M. H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM Journal of Scientific Computing, 14(5):1020--1033, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....forever, whereas a method like GCR(2) or GMRES(2) in which we minimize over two combined successive search directions, may lead to convergence, and this is mainly due to the fact that then complex eigenvalue components in the error can be effectively reduced. This point of view was taken in [60] for the construction of the Bi CGSTAB2 method. In the odd numbered iteration steps the Q polynomial is expanded by a linear factor, as in Bi CGSTAB, but in the even numbered steps this linear factor is discarded, and the Q polynomial from the previous even numbered step is expanded by a quadratic ....

M. H. Gutknecht. Variants of BICGSTAB for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020--1033, 1993.

....precisely, for k = m , with P the space of polynomials p of degree with p(0) 1, q k = p m; q k Gamma , with p such that kp(A)q k Gamma (A)r k k 2 is minima with respect to p 2 P for p = p m; 1. 3) There are numerous ways to implement this procedure, see, for instance, [11, 6, 10, 4], but basically the iteration steps of a BiCGstab method can be divided into two parts, namely, i) the BiCG part, in which the BiCG residual is implicitly updated using the short recursions of BiCG (ii) the polynomial part, in which the minimal residual polynomial is constructed and used to ....

M. H. Gutknecht, Variants of BiCGStab for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020--1033.

....Although some new algorithms for the implementation of Lanczos type product methods are obtained in the sequel, our purpose is to derive transpose free algorithms for Lanczos method itself. Thus, in passing, known algorithms for product methods will be recovered such as those given in [13] [19] and [23] Our derivation of the algorithms is based on the interpretation of Lanczos method in terms of formal orthogonal polynomials (FOP) Such polynomials were already known to Lanczos [21] but, although some considerations and algorithms were given in [2] and in [18] their systematical use ....

.... x k defined by r k = b Gamma A x k without using A . Thus, for the choice W k ( r k is not the residual of a Lanczos type product method. Among the most well known Lanczos type product methods are the CGS of Sonneveld [25] the BiCGSTAB of van der Vorst [26] and its variants [19]. Other methods of this type are given in [6] see [3] for a preliminary version) Even if the polynomials U k and V k can be arbitrarily chosen, some choices are more interesting than others. In any case, let us remark that all the algorithms simplify if 8k, U k j V k . The simplest choice for ....

M.H. Gutknecht, Variants of BiCGStab for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993) 1020--1033.

....B B B B B 2 1 0 2 1 1 0 2 . 1 . 1 0 2 1 C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B 1 1 . 1 C C C C C C C C C C C C C B B B B B B B B B B B B B 3 4 . 4 C C C C C C C C C C C C C This system was given by Gutknecht [27] and, for x 0 = 0 and y = 0; 0; 0; Gamma1; 1; Gamma( Gamma1) n Gamma1 ; 0) a breakdown occurs at the first iteration when n is even since (y; r 0 ) y; Ar 0 ) 0. When n is odd, oe 2 = 0 at the first iteration and a breakdown also occurs but for a different reason. For n = ....

M.H. Gutknecht, Variants of BICGSTAB for matrices with complex spectrum, to appear.

....subspace QMR QOR algorithms, the QMR method of Freund and Nachtigal [14] and the TFQMR and CGS methods due to Freund [11, 12] and Sonneveld [39] respectively. As another QMR QOR pair, we mention the QMRCGSTAB method of Chan et al. 5] and BICGSTAB developed by Van der Vorst [43] and Gutknecht [22]. 37 The QMR algorithm of Freund and Nachtigal proceeds exactly as described above, with the basis of the Krylov space generated by the look ahead Lanczos algorithm. The QOR counterpart of Freund and Nachtigal s QMR is the BCG algorithm, the iterates of which are characterized by 2 x 0 Km ....

