This dissertation introduces a novel fast direct solver and preconditioner for the solution of block tridiagonal linear systems that arise from the discretization of elliptic partial differential equations on a Cartesian product mesh, such as the variable-coefficient Poisson equation, the convection-diffusion equation, and the wave Helmholtz equation in heterogeneous media. The algorithm extends the traditional cyclic reduction method with hierarchical matrix techniques. The resulting method exposes substantial concurrency, and its arithmetic operations and memory consumption grow only log-linearly with problem size, assuming bounded rank of off-diagonal matrix blocks, even for problems with arbitrary coefficient structure. The method can be used as a standalone direct solver with tunable accuracy, or as a black-box preconditioner in conjunction with Krylov methods. The challenges that distinguish this work from other thrusts in this active field are the hybrid distributed-shared parallelism that can demonstrate the algorithm at large-scale, full three-dimensionality, and the three stressors of the current state-of-the-art multigrid technology: high wavenumber Helmholtz (indefiniteness), high Reynolds convection (nonsymmetry), and high contrast diffusion (inhomogeneity). Numerical experiments corroborate the robustness, accuracy, and complexity claims and provide a baseline of the performance and memory footprint by comparisons with competing approaches such as the multigrid solver hypre, and the STRUMPACK implementation of the multifrontal factorization with hierarchically semi-separable matrices. The companion implementation can utilize many thousands of cores of Shaheen, KAUST's Haswell-based Cray XC-40 supercomputer, and compares favorably with other implementations of hierarchical solvers in terms of time-to-solution and memory consumption.

This research focuses on implementing and improving a parallel solution framework for the linear static analysis of large structural models on PC clusters. The framework consists of two separate programs where the first one is responsible from preparing data for the parallel solution that involves partitioning, workload balancing, and equation numbering. The second program is a fully parallel nite element program that utilizes substructure based solution approach with direct solvers. The first step of data preparation is partitioning the structure into substructures. After creating the initial substructures, the estimated imbalance of the substructures is adjusted by iteratively transferring nodes from the slower substructures to the faster ones. Once the final substructures are created, the solution phase is initiated. Each processor assembles its substructure&#039 / s stiffness matrix and condenses it to the interfaces. The interface equations are then solved in parallel with a block-cyclic dense matrix solver. After computing the interface unknowns, each processor calculates the internal displacements and element stresses or forces. Comparative tests were done to demonstrate the performance of the solution framework.

Afin de s'adapter aux architectures multicoeurs et aux machines de plus en plus complexes, les modèles de programmations basés sur un parallélisme de tâche ont gagné en popularité dans la communauté du calcul scientifique haute performance. Les moteurs d'exécution fournissent une interface de programmation qui correspond à ce paradigme ainsi que des outils pour l'ordonnancement des tâches qui définissent l'application. Dans cette étude, nous explorons la conception de solveurs directes creux à base de tâches, qui représentent une charge de travail extrêmement irrégulière, avec des tâches de granularités et de caractéristiques différentes ainsi qu'une consommation mémoire variable, au-dessus d'un moteur d'exécution. Dans le cadre du solveur qr mumps, nous montrons dans un premier temps la viabilité et l'efficacité de notre approche avec l'implémentation d'une méthode multifrontale pour la factorisation de matrices creuses, en se basant sur le modèle de programmation parallèle appelé "flux de tâches séquentielles" (Sequential Task Flow). Cette approche, nous a ensuite permis de développer des fonctionnalités telles que l'intégration de noyaux dense de factorisation de type "minimisation de cAfin de s'adapter aux architectures multicoeurs et aux machines de plus en plus complexes, les modèles de programmations basés sur un parallélisme de tâche ont gagné en popularité dans la communauté du calcul scientifique haute performance. Les moteurs d'exécution fournissent une interface de programmation qui correspond à ce paradigme ainsi que des outils pour l'ordonnancement des tâches qui définissent l'application. Dans cette étude, nous explorons la conception de solveurs directes creux à base de tâches, qui représentent une charge de travail extrêmement irrégulière, avec des tâches de granularités et de caractéristiques différentes ainsi qu'une consommation mémoire variable, au-dessus d'un moteur d'exécution. Dans le cadre du solveur qr mumps, nous montrons dans un premier temps la viabilité et l'efficacité de notre approche avec l'implémentation d'une méthode multifrontale pour la factorisation de matrices creuses, en se basant sur le modèle de programmation parallèle appelé "flux de tâches séquentielles" (Sequential Task Flow). Cette approche, nous a ensuite permis de développer des fonctionnalités telles que l'intégration de noyaux dense de factorisation de type "minimisation de cAfin de s'adapter aux architectures multicoeurs et aux machines de plus en plus complexes, les modèles de programmations basés sur un parallélisme de tâche ont gagné en popularité dans la communauté du calcul scientifique haute performance. Les moteurs d'exécution fournissent une interface de programmation qui correspond à ce paradigme ainsi que des outils pour l'ordonnancement des tâches qui définissent l'application. / To face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. In this study we investigate the design of task-based sparse direct solvers which constitute extremely irregular workloads, with tasks of different granularities and characteristics with variable memory consumption on top of runtime systems. In the context of the qr mumps solver, we prove the usability and effectiveness of our approach with the implementation of a sparse matrix multifrontal factorization based on a Sequential Task Flow parallel programming model. Using this programming model, we developed features such as the integration of dense 2D Communication Avoiding algorithms in the multifrontal method allowing for better scalability compared to the original approach used in qr mumps. In addition we introduced a memory-aware algorithm to control the memory behaviour of our solver and show, in the context of multicore architectures, an important reduction of the memory footprint for the multifrontal QR factorization with a small impact on performance. Following this approach, we move to heterogeneous architectures where task granularity and scheduling strategies are critical to achieve performance. We present, for the multifrontal method, a hierarchical strategy for data partitioning and a scheduling algorithm capable of handling the heterogeneity of resources. Finally we present a study on the reproducibility of executions and the use of alternative programming models for the implementation of the multifrontal method. All the experimental results presented in this study are evaluated with a detailed performance analysis measuring the impact of several identified effects on the performance and scalability. Thanks to this original analysis, presented in the first part of this study, we are capable of fully understanding the results obtained with our solver.