Y. Saad and M. Shultz, "GMRES: A Generalized Minimum Residual Algorithm for Solving Nonsymmetric Linear Systems," SIAM J. Scientific Statistical Computing, Vol. 7, No. 3, July 1986, pp. 856--869.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... linear system Pu = g (6) has the same solution as that of (3) We shall prove in the remainder of the paper that P is indeed nonsingular and uniformly well conditioned, and that therefore (6) can be solved by using certain Krylov space type iterative acceleration methods, such as CG or GMRES [21]. We remark that if the bilinear form b( Delta; Delta) is symmetric, then the operator P is also symmetric with respect to b( Delta; Delta) In other words, the local multiplicative Schwarz operator P is symmetric if both P ij and P ji are included in its definition. Later, in this section, we ....

....that the operator P is uniformly bounded and its symmetric part, with respect to the inner product a( Delta; Delta) is uniformly positive definite, in the following theorem. This theorem provides the optimal convergence of several Krylov space iterative methods, including GCR [16] and GMRES [21] among others. Theorem 2. There exist positive constants H 0 , c(H 0 ) and C, independent of the mesh parameters h and H, such that if H H 0 , the operator P is uniformly bounded, i.e. kPuk a Ckuk a ; 8u 2 V h ; and its symmetric part is uniformly positive definite, i.e. a(P u; u) ckuk ....

Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856--869.

....methods of finding the approximate solution of the linear system. The proper selection and use of this method is crucial to the success of the overall solver [2] The most successful class are the Krylov iterative methods. Specifically, the preconditioned Generalized Minimum Residual (GMRES)[16] has proven to be effective for aerodynamic systems. We call these linear iterations the inner iterations. Over solving the linear system needs to be avoided for efficiency. A stopping criterion is 333.2 needed for the inner iterations. There are two considerations. First, we use a target ....

Saad Y and Schultz M. H. GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Computing, Vol. 7, pp 856--869, 1986.

....and indefinite or both, including discrete systems associated with Maxwell s equations. Throughout the 1980 s and 1990 s, much e#ort went into the development of Krylov subspace methods for the solution of more general linear algebraic equations. Methods such as GMRES, QMR, and BiCGStab [10, 45, 49, 93, 94] provide generalizations of the conjugate gradient method that apply for non Hermitian problems. However, the theory guaranteeing convergence behavior of non Hermitian matrix iterations is scarce. In spite of the lack of such theoretical results, non Hermitian Krylov subspace iterations are ....

Y. Saad and M.H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comp., 7:856--869, 1986. 146

....the (n Gamma 1) dimensional subspace. The fact that the image subspace q j may differ from the original subspace u j raises another problem. Iterative linear solvers of Krylov subspace type require that the operator is defined on its image subspace as well. Krylov subspace methods, as GMRES [12] and Bi CGSTAB methods [19, 13] subject to appropriate preconditioning, can cope with this difficulty: the preconditioner can be designed to map the image subspace to the original subspace, while the Krylov subspace solver keeps the approximate solutions of the linear system in the original ....

Y. Saad and M.H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869.

....for solving nonlinear problems. Key words. Nonlinear problems, Newton s method, Inexact Newton, Iterative methods. AMS subject classification. 65H10. 1. Introduction. Our goal in this paper is twofold. A number of iterative solvers for linear systems of equations, such as FOM [23] GMRES [26], GCR [31] Flexible GMRES [25] GMRESR [29] and GCRO [7] are in structure very similar to iterative methods for linear eigenproblems, like shift and invert Arnoldi [1, 24] Davidson [6, 24] and Jacobi Davidson [28] We will show that all these algorithms can be viewed as instances of an ....

....that take their updates to the approximate solution as a linear combination of previous directions p j . Preferable updates p k : P jk fl j p j are those for which b Gamma Ax k 1 , where x k 1 = x k p k , is minimal in some sense: e.g. kb Gamma Ax k 1 k 2 is minimal, as in GMRES [26] and GCR [31] or b Gamma Ax k 1 is orthogonal to the p j for j k, as in FOM or GENCG [23] or b Gamma Ax k 1 is quasi minimal , as in Bi CG [17] and QMR [11] Of course the distinction between preconditioning and acceleration is not a clear one. Acceleration techniques with a limited number ....

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Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869.

....will derive explicit expressions for left and right preconditioned correction equations. In each iteration step we need to solve a deflated correction equation (12) for a given e q and e (cf. 12) For the approximate solution of this equation we may use a Krylov subspace method, e.g. GMRES [17], or BiCGstab( 19] The rate of convergence and the efficiency of Krylov subspace methods is often improved by preconditioning. The identification of an effective preconditioner may be a problem. For instance, for interior eigenvalues the construction of an effective incomplete LU factorization ....