M. H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Stat. Comput., 14(5):1020--1033, 1993.

....because of its optimality over the solution space, fast convergence using preconditioners, small memory size and acceptable rounding off error properties. Recently, generalization of the CG method for nonsymmetric and or indefinite matrix with these remarkable properties is a hot research area [48, 55, 19, 63]. The MG method was realized to be quite an efficient method with a broad area of application from the midseventies because of its mesh independent and fast convergence, and the number of publications has grown rapidly. This thesis proposes the multigrid preconditioned conjugate gradient (MGCG) ....

Gutknecht, M. H., "Variants of Bi-CGSTAB for matrices with complex spectrum," SIAM J. Sci. Comput., vol. 14, pp. 1020--1033, 1993.

....subspace QMR QOR algorithms, the QMR method of Freund and Nachtigal [14] and the TFQMR and CGS methods due to Freund [11, 12] and Sonneveld [39] respectively. As another QMR QOR pair, we mention the QMRCGSTAB method of Chan et al. 5] and BICGSTAB developed by Van der Vorst [43] and Gutknecht [22]. 37 The QMR algorithm of Freund and Nachtigal proceeds exactly as described above, with the basis of the Krylov space generated by the look ahead Lanczos algorithm. The QOR counterpart of Freund and Nachtigal s QMR is the BCG algorithm, the iterates of which are characterized by x BCG 2 x 0 ....

M. H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Stat. Comput., 14(5):1020--1033, 1993.

....quite more complex than (1.7) and the right hand side of (1.11) is growing however small it may be with the iteration. In order to obtain short recurrences for nonsymmetric matrices A there are various other possibilities, among them . restarted or truncated versions, CGSTAB approaches [6, 13, 15] introduced by van der Vorst. However, all techniques mentioned above produce short recurrences, but do not fulfill the convergence properties (1.7) or (1.8) We will show in the following that we can enforce automatic termination of the sequence by allowing the matrix Z to be dependent on the ....

M.H. Gutknecht, Variants of BICGSTAB for matrices with complex spectrum, Research Report 91-14, Eidgenossische Technische Hochschule Zurich, Interdisziplinares Projektzentrum fur Supercomputing, ETH-Zentrum, CH-8092 Zurich, August 1991.

.... the full orthogonal method (FOM) and their variants such as restarted GMRES(q) 17] The second set of methods is based upon nonsymmetric Lanczos recursions, and includes the bi conjugate gradient method (BiCG) the quasi minimal residual method (QMR) and their variants such as BiCGSTAB(q) [9, 12, 22, 13, 19]. In this paper we study the convergence behavior of these methods using the MATLAB [14] templates discussed in [2] In section 2 we review briefly the GMRES FOM and the QMR BiCG methods. In section 3 we obtain a relationship between BiCG and FOM, and a weaker relationship between QMR and GMRES. ....

....may occur because in real arithmetic BiCGSTAB cannot handle complex eigenvalues very well. It fits linear GMRES polynomials at each iteration. These tests should be rerun using BiCGSTAB(2) which fits second order GMRES polynomials which can model pairs of complex eigenvalues in real arithmetic [13]. BiCGSTAB(2) was not available in the MATLAB Templates [2] Finally we consider QMR on each of these test problems. See figures 33 36. On these tests, to within error norms of size 10 Gamma9 there were no significant differences between the convergence in the normal and the nonnormal cases. ....

M. H. Gutknecht. Variants of BiCGSTAB for matrices with complex spectrum, SIAM J on Scientific and Statistical Computing, 14,(1993) pp. 1020--1033.

.... Then the finite element approximation u h;i 2 S h D (t n;i ) of the solution u i of (9) is required to satisfy A(u h;i ; OE) 0; 8OE 2 S h D (t n;i ) 13) The above system of nonlinear algebraic equations is solved by Newton s iteration, with the resulting linear systems solved by BiCGStab2 [13], preconditioned with the incomplete LU decomposition. For problems with varying time scales, ffit n should be selected dynamically. This is done with a standard approach as follows: Assume that the s stage SDIRK method (7) is of order p. Let b i (1 i s) be a set of parameters of an embedded ....

