C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8240 UC404, Sandia National Laboratories, Albuquerque, March 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computational experience has been gathered. In this section, we shall briefly highlight some of the recent developments and other methods not treated here. A survey of methods up to about 1991 can be found in Freund, Golub and Nachtigal [106] Two more recent reports by Meier Yang [151] and Tong [197] have extensive numerical comparisons among various methods, including several more recent ones that have not been discussed in detail in this book. Several suggestions have been made to reduce the increase in memory and computational costs in GMRES. An obvious one is to restart (this one is ....

C. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Rep. SAND91-8240, Sandia Nat. Lab., Livermore, CA, 1992.

....a solution, we set the right hand side to A times a random vector. The orsx, shermanx, poresx, and saylorx problems were all taken from the Harwell Boeing matrix collection. A comparative study of these problems for di#erent iterative methods using incomplete LU preconditioning can be found in [15]. 850 M. J. GROTE AND T. HUCKLE 0 500 1000 0 500 1000 nz = 9613 0 500 1000 0 500 1000 nz = 10948 FIG.4.Pores2: original matrix A (left) and sparse approximate inverse M with # =0.4(right) TABLE 5 Convergence results for poresx and saylorx: unpreconditioned (top) and preconditioned (middle ....

C. H. TONG, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Sandia report SAND91-8240, Sandia National Laboratories, Albuquerque, NM, 1992.

....aspect of the numerical behavior, the properties of CSBCG with respect to roundoff error. In exact arithmetic, CSBCG computes selected iterates of BCG and hence has the same convergence rate as BCG. Several more general illustrations of the effectiveness of the CSBCG method are given in [4] In [27], biconjugate gradient and many related methods [26] 28] 21] 11] 10] 12] 13] are compared on a series of test problems. All of our examples concern the model convection diffusion equation Gamma Deltau fiu x = 1 in Omega = 0; 1) Theta (0; 1) with the Dirichlet boundary condition u ....

....( 1.78) For this example, the standard BCG algorithm looks surprisingly good. Despite a very erratic and oscillatory convergence behavior, it is in fact working steadily toward convergence. This observation is consistent with the behavior of the BCG algorithm in the extensive tests of Tong in [27]. It is unknown to us how the iterates of the algorithm compare to those computed in exact arithmetic, but we doubt that they are close. On the other hand, failure to compute accurate direction vectors, poor approximation of a Krylov subspace, loss of orthogonality and near failures in the Lanczos ....

C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8402 UC-404, Sandia National Laboratories, Albuquerque, 1992.

....AZTEC package [HST95] It is an open question under what conditions this is true. ffl Despite the fact that SSOR as a stand alone iteration often converges more slowly than SOR (see [You71] p. 462, BBC 94] p. 12) it is sometimes considered as a preconditioner for nonsymmetric matrices ( Ton92] BBC 94] The merit of SOR as a preconditioner relative to SSOR as a preconditioner needs to be explored. In addition to the questions listed above, there are many others that need to be answered before the extent to which multi step multicolor SOR can be considered an effective ....

C. Tong. A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems. Technical Report SAND91-8240B/UC-404, Sandia National Laboratories, Albuquerque, New Mexico, and Livermore, California, September 1992.

....solving (1) is Krylov subspace or conjugate gradient type methods. So far, much work has been done; see, e.g. 12, 2, 3, 4, 42, 31, 32, 33, 35, 8, 9, 13, 19, 20, 29, 24, 25, 27] and many others. For survey papers, refer to [17, 36, 18, 14] for a numerically comparative study of some of them, see [37, 38]; for a synthesis and general description of available conjugate gradient type methods, see [34] Of these methods, one kind of the most successful schemes is based on the orthogonal projection, typified by GCR or ORTHOMIN [39, 42, 9] ORTHODIR and ORTHORES [42] FOM or Arnoldi s method [31] and ....

C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Technical Report Sand91-8240, Sandia National Laboratories, New Mexico 87185 and Livermore, California 94551, 1991.

....is studied. It is shown that the hybrid part can be selected to minimize the number of Newton steps rather than for the reduction of the computational costs for the linear Jacobian systems. It is not our purpose to compare hybrid BiCG methods with BiCG or QMR [12] This has been done in, e.g. [24, 16]. In the present paper, we will concentrate on approaches that help to reduce the effects of local rounding errors on the BiCG iteration coefficients in hybrid schemes. Locally accurate BiCG coefficients, i.e. coefficients that ensure at least local biorthogonality of the BiCG basis vectors, are ....

C. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8240, Sandia Nat. Lab., Livermore, 1992.

....good case in favor of the effectiveness of SPAI was already made in [2] and also in [3, 4] Here we present more evidence by studying matrices where ILU type methods either fail or have difficulty. Based on the extensive study of the convergence behavior of ILU preconditioned iterative methods in [14], we selected six matrices where ILU preconditioners either failed or required high levels of fill in (large k s in ILUT(k) independently of the iterative solver used, in order to achieve convergence in a small number of steps. These six matrices are listed in Table 1 and the results are ....

....in a small number of steps. These six matrices are listed in Table 1 and the results are displayed in Table 2 and are self explanatory 3 . The tolerance was set to 10 Gamma8 and the iterative method used was BICGSTAB 4 . A right hand side b of 1 s was used, but to better compare with [14] we also tried, for some of the matrices, a right hand side such that the solution is a random vector. No significant difference was observed. Although SPAI succeeded in some of the matrices where ILU type methods had failed, the SPAI preconditioner was significantly denser than the ILU type ....

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C.H. Tong, A Comparative Study of Preconditioned Lanczos Methods for Nonsymmetric Linear Systems, Sandia Report SAND91-8240, Sandia National Laboratory, January 1992.