....we have selected for all cases j max = 15, j min = 10 (the dimension of the subspace before and after implicit restart, respectively) and a fixed random real vector v 0 as an initial guess (cf. Alg. 2 and Alg. 4) As iterative solvers for the correction equation, we have considered full GMRES [17] with a maximum of m steps, denoted by GMRESm , and BiCGstab(2) 19] For BiCGstab(2) a maximum of 100 matrix multiplications was allowed. As stopping criterion for the iterative methods for the correction equation, we have used ke r i k 2 2 Gammaj ke r 0 k 2 , where e r 0 is the initial ....

Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869.

....system Ku = f (5) An important question is whether the land grid points should be included in this system, by including dummy equations. The consequences of this choice when solving (5) is the subject of the next section. 3 Solution method and parallelization. Krylov subspace methods like GMRES [5], Bi CGSTAB [11] and BiCGstab( 6] are powerful techniques for solving large and sparse nonsymmetric linear systems of equations. Apart from scalar operations, the methods comprise of inner products, vector updates, matrixvector multiplications and preconditioning operations. The vector update ....

Y. Saad and M.H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Cornput., 7:856 869, 1986

.... the class of hybrid Krylov methods [11] 4 Experimental Results Two waveform conjugate direction methods were implemented in the WR based device transient simulation program WORDS [2] the nonlinear WGCR algorithm described in Section 3 as well as a nonlin ear waveform GMRES (WGMRES) algorithm [11, 12]. The WORDS program uses red black block GaussSeidel WR and WRN [8] where the blocks correspond to vertical mesh lines; the corresponding Gauss Seidel preconditioner is used for the WGCR and WGMRES implementations. For all methods, the equations gov erning nodes in the same block are solved ....

Y. Saad and M. Schultz, "GMRES: A generalized mini- mum residual algorithm for solving nonsymmetric linear systems," SIAM J. Set. Statist. Coraput., vol. 7, pp. 856- 869, July 1986.

.... methods can be used to accelerate the convergence of WR, but as C is not self adjoint, a variant suitable for non self adjoint operators must be used [8, 9] One such method, shown in Algorithm 2, is waveform GMRES (WGMRES) an exten sion of the generalized minimum residual algorithm (GMRES) [16] to the space 271 Algorithm 2 (Waveform GMRES) 1. Start: Set r = I 2. Iterate: For k = 1,2, until satisfied do: hLk ( I )vk,vJ) j 1,2, k 9 (I g)v = i=x hi,v , v = 9 i h x, 3. Form approximate solution: o V y, where y minimizes ] fie To apply WGMRES to the ....

....that is required consists of large packets of information, i.e. entire waveforms. 4.2 Pointwise Newton (MRES In our experience, the most efficient serial algorithm for device transient simulation was the pointwise Newton GMRES algorithm. In this algorithm, block Jacobi preconditioned GMRES [16] is used to solve the linear systems arising at each Newton iteration of each timestep of an implicit integration formula applied to (1) The pointwise Newton GMRES method in pWORDS uses the same vertical line blocks as the waveform methods, but communication is re quired for each GMRES ....

Y. Saad and M. Schultz, "GMRES: A generalized mini- mum residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Statist. Cornput., vol. 7, pp. 856- 869, July 1986.

....dipoles, with the panels associated with the elements of acting as uniform monopoles (single layers) and the panels associated with acting as uniform dipoles (double layers) III. DIFFICULTIES WITH MULTIPOLE ACCELERATION Consider using a Krylov subspace based iterative algorithm, such as GMRES [10] to solve (10) at each timestep. The th iteration of the GMRES algorithm requires computing the matrix vector product , where is the th GMRES search direction. Since is dense, computing directly requires operations. However, forming is equivalent to computing potentials at points due to a ....

....acceleration always produces results matching those of the explicit calculations, independent of the condition number. A fairly complex 3 D interconnect example is presented here to demonstrate that the multipole accelerated surfacevolume method is necessary for large problems. The GMRES [10] iterative method without preconditioning is used to solve the linear systems (10) and (18) Polysilicon resistivity of cm is assumed for all conductors, and oxide permittivity of is assumed throughout space. All computations are performed on a 266 MHz DEC AXP3000 900 workstation with one gigabyte ....

Y. Saad and M. H. Schultz, "GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., vol. 7, pp. 856--869, 1986.

....sources and dipoles, with the panels associated with the elements of acting as uniform sources (single layers) and the panels associated with 9 acting as uniform dipoles (double layers) 2. 2 Multipole Acceleration Consider using a Krylov subspace based iterative algorithm, such as GMRES [3], to solve (6) at each timestep. The iteration of the GMRES algorithm requires computing the matrix vector product Hu u GMRES search direction. Since H dense, this costs operations. However, forming the matrix vector product in this case is equivalentto computing potentials at ....

....in Figure 1. For the FD BE method, the accelerated solution is indistinguishable from the explicit solution, while for the BE method, the accelerated results are clearly erroneous. 5 Application Experiments Two fairly realistic three dimensional interconnect examples are given here. The GMRES [3] iterative method is used for solving the linear systems in (6) and v2 v3 v4 v11 Figure 2: Clock driving five conductor bus, with ground wires nearby. 13) It is assumed that all conductors are polysilicon, with :02 Omega 0 cm, ffl = 12 both inside and outside the conductors, which ....

