Development of scalable linear solvers for engineering applications

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, particularly those with diffusive character, sparse linear systems with symmetric positive definite (SPD) matrices need to be solved, and multilevel methods represent common choices for the role of iterative solvers or preconditioners. The weak scalability showed by those techniques is one of the main reasons for their popularity, since it allows the solution of linear systems with growing size without requiring a substantial increase in the computational time and number of iterations. On the other hand, single-level preconditioners such as the adaptive Factorized Sparse Approximate Inverse (aFSAI) might be attractive for reaching strong scalability due to their simpler setup. In this thesis, we propose four multilevel preconditioners based on aFSAI targeting the efficient solution of ill-conditioned SPD systems through parallel computing. The first two novel methods, namely Block Tridiagonal FSAI (BTFSAI) and Domain Decomposition FSAI (DDFSAI), rely on graph reordering techniques and approximate block factorizations carried out by aFSAI. Then, we introduce an extension of the previous techniques called the Multilevel Factorization with Low-Rank corrections (MFLR) that ensures positive definiteness of the Schur complements as well as improves their approximation with the aid of tall-and-skinny correction matrices. Lastly, we present the adaptive Smoothing and Prolongation Algebraic MultiGrid (aSPAMG) preconditioner belonging to the adaptive AMG family that introduces the use of aFSAI as a flexible smoother; three strategies for uncovering the near-null space of the system matrix and two new approaches to dynamically compute the prolongation operator. We assess the performance of the proposed preconditioners through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other approaches such as aFSAI, ILU (ILUPACK), and BoomerAMG (HYPRE), showing that our new methods prove comparable, if not superior, in many test cases

Accelerating advanced preconditioning methods on hybrid architectures

Un gran número de problemas, en diversas áreas de la ciencia y la ingeniería, involucran la solución de sistemas dispersos de ecuaciones lineales de gran escala. En muchos de estos escenarios, son además un cuello de botella desde el punto de vista computacional, y por esa razón, su implementación eficiente ha motivado una cantidad enorme de trabajos científicos. Por muchos años, los métodos directos basados en el proceso de la Eliminación Gaussiana han sido la herramienta de referencia para resolver dichos sistemas, pero la dimensión de los problemas abordados actualmente impone serios desafíos a la mayoría de estos algoritmos, considerando sus requerimientos de memoria, su tiempo de cómputo y la complejidad de su implementación. Propulsados por los avances en las técnicas de precondicionado, los métodos iterativos se han vuelto más confiables, y por lo tanto emergen como alternativas a los métodos directos, ofreciendo soluciones de alta calidad a un menor costo computacional. Sin embargo, estos avances muchas veces son relativos a un problema específico, o dotan a los precondicionadores de una complejidad tal, que su aplicación en diversos problemas se vuelve poco práctica en términos de tiempo de ejecución y consumo de memoria. Como respuesta a esta situación, es común la utilización de estrategias de Computación de Alto Desempeño, ya que el desarrollo sostenido de las plataformas de hardware permite la ejecución simultánea de cada vez más operaciones. Un claro ejemplo de esta evolución son las plataformas compuestas por procesadores multi-núcleo y aceleradoras de hardware como las Unidades de Procesamiento Gráfico (GPU). Particularmente, las GPU se han convertido en poderosos procesadores paralelos, capaces de integrar miles de núcleos a precios y consumo energético razonables.Por estas razones, las GPU son ahora una plataforma de hardware de gran importancia para la ciencia y la ingeniería, y su uso eficiente es crucial para alcanzar un buen desempeño en la mayoría de las aplicaciones. Esta tesis se centra en el uso de GPUs para acelerar la solución de sistemas dispersos de ecuaciones lineales usando métodos iterativos precondicionados con técnicas modernas. En particular, se trabaja sobre ILUPACK, que ofrece implementaciones de los métodos iterativos más importantes, y presenta un interesante y moderno precondicionador de tipo ILU multinivel. En este trabajo, se desarrollan versiones del precondicionador y de los métodos incluidos en el paquete, capaces de explotar el paralelismo de datos mediante el uso de GPUs sin afectar las propiedades numéricas del precondicionador. Además, se habilita y analiza el uso de las GPU en versiones paralelas existentes, basadas en paralelismo de tareas para plataformas de memoria compartida y distribuida. Los resultados obtenidos muestran una sensible mejora en el tiempo de ejecución de los métodos abordados, así como la posibilidad de resolver problemas de gran escala de forma eficiente

