Using the VBARMS method in parallel computing

The paper describes an improved parallel MPI-based implementation of VBARMS, a variable block variant of the pARMS preconditioner proposed by Li, Saad and Sosonkina [NLAA, 2003] for solving general nonsymmetric linear systems. The parallel VBARMS solver can detect automatically exact or approximate dense structures in the linear system, and exploits this information to achieve improved reliability and increased throughput during the factorization. A novel graph compression algorithm is discussed that finds these approximate dense blocks structures and requires only one simple to use parameter. A complete study of the numerical and parallel performance of parallel VBARMS is presented for the analysis of large turbulent Navier-Stokes equations on a suite of three- dimensional test cases

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Implicit methods based on the Newton’s rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynolds-averaged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of three-dimensional test cases

VBARMS: A variable block algebraic recursive multilevel solver for sparse linear systems

Sparse matrices arising from the solution of systems of partial differential equations often exhibit a perfect block structure. It means that the nonzero blocks in the sparsity pattern are fully dense (and typically small), e.g., when several unknown quantities are associated with the same grid point. However, similar block orderings can be sometimes found also on general unstructured matrices by ordering consecutively rows and columns with a similar sparsity pattern. We also can treat some zero entries of the reordered matrix as nonzero elements to enlarge the blocks to improve the performance. The reordering results in linear systems with blocks of variable size in general. Our recently developed parallel package pVBARMS (parallel variable block algebraic recursive multilevel solver) for distributed memory computers takes advantage of these frequently occurring structures in the design of the multilevel incomplete LU factorization preconditioner. It maximizes computational efficiency and achieves increased throughput during the computation and improved reliability on realistic applications. The method detects automatically any existing block structure in the matrix without any users prior knowledge of the underlying problem, and exploits it to maximize computational efficiency. We proposed a study of performance comparison of pVBAMRS and other popular solvers on a set of general linear systems arising from different application field. We also report on the numerical and parallel scalability of the pVBARMS package for solving the turbulent, Reynolds-averaged, Navier-Stokes (RANS) equations

A hybrid recursive multilevel incomplete factorization preconditioner for solving general linear systems

In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems. The novelty of the proposed method is to combine factorization techniques of both implicit and explicit type, recursive combinatorial algorithms, multilevel mechanisms and overlapping strategies to maximize sparsity in the inverse factors and consequently reduce the factorization costs. Numerical experiments demonstrate the good potential of the proposed solver to precondition effectively general linear systems, also against other state-of-the-art iterative solvers of both implicit and explicit form

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