The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence.  BiCG is the basis of several very popular methods, such as BiCGStab.  The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time.  However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence.  Despite this apparent sensitivity, BiCG often converges well in practice.  This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly.  We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods. / Master of Science

Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.

Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.

<p>This work explores how randomization can be exploited to deliver sophisticated</p><p>algorithms with provable bounds for: (i) The approximation of matrix functions, such</p><p>as the log-determinant and the Von-Neumann entropy; and (ii) The low-rank approximation</p><p>of matrices. Our algorithms are inspired by recent advances in Randomized</p><p>Numerical Linear Algebra (RandNLA), an interdisciplinary research area that exploits</p><p>randomization as a computational resource to develop improved algorithms for</p><p>large-scale linear algebra problems. The main goal of this work is to encourage the</p><p>practical use of RandNLA approaches to solve Big Data bottlenecks at industrial</p><p>level. Our extensive evaluation tests are complemented by a thorough theoretical</p><p>analysis that proves the accuracy of the proposed algorithms and highlights their</p><p>scalability as the volume of data increases. Finally, the low computational time and</p><p>memory consumption, combined with simple implementation schemes that can easily</p><p>be extended in parallel and distributed environments, render our algorithms suitable</p><p>for use in the development of highly efficient real-world software.</p>

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For sequences of related linear systems, the computation of a preconditioner for every system can be expensive. Often a fixed preconditioner is used, but this may not be effective as the matrix changes. This research examines the benefits of both reusing and recycling preconditioners, with special focus on ILUTP and factorized sparse approximate inverses and proposes an update that we refer to as a sparse approximate map or SAM update. Analysis of the residual and eigenvalues of the map will be provided. Applications include the Quantum Monte Carlo method, model reduction, oscillatory hydraulic tomography, diffuse optical tomography, and Helmholtz-type problems. / Master of Science

Le sujet de cette thèse est le développement de méthodes itératives permettant la résolution degrands systèmes linéaires creux d'équations présentant plusieurs seconds membres simultanément. Ces méthodes seront en particulier utilisées dans le cadre d'une application géophysique : la migration sismique visant à simuler la propagation d'ondes sous la surface de la terre. Le problème prend la forme d'une équation d'Helmholtz dans le domaine fréquentiel en trois dimensions, discrétisée par des différences finies et donnant lieu à un système linéaire creux, complexe, non-symétrique, non-hermitien. De plus, lorsque de grands nombres d'onde sont considérés, cette matrice possède une taille élevée et est indéfinie. Du fait de ces propriétés, nous nous proposons d'étudier des méthodes de Krylov préconditionnées par des techniques hiérarchiques deux niveaux. Un tel pre-conditionnement s'est montré particulièrement efficace en deux dimensions et le but de cette thèse est de relever le défi de l'adapter au cas tridimensionel. Pour ce faire, des méthodes de Krylov sont utilisées à la fois comme lisseur et comme méthode de résolution du problème grossier. Ces derniers choix induisent l'emploi de méthodes de Krylov dites flexibles. / The topic of this PhD thesis is the development of iterative methods for the solution of large sparse linear systems of equations with possibly multiple right-hand sides given at once. These methods will be used for a specific application in geophysics - seismic migration - related to the simulation of wave propagation in the subsurface of the Earth. Here the three-dimensional Helmholtz equation written in the frequency domain is considered. The finite difference discretization of the Helmholtz equation with the Perfect Matched Layer formulation produces, when high frequencies are considered, a complex linear system which is large, non-symmetric, non-Hermitian, indefinite and sparse. Thus we propose to study preconditioned flexible Krylov subspace methods, especially minimum residual norm methods, to solve this class of problems. As a preconditioner we consider multi-level techniques and especially focus on a two-level method. This twolevel preconditioner has shown efficient for two-dimensional applications and the purpose of this thesis is to extend this to the challenging three-dimensional case. This leads us to propose and analyze a perturbed two-level preconditioner for a flexible Krylov subspace method, where Krylov methods are used both as smoother and as approximate coarse grid solver.

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.

No description available.