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

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

A novel AMG approach based on adaptive smoothing and prolongation for reservoir simulations

Reservoir models can easily incorporate millions or even billions of unknowns. Algebraic multigrid (AMG) methods are often the standard choice as iterative solvers or preconditioners for the solution of the resulting linear systems. These comprise a family of techniques built on a hierarchy of levels associated with decreasing size problems. In this way, optimality and efficiency are achieved by combining two complementary processes, i.e. relaxation and coarse-grid correction. One of the key factors defining a fast AMG method consists of capturing accurately the near-null space of the system matrix for the construction of suitable prolongation operators. In this work, we propose a novel AMG package, aSP-AMG, where aSP means \u201cadaptive Smoothing and Prolongation\u201d and the \u201cadaptive\u201d attribute implies that we follow the perspective of adaptive and bootstrap AMG. We construct a space of smooth vectors of limited size (test space) using the Lanczos method and introduce the factorized sparse approximate inverse (FSAI) as a smoother. This improves the smoothing capabilities of the aSP-AMG as FSAI is more effective than Jacobi and much sparser than Gauss-Seidel. Moreover, FSAI has been shown to be strongly scalable. The coarsening phase is carried out as in classical AMG, but the strength of connection is computed by means of the affinity based on the test space. Finally, three new techniques are developed for building the prolongation operator: i) ABF, running few iterations of the aFSAI algorithm; ii) LS-ABF, updating the ABF coefficients with a least squares minimization; iii) DPLS, considering a least-squares process only. The aSP-AMG performance is assessed through the solution of reservoir engineering problems including both fluid flow and geomechanical test cases. Comparisons are made with the FSAI and BoomerAMG preconditioners, showing that the new method is generally superior to both approaches

Recent advancements in preconditioning techniques for large size linear systems suited for high performance computing

The numerical simulations of real-world engineering problems create models with several millions or even billions of degrees of freedom. Most of these simulations are centered on the solution of systems of non-linear equations, that, once linearized, become a sequence of linear systems, whose solution is often the most time-demanding task. Thus, in order to increase the capability of modeling larger cases, it is of paramount importance to exploit the resources of High Performance Computing architectures. In this framework, the development of new algorithms to accelerate the solution of linear systems for many-core architectures is a really active research field. Our main focus is algebraic preconditioning and, among the various options, we elect to develop approximate inverses for symmetric and positive definite (SPD) linear systems, both as stand-alone preconditioner or smoother for AMG techniques. This choice is mainly supported by the almost perfect parallelism that intrinsically characterizes these algorithms. As basic kernel, the Factorized Sparse Approximate Inverse (FSAI) developed in its adaptive form by Janna and Ferronato is selected. Recent developments are i) a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity and ii) a novel AMG approach featuring the adaptive FSAI method as a flexible smoother as well as new approaches to adaptively compute the prolongation operator. In this latter work, a new technique to build the prolongation is also presented

A New Semi-Structured Algebraic Multigrid Method

Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display excellent parallelization properties. Since current architecture trends are favoring regular compute patterns to achieve high performance, the ability to express structure has become much more important. The hypre software library provides high-performance multigrid preconditioners and solvers through conceptual interfaces, including a semi-structured interface that describes matrices primarily in terms of stencils and logically structured grids. This paper presents a new semi-structured algebraic multigrid (SSAMG) method built on this interface. The numerical convergence and performance of a CPU implementation of this method are evaluated for a set of semi-structured problems. SSAMG achieves significantly better setup times than hypre's unstructured AMG solvers and comparable convergence. In addition, the new method is capable of solving more complex problems than hypre's structured solvers