Numerical models for the large-scale simulation of fault and fracture mechanics

The possible activation of pre-existing faults and the generation of new fractures in the subsurface may play a critical role in several fields of great social interest, such as the management and the exploitation of groundwater resources, especially in arid areas, the hydrocarbon recovery and storage, and the monitoring of the seismic activity in the Earth’s crust. The sliding and/or opening of a fault can create preferential leakage paths for the pore fluid escape, causing a matter of great concern in the process of storing fluids and hydrocarbons underground. The most challenging effect connected to a fault activation is the possible earthquake triggering. Many earthquakes associated with the production and injection of fluids have been recently reported. Similar issues arise also in the development of unconventional hydrocarbon reservoirs, that has recently experienced a dramatic increase thanks to the deployment of the “fracking” technology, which is based on the massive generation of fractures through the injection of fluids at high pressures. The use of this technique in densely populated areas has raised a large scientific debate on the possible connected environmental risks. The over-exploitation of fresh aquifers in arid regions has caused the generation of significant ground fissures. In this thesis, a novel formulation based on the use of Lagrange multipliers has been developed for the stable and robust numerical modeling of fault mechanics. A fault or fracture is simulated as a pair of inner surfaces included in a 3D geological formation where Lagrange multipliers are used to prescribe the contact constraints. The standard variational formulation of the contact problem with Lagrange multipliers is modified to take into account the energy dissipated by the frictional work along the activated fault portion. This term is computed by making use of the principle of maximum plastic dissipation, whose application defines the direction of the limiting shear stress vector. The novel approach has been verified against analytical solutions and applied in a number of real-world problems. In particular, we test the novel approach in four cases: (i) mechanics of two adjacent blocks, to investigate the numerical properties of the algorithm; (ii-iii) ground fractures due to groundwater withdrawal, with different geometries; (iv) fault reactivation in an underground reservoir subject to primary production and Underground Gas Storage cycles. The results are analyzed and commented. In the fourth case, the possible magnitude of the seismic events triggered by fault reactivation is computed, in order to evaluate whether underground human activities may generate seismicity. The application of the fault model to large-scale problems gives rise to a set of sparse discrete systems of linearized equations with a generalized non-symmetric saddle point structure. The second part of this thesis is devoted to the development of efficient algorithms for the iterative solution of this kind of system. We focus on a preconditioning technique, denoted as “constraint preconditioning”, which exploits the native block structure of the Jacobian. The quality and performance of the preconditioner relies on two steps: (i) the preconditioning of the leading block and (ii) the Schur complement computation. In this work, novel preconditioning techniques for the leading block based on a multilevel framework are developed and tested. The main idea behind the multilevel preconditioner is to improve the quality of the factorized approximate inverses borrowing the scheme of incomplete factorizations, thus introducing some sequentially in perfectly parallelizable algorithms. The proposed approach is robust, from a theoretical point of view, and very efficient in parallel environment. As to the latter point, i.e. the Schur complement computation, it can be done with the aid of different approximations. The main difference is whether the Jacobian is symmetrized or not. The computation can be founded on the FSAI approximation of the leading block inverse or on a physically-based block diagonal block algorithm. The Schur complement must be inverted, thus other possibilities come in. The approximate Schur complement can be inverted through FSAI, if symmetric, or an incomplete factorization, if non-symmetric, but it can also be solved exactly, thanks to a direct solver. The performances of the proposed algorithms are finally investigated and discussed in a set of real-world numerical examples