M. H. Gutknecht, Variants of BICGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput. 14, 1020(1993).

....time step Deltat = 0:001. With this small time step size, the time discretization error in our results can be ignored compared to the error from spatial discretization. The resulting system of algebraic equations from this implicit integration is solved at each time step using the BiCGstab2 [4, 15] iterative method (without special treatments such as scaling or preconditioning) until the mean square root of the residual is less than 10 Gamma8 . Four different types of meshes are used. The meshes and numerical results are described in the following sections. 10 5.1 Uniform Mesh Initial ....

M. H. Gutknecht. Variants of bicgstab for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020 -- 1033, 1993.

.... by Psi k (A) fl k A I ) Psi k Gamma1 (A) 74) where fl k is determined from a 1 dimensional minimization kr k k = min fl k k(fl k A I ) Psi k Gamma1 (A) Pi k (A)r 0 k: 75) The behavior of BiCGSTAB has been visualized very impressively by Fujino, Zhang and Mori [11] Gutknecht [14], Sleijpen and Fokkema [29] and Sleijpen, van der Vorst and Fokkema [30] generalized BiCGSTAB so that Psi k is determined by an dimensional minimization. These methods are called BiCGSTAB( 15 4.4. Methods based on the normal equations. Instead of solving the linear system (1) the two ....

M.H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020--1033, 1993.

....Bi CG performs well, the convergence of BiCGSTAB stagnates. This was observed specifically in case of discretized advection dominated PDE s. The stagnation is due to the fact that for this type of equations the matrix has almost pure imaginary eigenvalues. With his BiCGStab2 algorithm Gutknecht [5] attempted to avoid this stagnation. Here, we generalize the Bi CGSTAB algorithm further, and overcome some shortcomings of BiCGStab2. In some sense, the new algorithm combines GMRES(l) and Bi CG and profits from both. Key words. Bi conjugate gradients, non symmetric linear systems, CGS, ....

....and this may cause stagnation or even breakdown. As numerical experiments confirm, this is likely to happen if A is real and has nonreal eigenvalues with an imaginary part that is large relative to the real part. One may expect that second degree MR polynomials can better handle this situation. In [5] Gutknecht introduces a BiCGStab2 algorithm that employs such second degree polynomials. Although this algorithm is certainly an improvement in many cases, it may still su#er from problems in cases where Bi CGSTAB stagnates or breaks down. At every second step, Gutknecht corrects the first degree ....

[Article contains additional citation context not shown here]

M.H. Gutknecht, Variants of BiCGStab for matrices with complex spectrum, IPS Research Report No. 91-14, 1991.

....is exact and can be formulated as short recurrence at the same time. All other here discussed conjugate Krylov subspace methods have the same two desirable properties as well. CGS [70] minimal residual smoothing applied to the biconjugate gradients (MRS) 62, 63, 93] and BiCGSTAB approaches [36, 69, 79] fit into this class. Their classification is incomplete in table 1 because the determination of Z k is rather complex and would be beyond the scope of this paper. In particular, from the investigations of Barth and Manteuffel, see [6] follows for the QMR method by Freund and Nachtigal [28, 29] ....

....be derived from corollary 10. Well known techniques like Craig s method [11] the biconjugate gradients (BCG) 23, 47] Sonnefeld s CGS [70] Freund s and Nachtigal s QMR method [29] the biconjugate gradients smoothed by Schonauer s minimal residual algorithm [62, 63, 93] and BiCGSTAB approaches [36, 69, 79] introduced by van der Vorst can be formulated with short recurrences. This can be also obtained from corollary 10. Moreover, corollary 10 is the basis for constructing short recurrences for conjugate Krylov subspace methods [83] The more general theorem 9 offers the possibility to search for new ....

M.H. Gutknecht. Variants of BICGSTAB for matrices with complex spectrum. Research Report 91-14, Eidgenossische Technische Hochschule Zurich, Interdisziplinares Projektzentrum fur Supercomputing, ETH-Zentrum, CH-8092 Zurich, August 1991.

