R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....guess was always x 0 =0,r 0 =r 0 =b, and the stopping criterion #b Axm # 2 #b# 2 10 8 ,x m= Mym . 33) In Table 1 we present the convergence results for a variety of iterative methods and decreasing values of #. The algorithms for the various iterative methods were obtained from [2]. The convergence results without preconditioning are listed under M = I. We denote by CGNE the CG algorithm applied to the preconditioned normal equations AMM T A T y = b with x = MM T A T y. Aswereduce#and #AM I#F , the number of iterations drops significantly for all iterative ....

....In all numerical calculations the initial guess was x 0 =0,r 0 =r 0 =b, and unless specified the stopping criterion was as in (33) All the computations were done in double precision FORTRAN, partly on a Sparc10 Sun station and partly on an Iris 4D 35 SGI station. The FORTRAN code provided in [2] was used for GMRES. We begin with the convergence results for the five shermanx black oil simulators. This set consists of sherman1: a black oil simulator, shale barrier, 10 10 10 grid, 1 unknown, of size n = 1000, and with nz = 3750 nonzero elements. sherman2: a thermal simulation, steam ....

R. BARRET ET AL., Templates for the Solution of Linear Systems, SIAM, Philadelphia, PA, 1994.

.... k x k , kd k k, ae 0 = 1 p k = 0 repeat call MC(c k ; d k ) d k : d k A k c k ae 1 = c T k d k p k : c k ae 1 ae 0 p k ae 0 = ae 1 , ff = ae 0 p T k A k p k d k : d k Gamma ffA k p k x k : x k ffp k until kd k k g For nonsymmetric problems we use bi cg stab (cf. [3]) BCGS(x k ; b k ) f d k = b k Gamma A k x k , kd k k, ae 1 = ff = 1 p k = 0, v k = 0, w k = d k repeat ae 0 = ae 1 , ae 1 = d T k w k p k : d k ae 1 ae 0 ff (p k Gamma v k ) call MC(p k ; p k ) v k = A k p k ff = ae 1 v T k d k s k = d k Gamma ffv k call ....

R. BARRET, M. BERRY, T. CHAN, J. DEMMEL, J. DONATO, J. DONGARRA, V. EI- JKHOUT, R. POZO, C. ROMINE, AND H. VAN DER VORST, Templates for the solution of linear systems, templates@cs.utk.edu, 1993.

....because of the relatively simple geometries considered. The Q method results in a system of nonlinear algebraic equations which are linearized by means of the Newton Raphson method, and the resulting linear equations are solved by means of an efficient implementation of the BiCGSTAB technique [4] on an MIMD computer. Furthermore, the algorithm significantly reduces the number of messages per cycle and enables the overlapping of these messages with the computations. The efficiency of the algorithm has been assessed in the MSSCLA model analyzed here and for other systems of pdes, and is ....

....other cases, two communication steps could be joined into only one if the first value is not required before the second one. However, very few communications in the BiCGSTAB algorithm can be joined and overlapped with computations following the above procedure. Following the ideas presented in [4] for the Conjugate Gradient method, a reordering of the BiCGSTAB algorithm which allows for the overlapping of all the message exchanges with current calculations and which significantly reduces the number of messages required, has been developed. The proposed algorithm is shown in Table 2. In ....

Barret, R., et al., Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1994.

....to implement on a parallel architecture, especially for unstructured matrices. Indeed, the application of the preconditioner in the iteration phase requires the solution of triangular systems at each step, which is difficult to parallelize because of the recursive nature of the computation (see [2], section 4.4.4) Our aim is to find an inherently parallel preconditioner, which retains the convergence properties of incomplete LU . A natural way to achieve parallelism is to compute an approximate inverse M of A, such that AM I in some sense. The evaluation of My is then easy to ....

....if the problem results from the discretization of a partial differential equation, it is generally meaningful to look for a sparse approximate inverse. Polynomial preconditioners with M = p (A) are inherently parallel, but do not lead to as much improvement in the convergence as incomplete LU (see [2], section 3.5) A different approach is to minimize kAM Gamma Ik. Yet we cannot confine the whole spectrum of AM to the vicinity of 1, or else we would be minimizing the 1 norm or the 2 norm. This would be too expensive, since M = 0 is a better solution as long as kAM Gamma Ik 1. To ensure ....

