A Bramble-Pasciak-like method with applications in optimization

Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from partial differential equations and in the area of Optimization such problems are ubiquitous.\ud In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples

Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

In this paper, we study a class of tuned preconditioners that will be designed to accelerate both the DACG-Newton method and the implicitly restarted Lanczos method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. These tuning strategies are based on low-rank modifications of a given initial preconditioner. We present some theoretical properties of the preconditioned matrix. We experimentally show how the aforementioned methods benefit from the acceleration provided by these tuned/deflated preconditioners. Comparisons are carried out with the Jacobi-Davidson method onto matrices arising from various large realistic problems arising from finite element discretization of PDEs modeling either groundwater flow in porous media or geomechanical processes in reservoirs. The numerical results show that the Newton-based methods (which includes also the Jacobi-Davidson method) are to be preferred to the - yet efficiently implemented - implicitly restarted Lanczos method whenever a small to moderate number of eigenpairs is required. \ua9 2016 John Wiley & Sons, Ltd

Null-space preconditioners for saddle point systems

The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive-definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization, and compare their performance with the equivalent Schur-complement based preconditioners. We also describe how to apply the non-symmetric preconditioners proposed using the conjugate gradient method (CG) with a non-standard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Fast nonlinear solvers in solid mechanics

La thèse a pour objectif le développement de méthodes performantes pour la résolution de problèmes non linéaires ne mécanique des solides. Il est coutume d'utiliser une méthode de type Newton qui conduit à la résolution d'une séquence de systèmes linéaires. De plus, la prise en compte des relations linéaires imposées à l'aide de multiplicateurs de Lagrange confère aux matrices une structure de point-selle. Dans un cadre plus général, nous proposons, étudions et illustrons deux classes d'enrichissement de préconditionneurs (limited memory preconditioners) pour la résolution de séquences de systèmes linéaires par une méthode de Krylov. La première est un extension au cas symétrique indéfini d'une méthode existante, développée initialement dans le cadre symétrique défini positif. La seconde est plus générale dans le sens où elle s'applique aus systèmes non symétriques. Ces deux familles peuvent être interprétées comme des variantes par blocs de formules de mise à jour utilisées dans différentes méthodes d'optimisation. Ces techniques ont été développées dans le logiciel de mécanique des solides Code_Aster (dans un environnement parallèle distribué via la bibliothèque PETSc) et sont illustrées sur plusieurs études industrielles. Les gains obtenus en terme de coût de calcul sont significatifs (jusqu'à 50%), pour un surcoût mémoire négligeable.The thesis aims at developing efficient numerical methods to solve nonlinear problems arising un solid mechanics. In this field, Newton methods are currently used, requiring the solution of a sequence of linear systems. Furthermore, the imposed linear relations are dualized with the Lagrange multipliers, leading to matrices with a saddle point structure. In a more general framework, we propose two classes of preconditioners (named limited memory preconditioners) to solve sequences of linear systems with a Krylov subspace method. The first class is based on an extension of a method initially developed for symmetric positive definite matrices to the symmetric indefinite case. Both families can be interpreted as block variants of updating formulas used in numerical optimization. They have been implemented into the Code_Aster solid mechanics software (in a parallel distributed environement using the PETSc library). These new preconditioning strategies are illustrated on several industrial applications. We obtain significant gains in computational cost (up to 50%) at a marginal overcost in memory

