A robust multilevel approximate inverse preconditioner for symmetric positive definite matrices

The use of factorized sparse approximate inverse (FSAI) preconditioners in a standard multilevel framework for symmetric positive definite (SPD) matrices may pose a number of issues as to the definiteness of the Schur complement at each level. The present work introduces a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity. The multilevel FSAI algorithm is further enhanced by introducing descending and ascending low-rank corrections, thus giving rise to the multilevel FSAI with low-rank corrections (MFLR) preconditioner. The proposed algorithm is investigated in a number of test problems. The numerical results show that the MFLR preconditioner is a robust approach that can significantly accelerate the solver convergence rate preserving a good degree of parallelism. The possibly large set-up cost, mainly due to the computation of the eigenpairs needed by low-rank corrections, makes its use attractive in applications where the preconditioner can be recycled along a number of linear solves

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 multilevel Schur complement preconditioner for complex symmetric matrices

This paper describes a multilevel preconditioning technique for solving complex symmetric sparse linear systems. The coefficient matrix is first decoupled by domain decomposition and then an approximate inverse of the original matrix is computed level by level. This approximate inverse is based on low rank approximations of the local Schur complements. For this, a symmetric singular value decomposition of a complex symmetric matix is used. Using the example of Maxwell's equations the generality of the approach is demonstrated

A multilevel Schur complement preconditioner with ILU factorization for complex symmetric matrices

This paper describes a multilevel preconditioning technique for solving complex symmetric sparse linear systems. The coefficient matrix is first decoupled by domain decomposition and then an approximate inverse of the original matrix is computed level by level. This approximate inverse is based on low rank approximations of the local Schur complements. For this, a symmetric singular value decomposition of a complex symmetric matix is used. The block-diagonal matrices are decomposed by an incomplete LDLT factorization with the Bunch-Kaufman pivoting method. Using the example of Maxwells equations the generality of the approach is demonstrated