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54
Preconditioning techniques for large linear systems: A survey
- J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 192 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
pARMS: a parallel version of the algebraic recursive multilevel solver
- Numer. Linear Algebra Appl
, 2003
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Multilevel preconditioners constructed from inverse-based ILUs
, 2004
"... This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination prod ..."
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Cited by 32 (9 self)
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This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of “templates” for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.
Preconditioning Helmholtz linear systems
, 2009
"... Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILU-based preconditioners ineffective. This paper present ..."
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Cited by 22 (1 self)
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Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILU-based preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).
Multilevel ILU with reorderings for diagonal dominance
- SIAM J. Sci. Comput
, 2005
"... This paper presents a preconditioning method based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivot ..."
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Cited by 17 (7 self)
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This paper presents a preconditioning method based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivoting version of the incomplete LU factorization. This leads to an effective incomplete factorization preconditioner for general nonsymmetric, irregularly structured, sparse linear systems.
A PARALLEL MULTISTAGE ILU FACTORIZATION BASED ON A HIERARCHICAL GRAPH DECOMPOSITION
"... PHIDAL (Parallel Hierarchical Interface Decomposition ALgorithm) is a parallel incomplete factorization method which exploits a hierarchical interface decomposition of the adjacency graph of the coefficient matrix. The idea of the decomposition is similar to that of the well-known wirebasket techni ..."
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Cited by 15 (1 self)
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PHIDAL (Parallel Hierarchical Interface Decomposition ALgorithm) is a parallel incomplete factorization method which exploits a hierarchical interface decomposition of the adjacency graph of the coefficient matrix. The idea of the decomposition is similar to that of the well-known wirebasket techniques used in domain decomposition. However, the method is devised for general, irregularly structured, sparse linear systems. This paper describes a few algorithms for obtaining good quality hierarchical graph decompositions and discusses the parallel implementation of the factorization procedure. Numerical experiments are reported to illustrate the scalability of the algorithm and its effectiveness as a general purpose parallel linear system solver.
Robust parameter-free algebraic multilevel preconditioning
"... To precondition large sparse linear systems resulting from the discretization of second-order elliptic partial di erential equations, many recent works focus on the so-called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partit ..."
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Cited by 14 (0 self)
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To precondition large sparse linear systems resulting from the discretization of second-order elliptic partial di erential equations, many recent works focus on the so-called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown to be both robust and e cient in several circumstances, leading to iterative solution schemes of optimal order of computational complexity. Now, despite the procedure is essentially algebraic, previous works focus generally on a speci c context and consider schemes that use classical grid hierarchies with characteristic mesh sizes h; 2h; 4h, etc. Therefore, these methods require some extra information besides the matrix of the linear system and lack of robustness in some situations where semi-coarsening would be desirable. In this paper, we develop a general method that can be applied in a black box fashion to a wide class of problems, ranging from 2D model Poisson problems to 3D singularly perturbed convection–di usion equations. It is based on an automatic coarsening process similar to the one used in the AMG method, and on coarse grid matrices computed according to a simple and cheap aggregation principle. Numerical experiments illustrate the e ciency and the robustness of the proposed approach. Copyright? 2002 John Wiley & Sons, Ltd. KEY WORDS: iterative methods; convergence; preconditioning
On Preconditioning Schur Complement And Schur Complement Preconditioning
, 2000
"... . We study two implementation strategies to utilize Schur complement technique in multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. The first strategy constructs a RILUM to precondition the original matrix. The second strategy solves the first ..."
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Cited by 11 (3 self)
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. We study two implementation strategies to utilize Schur complement technique in multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. The first strategy constructs a RILUM to precondition the original matrix. The second strategy solves the first Schur complement matrix using the lower level parts of the RILUM as the preconditioner. We discuss computational and memory costs of both strategies and the potential effect on grid independent convergence rate of RILUM with different implementation strategies. Key words. sparse matrices, Schur complement, RILUM, preconditioning techniques. AMS subject classifications. 65F10, 65N06. 1. Introduction. In this paper, we discuss the issue of implementing a class of multilevel recursive incomplete LU (RILUM) preconditioners. These preconditioners were first reported and implemented in [40]. RILUM is a general framework for constructing robust multilevel preconditioning techniques based on blo...
A greedy strategy for coarse-grid selection
, 2006
"... Efficient solution of the very large linear systems that arise in numerical modelling of real-world applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considere ..."
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Cited by 10 (4 self)
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Efficient solution of the very large linear systems that arise in numerical modelling of real-world applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered, much recent interest has been in the development of purely algebraic approaches that may be applied in many situations, without problem-specific tuning. Here, we consider an algebraic approach to finding the fine/coarse partitions needed in multilevel approaches. The algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms, multilevel block factorization and algebraic multigrid. While no guarantee on the rate of coarsening is given, the splitting is shown to always yield an effective preconditioner in the two-level sense. Numerical performance of two-level and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations, and the advantages of each of these two algorithmic settings are explored. 1
