’évaluer la performance de nos stratégies numériques, nous avons considéré un solveur parallèle hybride (direct/itérative) pleinement fonctionnel nommé MaPHyS pour la résolution des systèmes linéaires creux, et nous proposons des solutions numériques pour concevoir une version tolérante aux pannes du solveur. Le solveur étant hybride, nous nous concentrons dans cette étude sur l’étape de résolution itérative, qui est souvent l’étape dominante dans la pratique. Les solutions numériques proposées comportent deux volets. A chaque fois que cela est possible, nous exploitons la redondance de données entre les processus du solveur pour effectuer une régénération exacte des données en faisant des copies astucieuses dans les processus. D’autre part, les données perdues qui ne sont plus disponibles sur aucun processus sont régénérées grâce à un mécanisme d’interpolation. / As the computational power of high performance computing (HPC) systems continues to increase by using huge number of cores or specialized processing units, HPC applications are increasingly prone to faults. This study covers a new class of numerical fault tolerance algorithms at application level that does not require extra resources, i.e., computational unit or computing time, when no fault occurs. Assuming that a separate mechanism ensures fault detection, we propose numerical algorithms to extract relevant information from available data after a fault. After data extraction, well chosen part of missing data is regenerated through interpolation strategies to constitute meaningful inputs to numerically restart the algorithm. We have designed these methods called Interpolation-restart techniques for numerical linear algebra problems such as the solution of linear systems or eigen-problems that are the inner most numerical kernels in many scientific and engineering applications and also often ones of the most time consuming parts. In the framework of Krylov subspace linear solvers the lost entries of the iterate are interpolated using the available entries on the still alive nodes to define a new initial guess before restarting the Krylov method. In particular, we consider two interpolation policies that preserve key numerical properties of well-known linear solvers, namely the monotony decrease of the A-norm of the error of the conjugate gradient or the residual norm decrease of GMRES. We assess the impact of the fault rate and the amount of lost data on the robustness of the resulting linear solvers.For eigensolvers, we revisited state-of-the-art methods for solving large sparse eigenvalue problems namely the Arnoldi methods, subspace iteration methods and the Jacobi-Davidson method, in the light of Interpolation-restart strategies. For each considered eigensolver, we adapted the Interpolation-restart strategies to regenerate as much spectral information as possible. Through intensive experiments, we illustrate the qualitative numerical behavior of the resulting schemes when the number of faults and the amount of lost data are varied; and we demonstrate that they exhibit a numerical robustness close to that of fault-free calculations. In order to assess the efficiency of our numerical strategies, we have consideredan actual fully-featured parallel sparse hybrid (direct/iterative) linear solver, MaPHyS, and we proposed numerical remedies to design a resilient version of the solver. The solver being hybrid, we focus in this study on the iterative solution step, which is often the dominant step in practice. The numerical remedies we propose are twofold. Whenever possible, we exploit the natural data redundancy between processes from the solver toperform an exact recovery through clever copies over processes. Otherwise, data that has been lost and is not available anymore on any process is recovered through Interpolationrestart strategies. These numerical remedies have been implemented in the MaPHyS parallel solver so that we can assess their efficiency on a large number of processing units (up to 12; 288 CPU cores) for solving large-scale real-life problems.

Nous nous intéressons dans cette thèse, à l’étude des méthodes itératives pour la résolutiond’équations matricielles de grande taille : Lyapunov, Sylvester, Riccati et Riccatinon symétrique.L’objectif est de chercher des méthodes itératives plus efficaces et plus rapides pour résoudreles équations matricielles de grande taille. Nous proposons des méthodes itérativesde type projection sur des sous espaces de Krylov par blocs Km(A, V ) = Image{V,AV, . . . ,Am−1V }, ou des sous espaces de Krylov étendus par blocs Kem(A, V ) = Image{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V } . Ces méthodes sont généralement plus efficaces et rapides pour les problèmes de grande dimension. Nous avons traité d'abord la résolution numérique des équations matricielles linéaires : Lyapunov, Sylvester, Stein. Nous avons proposé une nouvelle méthode itérative basée sur la minimisation de résidu MR et la projection sur des sous espaces de Krylov étendus par blocs Kem(A, V ). L'algorithme d'Arnoldi étendu par blocs permet de donner un problème de minimisation projeté de petite taille. Le problème de minimisation de taille réduit est résolu par différentes méthodes directes ou itératives. Nous avons présenté ainsi la méthode de minimisation de résidu basée sur l'approche global à la place de l'approche bloc. Nous projetons sur des sous espaces de Krylov étendus Global Kem(A, V ) = sev{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Nous nous sommes intéressés en deuxième lieu à des équations matricielles non linéaires, et tout particulièrement l'équation matricielle de Riccati dans le cas continu et dans le cas non symétrique appliquée dans les problèmes de transport. Nous avons utilisé la méthode de Newtown et l'algorithme MINRES pour résoudre le problème de minimisation projeté. Enfin, nous avons proposé deux nouvelles méthodes itératives pour résoudre les équations de Riccati non symétriques de grande taille : la première basée sur l'algorithme d'Arnoldi étendu par bloc et la condition d'orthogonalité de Galerkin, la deuxième est de type Newton-Krylov, basée sur la méthode de Newton et la résolution d'une équation de Sylvester de grande taille par une méthode de type Krylov par blocs. Pour toutes ces méthodes, les approximations sont données sous la forme factorisée, ce qui nous permet d'économiser la place mémoire en programmation. Nous avons donné des exemples numériques qui montrent bien l'efficacité des méthodes proposées dans le cas de grandes tailles. / In this thesis, we focus in the studying of some iterative methods for solving large matrix equations such as Lyapunov, Sylvester, Riccati and nonsymmetric algebraic Riccati equation. We look for the most efficient and faster iterative methods for solving large matrix equations. We propose iterative methods such as projection on block Krylov subspaces Km(A, V ) = Range{V,AV, . . . ,Am−1V }, or block extended Krylov subspaces Kem(A, V ) = Range{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. These methods are generally most efficient and faster for large problems. We first treat the numerical solution of the following linear matrix equations : Lyapunov, Sylvester and Stein matrix equations. We have proposed a new iterative method based on Minimal Residual MR and projection on block extended Krylov subspaces Kem(A, V ). The extended block Arnoldi algorithm gives a projected minimization problem of small size. The reduced size of the minimization problem is solved by direct or iterative methods. We also introduced the Minimal Residual method based on the global approach instead of the block approach. We projected on the global extended Krylov subspace Kem(A, V ) = Span{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Secondly, we focus on nonlinear matrix equations, especially the matrix Riccati equation in the continuous case and the nonsymmetric case applied in transportation problems. We used the Newton method and MINRES algorithm to solve the projected minimization problem. Finally, we proposed two new iterative methods for solving large nonsymmetric Riccati equation : the first based on the algorithm of extended block Arnoldi and Galerkin condition, the second type is Newton-Krylov, based on Newton’s method and the resolution of the large matrix Sylvester equation by using block Krylov method. For all these methods, approximations are given in low rank form, wich allow us to save memory space. We have given numerical examples that show the effectiveness of the methods proposed in the case of large sizes.

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This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems. For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders. Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework. Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.