[Article contains additional citation context not shown here]

R. Barret et al.. Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1994.

....= b: 3) Once the solution y of (3) has been obtained, the solution of (1) is given by x = Gy. The choice between left or right preconditioning is often dictated by the choice of the iterative method. It is also possible to use both forms of preconditioning at once (split preconditioning) see [3] for further details. Note that in practice it is not required to compute the matrix product GA (or AG) explicitly, because conjugate gradient type methods only necessitate the coefficient matrix in the form of matrix vector multiplies. Therefore, applying the preconditioner within a step of a ....

....Furthermore, preconditioners that are very efficient in a scalar computing environment may show poor performance on vector and parallel machines, and conversely. Approximate Inverse Preconditioning 3 A number of preconditioning techniques have been proposed in the literature (see, e.g. 2] [3] and the references therein) While it is generally agreed that the construction of efficient general purpose preconditioners is not possible, there is still considerable interest in developing methods which will perform well on a wide range of problems and are well suited for state of the art ....

[Article contains additional citation context not shown here]

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

....not based on the Krylov subspace, 7 such as the Chebyshev iterative method. In our research on preconditioners reported in Chapter 10 we used the preconditioned conjugate gradient squared method. An excellent introduction to both the stationary and nonstationary iterative methods is given in [9]. The nonstationary method most effective for a given problem or problem class depends on the properties of the matrices involved (for example symmetry and positive definiteness) the availability of the eigenvalues of the matrices, and the amount of storage available. Making the best selection ....

....the availability of the eigenvalues of the matrices, and the amount of storage available. Making the best selection for a given problem domain usually depends on experience gained via trial and error using different methods. A useful guide to begin with (Figure 2 1) is given in Barret et al. [9]. n n n n n y y y y y Is storage at a premium Is the transpose available Are the outer Is the matrix symmetric known eigenvalues Try GMRES Try CGS or Bi CGSTAB Try QMR Try MinRes or CG Try CG Try Chebyshev or CG with long restart Is the matrix definite Figure 2 1. The iterative methods ....

R. Barret, M. Berry, T. Chan, J. Dimmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

....error. This includes the methods of Lanczos, Arnoldi, conjugate gradients, GMRES, CGS, QMR, and an alphabet soup of other algorithms. The following sections discuss some of the key early steps in the development of the KMP family. A reader interested in recent developments could begin with [24] [3], and the papers in this volume. 3 This work was supported by the National Science Foundation under Grant CCR 95 03126 y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 (oleary cs.umd.edu) 2 O Leary 2 The Beginnings ....

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1993. 6 O'Leary

....is utilized as in the section above. FEM with continuous, piecewise linear elements is chosen for solving the elliptic problem. The linear system of equations is solved by an iterative method for unsymmetric systems, namely BiCGSTAB [24] It is preconditioned by simple Jacobian precondioning [4]. Error estimation is performed corresponding to results from Babuska Rheinboldt and Bank Weiser [1, 3] In order to advect a passive tracer variable, the scheme for passive advection from the previous section is adopted. As displacements are known, the only operation required for tracer ....

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1993.

.... are state of the art iterative solvers developed mostly in the last fifteen years that may be used, among other things, to solve for the stationary distribution of Markov chains [30] A concise discussion on popular Krylov subspace methods and the motivation behind preconditioning may be found in [4]. In this study, we consider the methods Generalized Minimum RESidual (GMRES) Direct Quasi GMRES (DQGMRES) BiConjugate Gradient (BCG) Conjugate Gradient Squared (CGS) BiConjugate Gradient Stabilized (BCGStab) and Quasi Minimal Residual (QMR) with Incomplete LU (ILU) factorization ....

....[10] provide a detailed explanation of each test problem, the nonzero plots of the underlying matrices, information about the matrices and the partitionings, and the complete results. 2. Implementation Framework. In this study, we experiment with the (point) successive overrelaxation (SOR) method [30, 4], which is a stationary iterative method, two types of two level iterative methods, block SOR (BSOR) 30, 27, 22] and IAD [28, 32, 30, 11] and the Krylov subspace methods GMRES, DQGMRES, BCG, CGS, BCGStab, and QMR (see [4, 27] and the references therein) 2.1. Partitioning Techniques. When ....

[Article contains additional citation context not shown here]

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

....when the preconditioner is applied, we will solve the preconditioned linear system AM Gamma1 z = b; where x = M Gamma1 z; 2. 6) where the matrix M Gamma1 is some approximation to the matrix A Gamma1 . The relation (2. 6) represents so called explicit preconditioning (see, e.g. [1]) Its application in the iteration cycle of GMRES reduces to matrix vector products. On the contrary, a preconditioner is implicit if its application requires a solution of the special linear system within each step of the iterative method. It was realized that straightforward implementation of ....