....solver the combination was called a Newton Krylov Schwarz method in [5] and that term is now common. In this work we use the Krylov method BiCGSTAB [15, 19, 30] as the linear solver with an additive Schwarz preconditioner. Other low storage transpose free Krylov methods for nonsymmetric methods [14, 16] were tested in the early stages of this project along with BiCGSTAB(2) using the implementation from [31] Preconditioning (from the right in our case) which is critical to good performance, replaces (2.4) with the equivalent system F 0 ## c #Ms = F ## c # (2.8) andthensetss = Ms. The ....

M. H. GUTKNECHT, Variants of BICGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020--1033.

.... I ) Psi k Gamma1 (A) Pi k (A)r 0 k: 102) There are recent convergence estimates by Chan and Szeto [10] based on the BCG result, equation (73) Sleijpen and van der Vorst investigated the behavior for finite precision arithmetic and proposed techniques to maintain the convergence [45] Gutknecht [29], Sleijpen and Fokkema [44] and Sleijpen, van der Vorst and Fokkema [46] generalized BiCGSTAB so that Psi k is determined by an dimensional minimization. These methods are called BiCGSTAB( QMR. The quasi minimal residual method (QMR) was developed by Freund and Nachtigal [22] It uses Z k = ....

M.H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020--1033, 1993.

....and fi j have the same values as those generated by Bi CG and CGS. Hence, they can be used to extract eigenvalue approximations for the eigenvalues of A (see Bi CG) Bi CGSTAB can be viewed as the product of Bi CG and GMRES(1) Of course, other product methods can be formulated as well. Gutknecht [41] has proposed BiCGSTAB2, which is constructed as the product of Bi CG and GMRES(2) 4.5.6 Bi CGSTAB2 and variants The residual r k = b Gamma Ax k in the Bi Conjugate Gradient method, when applied to Ax = b with start x 0 can be written formally as P k (A)r 0 , where P k is a k degree ....

....forever, whereas a method like GCR(2) or GMRES (2) in which we minimize over two combined successive search directions, may lead to convergence, and this is mainly due to the fact that then complex eigenvalue components in the error can be effectively reduced. This point of view was taken in [41] for the construction of the BiCGSTAB2 method. In the odd numbered iteration steps the Q polynomial is expanded by a linear factor, as in BiCGSTAB, but in the even numbered steps this linear factor is discarded, and the Q polynomial from the previous even numbered step is expanded by a quadratic 1 ....

M. H. Gutknecht. Variants of BICGSTAB for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020--1033, 1993.

....solver the combination was called a Newton Krylov Schwarz method in [5] and that term is now common. In this work we use the Krylov method BiCGSTAB [15, 19, 30] as the linear solver with an additive Schwarz preconditioner. Other low storage transpose free Krylov methods for nonsymmetric methods [14, 16] were tested in the early stages of this project along with BiCGSTAB(2) using the implementation from [31] Preconditioning (from the right in our case) which is critical to good performance, replaces (2.4) with the equivalent system F 0 ( c )Ms = GammaF ( c ) 2.8) and then sets s = Ms. ....

M. H. GUTKNECHT, Variants of BICGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020--1033.

....the range of iterative methods in these classes which are in common use will not be given here as they are widely available in the literature. Algorithms for each method used in these experiments, together with a general description of their properties and relevant references, can be found in [1] [8] and [15] All calculations were done in FORTRAN on a Sun Sparc 10 workstation. The parameters in the ODE (2.1) were chosen to have the values ff = 0:4, fi = 0:32 and = p 0:84 to correspond with those used in [4] The best method (as described below) from each of the above classes was ....

....III) a wide range of methods have been proposed and several were tested. The biconjugate gradient method Bi CG [5] had convergence behaviour very similar to that of QMR, the quasi minimal residual method [1] with both methods performing poorly compared to Gutnecht s stabilised version BiCGStab2 [8]. Note that with this version of QMR no look ahead Lanczos steps [6] are performed. Further improvement was obtained by implementing the BiCGstab(l) algorithm of Sleijpen and Fokkema [15] With l = 1, this gives Bi CGSTAB and with l = 2, the method is equivalent to BiCGStab2 in exact arithmetic, ....