Fast Iterative Reconstruction in X-Ray Tomography Using Polar Coordinates

RÉSUMÉ Notre objectif est de réduire l’espace mémoire nécessaire ainsi que le temps de reconstruction excessif des méthodes itératives en imagerie à rayons X. En général, les méthodes itératives permettent d’obtenir une meilleure qualité de reconstruction que celles obtenues par FBP. Cela est dû à l’utilisation d’un modèle plus précis dans le processus de reconstruction. En effet, le modèle prend en compte le bruit et peut introduire un certain a priori sur l’image à reconstruire. Le problème peut alors être résolu par des techniques d’optimisation. En reconstruction d’images par méthodes itératives, la taille importante de la matrice de projection joue un rôle prédominant dans la mémoire requise par ce type de méthodes. Le temps de reconstruction, quant à lui, est rallongé par le grand nombre d’opérations de projection et rétroprojection. Ces aspects nécessitent une attention particulière lors de reconstructions par approches itératives. L’objectif de cette maîtrise a été de s’attaquer à ces deux aspects en développant une technique efficace de reconstruction d’images médicales. L’hypothèse de rayons X monochromatiques est utilisée et l’invariance en coordonnées polaires des tomographes commerciaux est considérée. En effet, l’utilisation de coordonnées polaire pour représenter l’objet permet d’obtenir une certaine redondance dans les coefficients de la matrice de projection. Celle-ci est parcimonieuse et a une structure bloc-circulante, ce qui mène à une réduction significative de l’espace mémoire nécessaire pour la stocker. Mais ce type de représentation entraîne des questions de qualité de reconstruction ainsi que des questions sur les aspects numériques des méthodes. Ce travail aborde les problèmes soulevés. Comme mentionné plus haut, le temps de reconstruction en tomographie rayons X est essentiellement déterminé par le temps de calcul des opérations de projection et rétroprojection qui sont réalisées à chaque itération. La parallélisation des calculs permet de réduire le temps de reconstruction de manière significative, ceci est abordé ici. De plus, la conception de préconditionneurs adaptés à la fonction objectif entraîne une amélioration de la vitesse de convergence des méthodes itératives. Ce travail consiste en une étude préliminaire sur les performances de reconstruction d’images tomographiques en se basant sur une représentation polaire des objets à imager. Les résultats obtenues dans ce mémoire peuvent être utilisés pour la reconstruction d’images cliniques 3D. Il est aussi possible d’étendre les algorithmes développés ici à un modèle polychromatique et aussi de réduire les artefacts métalliques.----------ABSTRACT We aim at reducing the high memory need and the long reconstruction time of the iterative methods for reconstructing the X-ray tomography images. In general, iterative methods are capable of providing a higher quality reconstructed image compared to those obtained through filtered backprojection. This is because in the iterative methods, a more accurate model is used in the reconstruction process. The model used in this technique accounts for the noise and can incorporate some prior knowledge on the image and therefore can provide images with higher quality compared to those obtained using the filtered backprojection technique. The reconstruction problem can then be solved using optimization techniques. In using iterative methods for image reconstruction, the large size of the projection matrix is the main cause of having high memory need in this method. Moreover, requiring to perform projection and backprojection operations numerous times is the main reason for the long reconstruction time. These problems need to be addressed properly for wider adoption of the iterative approaches in image reconstuction. The work presented herein aims at addressing these problems by developing an efficient technique which makes reconstruction of clinical size images possible. This will be done in a simple framework under the assumption of a monochromatic X-ray source. The objective is fulfilled by considering the fact that the geometry of commercial tomographs is invariant in polar coordinates. Using polar coordinates for representing the object, the coefficients of the projection matrix will be highly redundant. The matrix is also very sparse and has a block-circulant structure. Consequently, using polar coordinates for representing the object leads to a significant decrease in memory requirement. There are some questions associated with this type of representation which include numerical efficiency of the reconstruction process using this type of representation and actual quality of reconstructed image. This work tries to study and address these questions. As already mentioned, reconstruction time of tomography problems is mainly determined by the computation time of projection and backprojection operations that need to be performed at each iteration. The parallel implementation of these operations can reduce the reconstruction time significantly and is addressed here. Moreover, by designing preconditioners tailored to the structure of the objective function a sufficient increase in the convergence speed of iterative methods was achieved. The current work is a preliminary study on the efficiency of using polar coordinates for representing the object and reconstructing the tomography images. The results which have been obtained in this work can now be used for developing the 3D reconstruction of clinical data. We can also use the developed algorithms in this work to expand the current framework t

Preconditioners for iterative solutions of large-scale linear systems arising from Biot's consolidation equations

Ph.DDOCTOR OF PHILOSOPH

Preconditioners for Soil-Structure Interaction Problems with Significant Material Stiffness Contrast

Ph.DDOCTOR OF PHILOSOPH