....solving (1) is Krylov subspace or conjugate gradient type methods. So far, much work has been done; see, e.g. 12, 2, 3, 4, 41, 31, 32, 33, 35, 8, 9, 13, 18, 19, 29, 24, 25, 27] and many others. For survey papers, refer to [16, 36, 17, 14] for a numerically comparative study of some of them, see [37, 38]; for a synthesis and a general description of available conjugate gradient type methods, see [34] Of these methods, one kind of the most successful schemes is based on the orthogonal projection process, typified by GCR or ORTHOMIN [39, 41, 9] FOM or Arnoldi s method [31] and GMRES [35] There ....

C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Technical Report Sand91-8240, Sandia National Laboratories, New Mexico 87185 and Livermore, California 94551, 1991.

....of a solution, we set the right hand side to A times a random vector. The orsx, shermanx, poresx, and saylorx problems were all taken from the Harwell Boeing matrix collection. A comparative study of these problems for different iterative methods using incomplete LU preconditioning can be found in [15]. In the final part of this section we shall consider several large problems. The right hand side was always provided, the initial guess x 0 = 0, and the tolerance as in (33) We start with four typical problems from Centric Engineering: 0 500 1000 0 500 1000 nz = 9613 0 500 1000 0 500 1000 nz ....

C. H. Tong. A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Sandia report SAND91-8240, Sandia National Lab., January 1992.

....bbicg algorithm, proposed and described in [19] is a block generalization of bicg [4] In exact arithmetic, if bbicg does not terminate before k = dn=se steps, then the solution becomes available at step k [19, Th. 1] Even though the underlying bicg method can be effective in several cases [10, 16, 17, 37], there are some drawbacks that seem to be inherited by the block version. These are erratic convergence behavior, possible division by zero, and sensitivity to the choice of the auxiliary starting vector. An organizational drawback is the use of the transpose of A. Recent research has been ....

....other methods, mhgmres suffered a milder performance degradation, while bbicg failed to converge (compare with Table 6. Experiments with matrices from the Harwell Boeing collection [2] We present results from experiments with matrices pde 9511, orsreg 1 and sherman4. It was observed in studies [16, 37] that for these matrices bicg had better overall performance than restarted gmres. Table 11 reports the CPU time for all methods except mj3, using right hand sides with random elements. mhgmres returns the best performance ITERATIVE METHOD FOR SYSTEM WITH SEVERAL RIGHT HAND SIDES 15 Table 10 ....

C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8240 UC404, Sandia National Laboratories, Albuquerque, March 1992.

....related to the CGS algorithm. Based on these algorithms and the ones mentioned above, a variety of other similar algorithms have been and are being developed that are combinations of the above ideas, such as QMRS [20] or QMRCGSTAB [7] which behave similarly to some of the methods considered above [45]. Two methods that have not received much attention for solving linear systems are Broyden s method [4] and the EN method [13] which was developed fairly recently. The family of Broyden methods has suffered from a bad reputation for solving linear systems, but recent efforts [8] have shown that ....

C. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report, Center for Computational Engineering, Sandia National Laboratories, Livermore, CA, 1992.

....a solution, we set the righthand side to A times a random vector. The oilgenx, shermanx, poresx, and saylorx problems were all taken from the Harwell Boeing matrix collection. A comparative study of these problems for different iterative methods using incomplete LU preconditioning can be found in [13]. In the final part of this section we shall consider a selection of very large problems. The right hand side was always provided, the initial guess x 0 = 0, and the tolerance as in (33) We start with three typical problems from Centric Engineering : P1: incompressible flow in a pressure driven ....

C.H. Tong. A Comparative Study of Preconditioned Lanczos Methods for Nonsymmetric Linear Systems, Sandia report SAND91-8240, Sandia National Laboratories, January 1992.

....maximum number of iterations has been reached. We note that although the example is contrived, it does justify the implementation of a QMRCGSTAB type method. We finally observe that experiments using several of the methods discussed herein, albeit using another naming convention, were presented in [15]. 5. Conclusions and future work. We have derived two QMR variants of Bi CGSTAB. Our motivation for these methods was to inherit any potential improvements on performance Bi CGSTAB offers over CGS, while at the same time provide a smoother convergence behavior. TABLE 2 Example 5: Correct ....

C. H. TONG, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Rep. SAND91-8240 UC404, Sandia National Laboratories, Albuquerque, March 1992.

....(e.g. CGS, BiCGSTAB, QMR, CSBCG [19, 21, 6, 2] each of which was specially designed to overcome some of its inherent difficulties (the need for adjoint matrix vector product, potential breakdowns, erratic convergence behavior, etc. However, it has been observed by Bank and Chan [2] and Tong [20] that, in finite precision arithmetic, BiCG remains competitive (in terms of convergence and convergence rates) especially when coupled with no or relatively poor preconditioners. One of major concerns in using BiCG is the two types of potential breakdown problems, which can cause numerical ....

....of potential breakdown problems, which can cause numerical instability. In addition, in finite precision arithmetic the biorthogonality is lost, as is experienced by other Lanczos type algorithms. In spite of these difficulties, BiCG often exhibits exceptional numerical robustness in practice [2, 20]. On the other hand, convergence of the preconditioned conjugate gradient method with inexact matrix vector multiplication has been observed and studied by Golub and Overton [7, 8] in the context of inner outer iterations. It seems that the convergence of the BiCG (or CG) residuals is not ....

C. H. Tong, A Comparative Study of Preconditioned Lanczos Methods for Nonsymmetric Linear Systems, Tech. Report SAND91-9240B, Sandia National Lab., Livermore, 1992.

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C. H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8240 UC404, Sandia National Laboratories, Albuquerque, March 1992.

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