Threshold Ordering for Preconditioning Nonsymmetric Problems

Numerical experiments are presented whereby the e ect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are di erent variants of incomplete factorizations. It is shown that reorderings for direct methods, such as Reverse Cuthill-McKee and Minimum Degree, can be very bene cial. The bene t can be seen in the reduction of the number of iterations and also in measuring the appropriate deviation of the preconditioned operator from the identity.

Computer Methods in Applied Mechanics and Engineering, 1986

The task of making an incomplete factorization of the finite element stiffness matrix using only element matrices is concerned. We present a technique for realizing this and obtain a method which requires an amount of core storage that is independent of the number of unknowns in the discrete model, i.e., of the mesh size parameter. On the other hand datatransfers from/ to secondary storage and more arithmetic operations than in a corresponding completely-in-core method are required. For many problems solved in practice even thetotalrequirementofstorageforthestiffnessandpreconditioningmatricesisindependentofthesizeofthe mesh. Theoretical estimates of the rate of convergence of the corresponding preconditioned conjugate gradient method are derived for a model problem and a number of test examples are examined.

Simulation with models based on partial differential equations require very often the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present some ideas for updating approximate inverse preconditioners in factorized form. Computational costs, reorderings and implementation issues are considered.

BIT Numerical Mathematics, 2010

In this paper, we consider level-based preconditioning, which is one of the basic approaches to incomplete factorization preconditioning of iterative methods. It is well-known that while structure-based preconditioners can be very useful, excessive memory demands can limit their usefulness. Here we present an improved strategy that considers the individual entries of the system matrix and restricts small entries to contributing to fewer levels of fill than the largest entries. Using symmetric positive-definite problems arising from a wide range of practical applications, we show that the use of variable levels of fill can yield incomplete Cholesky factorization preconditioners that are more efficient than those resulting from the standard level-based approach. The concept of level-based preconditioning, which is based on the structural properties of the system matrix, is then transferred to the numerical incomplete decomposition. In particular, the structure of the incomplete factorization determined in the symbolic factorization phase is explicitly used in the numerical factorization phase. Further numerical results demonstrate that our level-based approach can lead to much sparser but efficient incomplete factorization preconditioners.

Computing, 1994

Circulant Block-Factorization Preconditioners for Elliptic Problems. New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case of y-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to a y-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner is O(N log N) (based on FFT), where N is the number of unknowns. That is, the algorithm is nearly optimal Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.

Journal of Applied Mathematics and Computing, 2004

We develop in this paper some preconditioners for sparse nonsymmetric M-matrices, which combine a recursive two-level block ILU factorization with multigrid method, we compare these preconditioners on matrices arising from discretized convection-diffusion equations using upwind finite difference schemes and multigrid orderings, some comparison theorems and experiment results are demonstrated.

Numerical Linear Algebra with Applications

Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in 2D and 3D with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems while multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

SIAM Journal on Scientific Computing, 2007

The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: We incorporate a nonsymmetric permutation to obtain a large diagonal, we use a more general parametrization for TPABLO, and we use a block Gauss-Seidel preconditioner which can be implemented to have the same execution time as the corresponding block Jacobi preconditioner. Experiments are presented showing that for certain classes of matrices, the block Gauss-Seidel preconditioner used with the system permuted with the new algorithm can outperform the best ILUT preconditioners in a large set of experiments. ).

SIAM Journal on Matrix Analysis and Applications, 2017

For general symmetric positive definite (SPD) matrices, we present a framework for designing effective and robust black-box preconditioners via structured incomplete factorization. In a scaling-and-compression strategy, off-diagonal blocks are first scaled on both sides (by the inverses of the factors of the corresponding diagonal blocks) and then compressed into low-rank approximations. ULV-type factorizations are then computed. A resulting prototype preconditioner is always positive definite. Generalizations to practical hierarchical multilevel preconditioners are given. Systematic analysis of the approximation error, robustness, and effectiveness is shown for both the prototype preconditioner and the multilevel generalization. In particular, we show how local scaling and compression control the approximation accuracy and robustness, and how aggressive compression leads to efficient preconditioners that can significantly reduce the condition number and improve the eigenvalue clustering. A result is also given to show when the multilevel preconditioner preserves positive definiteness. The costs to apply the multilevel preconditioners are about O(N), where N is the matrix size. Numerical tests on several ill-conditioned problems show the effectiveness and robustness even if the compression uses very small numerical ranks. In addition, significant robustness and effectiveness benefits can be observed as compared with a standard rank structured preconditioner based on direct off-diagonal compression.

2010

This work analyzes the influenc of matrices reordering algorithms on solving linear systems using non-stationary iterative methods GMRES and Conjugate Gradient, both with and without preconditioning. The algorithms referenced most often in the literature for the reordering of matrices are Reverse Cuthill-McKee (RCM), Gibbs-Poole-Stockmeyer (GPS), Nested Dissection (ND) and Spectral (ES). We analyze these algorithms and propose some modification comparing their solution qualities (minimizing bandwidth and minimizing envelope) and CPU times. Moreover, the linear systems associated with sparse matrices are solved via preconditioned Krylov-type iterative methods considering the incomplete LU factorization preconditioners. For the computational tests, we consider a set of structurally symmetric matrices that can come from various field of knowledge. We conclude that the reordering of matrices, in most cases, reduces the number of iterations in the iterative methods, but that reducing the...