Y. Saad and M. H. Schultz. Gmres: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Stat. Comp., 7:856--869, 1986.

....requires too much user intervention and, more importantly, results in errors that are difficult to control and quantify. Iterative schemes, combined with a sparsification algorithm to compute dense matrix vector products efficiently, can be effective for solving large BEM systems. In [24] GMRES [17], a Krylovsubspacebased iterative method, was combinedwith a fast multipole [8] algorithm for substrate resistance extraction. However, accuracy is compromised since the multipole algorithm cannot handle substrate edge effects. A more serious difficulty is that Krylov subspace iterative methods ....

....panel interactions are not well approximated by moment matching, they are computed directly. IV. COMPUTATIONAL RESULTS We present numerical experiments comparing two iterative methods for solving (2) our new multigrid (MG) algorithm and the standard Generalized Minimal RESidual algorithm (GMRES [17]) without preconditioning. Since (2) results from a first kind integral operator (1) the smallest eigenvalues of the matrix approach zero with increasing mesh refinement [12] and becomes more ill conditioned. It is well known that Krylov subspace based iterative methods such as GMRES or CG ....

Y. Saad and M. H. Schultz. Gmres: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Stat. Comp., 7:856--869, 1986.

....kinds of interfaces. In all examples we choose oe 1 = 1 and oe 2 = 1000. For the numerical solution of the integral equation we use a recent Nystrom algorithm [17] which relies on 16 point Gauss Legendre quadrature with aposteriori refinement, solution of systems of linear equations with the GMRES [20] iterative solver, an adaptive method for evaluation of layer potentials close to their sources, and Fast Multipole Method [14 16] acceleration of matrix vector multiplication. Initially, we place eight Gaussian segments per disk. For each stage of adaptive refinement we increase the number of ....

Y. Saad and M. H. Schultz, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856-869 (1986).

....in the Legendre expansion are slowly decaying will be subdivided, and the integral equations are solved again. The final accuracy is determined by using convergence studies explained in [5] and [4] The linear system of equations in each refinement step is solved with the GMRES iterative solver [17] and the iterations are terminated when the residual is less than 10 13 . For simplicity we allow the possibility of negative crack opening displacement. The computed e#ective moduli then become those of a material where the cracks are not closed prior to loading, but have a small initial ....

Y. Saad and M. H. Schultz, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856--869.

....required that the discretized integral equation should be satisfied pointwise at all N quadrature nodes. This gives rise to a system of N linear equations for the N unknown values of u r . 4. The system of N linear equations is solved with any suitable method. We choose the GMRES iterative solver [17]. Wherever the interfaces are smooth, the potential u r will be polynomial like and Gaussian quadrature will do fine in the Nystrom scheme. Close to a corner the the potential u r will not be polynomial like. Equations (3) and (4) suggest that u r should be represented by a power series ....

....bounds for extreme cases [22] The typical accuracy in # e# for non trivial but well conditioned problems involving smooth interfaces is twelve digits [20, 21] We use the evaluation techniques of Section 5 to solve equation (11) adaptively with a Nystrom scheme. The GMRES iterative solver [17] is used for the system of linear equations. 7 Table 1: E#ective conductivity # e# of the square array of prisms depicted in Figure 3. The background material has conductivity # 1 = 1. The prisms have conductivity # 2 = 100 and their area fraction is p 2 = 0.49. In the table points denote the ....

Y. Saad and M. H. Schultz, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856-869 (1986).

....quadrature panels that do not contain crack tips we use 16 point Gauss Legendre quadrature. On quadrature panels that do contain crack tips we use a 16 point quadrature rule based on interpolation with weighted polynomials. The linear system of equations is solved with the GMRES iterative solver [14] and the iterations are terminated when the residual is less than 10 14 . In the numerical examples below we start with 192 uniformly distributed discretization points, solve, refine adaptively, and solve again until the convergence of the stress intensity factors stops. This typically happens at ....

Saad, Y. and Schultz, M.H., GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 1986, 7, 856-869.

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Y. Saad and M. Shultz, "GMRES: A Generalized Minimum Residual Algorithm for Solving Nonsymmetric Linear Systems," SIAM J. Scientific Statistical Computing, Vol. 7, No. 3, July 1986, pp. 856--869.

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Y. Saad and M. H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computating, 7(3):856--869, July 1986.

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Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856 (1986).

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Y. Saad and M. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), pp. 856-869.

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Y. Saad and M. H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856--869, 1986.

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Yousef Saad and Martin H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7:856--869, 1986.

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Y. Saad and M. H. Schultz, GMRES: a generalized minimum residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.

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Y. Saad and M. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.

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Y. Saad and M. H. Schultz. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856--869, 1986.

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