In this paper we present an approach to the nonlinear model reduction based on representing the nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a 'training input'. Computational results and performance data are presented for a nonlinear circuit and a micromachined fixed-fixed beam example. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or the recently developed quadratic reduction techniques. Finally, it is shown tat the proposed technique is computationally inexpensive, and that the models can be constructed 'on-the-fly', to accelerate simulation of the system response. / Singapore-MIT Alliance (SMA)

The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

Small signal stability has always been an important concern for analog designers. Recent advances such as the Loop Finder algorithm allows designers to detect and identify local, potentially unstable return loops without the need to identify and add breakpoints. However, this method suffers from extremely high time and memory complexity and thus cannot be scaled to very large analog circuits. In this research work, we first take an in-depth look at the loop finder algorithm so as to identify certain key enhancements that can be made to overcome these shortcomings. We next propose pole discovery and impedance computation methods that address these shortcomings by exploring only a certain region of interest in the s-plane. The reduced time and memory complexity obtained via the new methodology allows us to extend automatic stability checking to much larger circuits than was previously possible.

Computational methods for reducing the complexity of Finite Element (FE) models in structural dynamics are usually based on modal analysis. Classical approaches such as modal truncation, static condensation (Craig-Bampton, Guyan), and component mode synthesis (CMS) are available in many CAE tools such as ANSYS. In other disciplines, different techniques for Model Order Reduction (MOR) have been developed in the previous 2 decades. Krylov subspace methods are one possible choice and often lead to much smaller models than modal truncation methods given the same prescribed tolerance threshold. They have become available to ANSYS users through the tool mor4ansys. A disadvantage is that neither modal truncation nor CMS nor Krylov subspace methods preserve properties important to control design. System-theoretic methods like balanced truncation approximation (BTA), on the other hand, are directed towards reduced-order models for use in closed-loop control. So far, these methods are considered to be too expensive for large-scale structural models. We show that recent algorithmic advantages lead to MOR methods that are applicable to FE models in structural dynamics and that can easily be integrated into CAE software. We will demonstrate the efficiency of the proposed MOR method based on BTA using a control system including as plant the FE model of a machine tool.

We consider large and sparse eigenproblems where the spectrum exhibits special symmetries. Here we focus on Hamiltonian symmetry, that is, the spectrum is symmetric with respect to the real and imaginary axes. After briefly discussing quadratic eigenproblems with Hamiltonian spectra we review structured Krylov subspace methods to aprroximate parts of the spectrum of Hamiltonian operators. We will discuss the optimization of the free parameters in the resulting symplectic Lanczos process in order to minimize the conditioning of the (non-orthonormal) Lanczos basis. The effects of our findings are demonstrated for several numerical examples.

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory $\mathcal{H}_2$ optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are \emph{optimal} with respect to an $\mathcal{H}_2\otimes\mathcal{L}_2$ joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications.

Studijų metu gerai susipažinęs su dizaino sąvoka pastebėjau, kad ji apima įvairią kūrybą, gerokai daugiau nei iki tol įsivaizdavau. Anglakalbių šalių žodžio design supratimas dar gerokai platesnis. Ten grafikos dizaineris užsiima platesniu darbų spektru, kurį taip pat įvardija kaip design ar tai būtų logotipo kūrimas, ar interaktyvus skaitmeninis objektas. Bebaigdamas savo studijas pastebiu, kad didžioji dalis darbų buvo susietų su pramoniniu, interjero dizainu ir vienokia ar kitokia spauda. Dėl šios priežasties nusprendžiau savarankiškai gilinti dizaino įgūdžius, techninius gebėjimus platesniame kompiuterinių programų rate. Todėl bakalauro darbe, kurį įvardinau judančia grafika (angl. Motion graphics), pasitelkiau šiandien plačiai skaitmenine kūryba užsiimančių menininkų naudojamas programas. Motion graphics (judanti grafika) - tai grafika, kuriai pritaikoma video arba/ir animavimo technologija, sukurianti judesio iliuziją ar besikeičiančią objekto išvaizdą. Judanti grafika dažniausiai derinama su garsu naudojant įvairiuose multimedijos projektuose. Judanti grafika dažniausiai rodoma pasinaudojant elektroninės medijos technologijomis. Terminas naudojamas atskiriant nejudančią grafiką nuo grafikos keičiančios išvaizdą laike. Žinoma, mano dailės ir dizaino specialybė orientuota į mokytojo darbą, todėl filmuko kūrimas naudojant flash technologiją, video ir audio montavimo programas buvo puikus pasirinkimas tobulinant savo kaip dizaino specialisto gebėjimus. Taip pat plečiant... [toliau žr. visą tekstą] / For my bachelor degree work I created motion graphics short films which are suited to be shown at schools for children, as well the creation of them are written down in this work and can be used for school as a methodical instructions how these kind of films can be made. My work inspired by Ivan Krylov - a classic well known in the world Russian fables writer. This way my motion graphics films have a deep educational value. To create them I mainly used Adobe Photoshop, Adobe Flash, Sony Vegas and Audacity computer software. My work is great methodical instructions for young art and design teachers who want to teach their students how to use computer software basics for creative tasks. To create my motion graphics films I used computer software which is highly used by today's professional digital artists.

Stuttgart, Univ., Diss., 2005.

Universiẗat, Diss., 2003--Stuttgart.

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.