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Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches. [ABSTRACT FROM AUTHOR]

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, sparse linear systems with symmetric positive definite matrices need to be solved, and algebraic multigrid (AMG) methods represent common choices for the role of iterative solvers or preconditioners. The reason for their popularity relies on the fast convergence that these methods provide even in the solution of large size problems, which is a consequence of the AMG ability to reduce particular error components across their multilevel hierarchy. Despite carrying the name "algebraic," most of these methods still make assumptions on additional information other than the global assembled matrix, such as the knowledge of the operator's near kernel, which limits their applicability as black-box solvers. In this work, we introduce a novel AMG approach featuring the adaptive factored sparse approximate inverse (aFSAI) method as a flexible smoother, as well as three new approaches to adaptively compute the prolongation operator. We assess the performance of the proposed AMG through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other methods such as the aFSAI and BoomerAMG preconditioners, showing that our new method proves to be superior to the first strategy and comparable to the second one, if not better, as in the solution of linear elasticity models. [ABSTRACT FROM AUTHOR]

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size linear systems, especially when accurate results are sought for derived variables, like stress or deformation fields. Such a task represents the most time-consuming kernel, and motivates the development of robust and efficient linear solvers for these applications. On the one hand, direct solvers are robust and easy to use, but their computational complexity in the best scenario is superlinear, which limits applicability according to the problem size. On the other hand, iterative solvers, in particular those based on algebraic multigrid (AMG) preconditioners, can reach up to linear complexity, but require more knowledge from the user for an efficient setup, and convergence is not always guaranteed, especially in ill-conditioned problems. In this work, we present a novel AMG method specifically tailored for ill-conditioned structural problems. It is characterized by an adaptive factored sparse approximate inverse (aFSAI) method as smoother, an improved least-squared based prolongation (DPLS) and a method for uncovering the near-null space that takes advantage of an already existing approximation. The resulting linear solver has been applied in the solution of challenging linear systems arising from real-world linear elastic structural problems. Numerical experiments prove the efficiency and robustness of the method and show how, in several cases, the proposed algorithm outperforms state-of-the-art AMG linear solvers. Even more important, the results show how the proposed method gives good results even assuming a default setup, making it fully adoptable also for non-expert users and commercial software. • An Adaptive AMG linear solver for ill-conditioned structural problems is presented. • Results on more than 10 realistic examples prove its robustness and high efficiency. • Its performance is comparable or superior to other renowned state-of-art solvers. • It does not require tuning of user-parameters to achieve excellent performances. [ABSTRACT FROM AUTHOR]