M.H. Gutknecht, Variants of BICGSTAB for Matrices with Complex Spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020-1033.

....a different way, as in QMR [4] or [2] A weak point in BiCGSTAB is that it introduces one more break down possibillity on top of these, namely when the GMRES(1) part, or rather GCR(1) part, of the algorithm stagnates. This may happen, for instance in advection dominated pde problems. Gutknecht [5] has suggested to combine Bi CG with GCR(2) BiCGSTAB2 to overcome this problem. If this idea is implemented in a different manner, a very competitive BiCGSTAB variant is obtained, which is easy to implement. This is a particular class of the broader family BICGSTAB( introduced in [9] In [9] ....

....forever, whereas a method like GCR(2) or GMRES(2) in which we minimize over two combined successive search directions, may lead to convergence, and this is mainly due to the fact that then complex eigenvalue components in the error can be effectively reduced. This point of view was taken in [5] for the construction of the BiCGSTAB2 method. In the odd numbered iteration steps the Q polynomial is expanded by a linear factor, as in BiCGSTAB, but in the even numbered steps this linear factor is discarded, and the Q polynomial from the previous evennumbered step is expanded by a quadratic 1 ....

M. H. Gutknecht, Variants of BICGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020--1033.

....Bi CGSTAB, a combination of Bi CG and GCR(1) often converges much faster than a method like Bi CG. But, especially in case of discretized partial differential equations with large advection terms, Bi CG is often superior: the GCR(1) part may lead to stagnation in such a situation. Gutknecht [2] proposed to combine Bi CG and GCR(2) to overcome this problem. We implemented this idea in a different manner, obtaining competitive variants that are efficient and easy to implement and surprisingly stable. Our BiCGstab( approach allows combinations with steps of any Krylov subspace method ....

M.H. Gutknecht, Variants of BiCGStab for Matrices with Complex Spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020--1033.

.... proposed 1993 by Zhang[15] where a 2 dimensional minimization is incorporated in each step, and which was therefore called BICGxMR2 in [7] In one version, which independently was also proposed by Cao [1] and Gutknecht[6] the implementation only requires a one line modification of BICGSTAB2 [5], which does such a minimization in every other step. But Zhang also proposed a version called GPBICG that is fully based on coupled two term recursions and can therefore be expected to have a better roundoff behavior. It was re cently shown in [8] that under certain assumptions Krylov solvers ....

....: 1 Xl l)tl( s) where ) lq 1 is chosen such that I]r q 11] IIt 1(A)p 1(A)r011 is minimized. Note that in the case of real data the polynomial t is real valued and therefore all its zeros 1 Xk, k = 1, l) are real, although the spectrum of A may be complex. In contrast, in BICGST B2 [5] in every other step two new zeros are chosen by a two dimensional residual minimization rule, and these zeros can be complexconjugate. The recurrences, which impose automatically the normalization t l(0) 1, are , if is even, tl l( q tl( q (1 if 1 is odd. 9) 10) Here, lq 1 is ....

[Article contains additional citation context not shown here]

M.H. Gutknecht. Variants of BiCGSTAB for matrices with complex spectrum. SIAM Journal on Scientific and Statistical Computing, 14(5):1020-1033, 1993.

....Moreover, they remain xed for all subsequent polynomials of the set. The rst disadvantage is avoided if the linear factors are replaced by quadratic factors that are attached every other step; they allow a twodimensional residual minimization in every other step, as suggested in BiCGStab2 [25]. The second disadvantage is overcome if the second set is chosen to satisfy a three term recurrence or a pair of coupled two term recurrences (which can be used for a two dimensional residual minimization in every step) as suggested by Zhang in his GPBI CG algorithm [49] an equivalent form of ....

M. H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020-1033, 1993.

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M. H. Gutknecht. Variants of BiCGStab for matrices with complex spectrum. SIAM Journal of Scientific Computing, 14(5):1020--1033, 1993.

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M. H. Gutknecht. Variants of BICGSTAB for matrices with complex spectrum. SIAM J. Sci. Comput., 14:1020--1033, 1993.

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