....approximate the exact L and U factors of A. Applying the preconditioner requires the solution of two sparse triangular systems (the forward and backward solves) Other notable examples of preconditioners include the ILQ, SSOR and ADI preconditioners (an initial information can be found, e.g. in [1]) 3 Numerical experiment description We show the behaviour of the preconditioned GMRES method on a real world problem. We have chosen a typical matrix which arises in the circuit simulation (in our case we used the matrix from the 32 bit adder design (see [3] Its structural pattern is shown ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

....to implement on a parallel architecture, especially for unstructured matrices. Indeed, the application of the preconditioner in the iteration phase requires the solution of triangular systems at each step, which is difficult to parallelize because of the recursive nature of the computation (see [1], section 4.4.4) Our aim is to find an inherently parallel preconditioner, which retains the convergence properties of incomplete LU . A natural way to achieve parallelism is to compute an approximate inverse M of A, such that AM I in some sense. The evaluation of My is then easy to ....

....if the problem results from the discretization of a partial differential equation, it is generally meaningful to look for a sparse approximate inverse. Polynomial preconditioners with M = p (A) are inherently parallel, but do not lead to as much improvement in the convergence as incomplete LU (see [1], section 3.5) We propose a different approach, and try to minimize kAM Gamma Ik. Yet we cannot confine the whole spectrum of AM to the vicinity of 1, or else we would be minimizing the 1 or 2 norm which would be too expensive, since M = 0 is a better solution as long as kAM Gamma Ik 1. ....

R. Barret et al.. Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1994.

....experiments. In this section we present the results of numerical experiments on a range of problems from the Harwell Boeing collection [6] and from Tim Davis collection [5] A comparison between preconditioners based on Frobenius norm minimization and the no fill ILU(0) preconditioner (see [1]) was carried out in [8] Here we make a similar comparison between ILU(0) and the approximate inverse preconditioner based on the biconjugation process. Additional experiments can be found in [3] together with implementation details. On input, all our codes for the computation of the ....

....linear system was computed from the solution vector x of all ones. We present results for the biconjugate gradient method (denoted BCG in the tables) the conjugate gradient squared method (CGS) and GMRES with no restarting and Householder orthogonalization (denoted GMR in the tables) See [1] for a description of these algorithms. We deliberately avoided the use of restarting with GMRES because we are interested in the effect of preconditioning on the rate of convergence rather than in the performance of a particular solver. However, in practical applications restarting should be used ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

.... are state of the art iterative solvers developed mostly in the last fifteen years that may be used, among other things, to solve for the stationary distribution of Markov chains [34] A concise discussion on popular Krylov subspace methods and the motivation behind preconditioning may be found in [4]. In this study, we consider the methods Generalized Minimum RESidual (GMRES) Direct Quasi GMRES (DQGMRES) BiConjugate Gradient (BCG) Conjugate Gradient Squared (CGS) BiConjugate Gradient Stabilized (BCGStab) and Quasi Minimal Residual (QMR) with Incomplete LU (ILU) factorization ....

....model correct to full machine precision when the model itself contains errors. However, a direct method (such as Gaussian elimination) is obligated to continue until the final operation has been performed. In this study, we experiment with the (point) successive overrelaxation (SOR) method [34, 4], a stationary iterative method. Moreover, two types of two level iterative methods are considered: block SOR (BSOR) 34, 28, 22] and IAD [36, 32, 34, 10] As for projection techniques, we choose to implement and experiment with the Krylov subspace methods GMRES [29] DQGMRES [30] BCG [4] CGS ....

[Article contains additional citation context not shown here]

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, PA, 1994.

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

No context found.

Barret, R., Berry, M., Dongarra, J., Pozzo, R.: Templates for the Solution of Linear Systems, 2nd Edition. SIAM Philadelphia (1995)

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, PA, 1994.

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Donagarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the solution of linear systems. SIAM, Philadelphia, 1994.

No context found.

Barret R., Berry M., Chan T., Demmel J., Donato J., Dongarra J., Eijkhout V., Pozo R., Romine C., and van der Vorst H. (1993) Templates for the solution of linear systems. templates@cs.utk.edu.

No context found.

Barret, R. et al., Templates for the Solution of Linear Systems, SIAM, Philadelphia, 1994.

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems. SIAM, Philadelphia, 1994.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC