R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... solver of Cimmino type [5] particularly suited for parallel computation (truncated or Inexact version of the Quasi Newton method, in the sense of [7] In the recent years there has been a growing of interest in the row projection methods for solving large sparse nonsymmetric linear systems [1,3,15,16]. Several numerical experiences have shown that they may be competitive with other iterative solvers, in particular when the Conjugate Gradient (CG) acceleration is employed. In [16] and [17] some preliminary numerical results in solving large sparse systems of nonlinear equations by an Inexact ....

Bramley R., and Sameh, A., Row projections methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13 (1), 168-193, 1992.

....We study the implementation in multiprocessor environments of block iterative methods from the class of row projection methods . This class of iterative methods is widely used in the frame of image reconstruction, for example, and is recognized to be quite robust (cf. Kamath and Sameh (1988) Bramley and Sameh (1992)) We have developed a block version of the method of Cimmino (see Cimmino (1939) for the solution of consistant sparse linear systems Ax = b (3) where A is an m Theta n sparse matrix (m n) and b is an m vector. In this block iterative method, A is partitioned in a row oriented way 0 B ....

R. Bramley and A. Sameh, (1992), Row projection methods for large nonsymmetric linear systems, SIAM J. Scientific and Statistical Computing, 13, 168--193.

....b as was noted by Bjorck and Elfving [21] and this would correspond to taking directions d k along the columns of the matrix. This class of methods regained interest in the 70s and 80s with the work of Bjorck and Elfving [21] Tanabe [165] and later Kamath and Sameh [101] and Bramley and Sameh [23]. However, one dimensional projections methods of a different type, based on very general definitions of norms were very popular in the late 50s. Here, we mention the work of Gastinel among others. Gastinel s approach [78] consisted of defining generating vectors for norms. Consider an arbitrary ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992. 20

....algorithm, has proven quite effective (for example, Arioli, Duff, Ruiz and Sadkane 1995) As in the point case, we can sacrifice parallelism a little and gain faster convergence through the use of a block Gauss Seidel method. The counterpart to block Cimmino is then block Kacmarz (for example, Bramley and Sameh 1992). 15 In the framework of the solution of partial differential equations, a more general technique for combining direct and iterative methods is to use domain decomposition (Chan and Mathew 1994, Smith, Bjorstad and Gropp 1996) where the problem is divided into separate domains (either ....

Bramley, R. and Sameh, A. (1992), `Row projection methods for large nonsymmetric linear systems', SIAM J. Scientific and Statistical Computing 13, 168--193.

....1.2. Related work. Several robust preconditioning techniques targeted at general sparse systems have been proposed in recent years. These include preconditioners based on incomplete orthogonalization, such as incomplete LQ [45] and incomplete QR [55] 8] row projection methods [1] 2] [14], and sparse approximate inverse techniques based on adaptive Frobenius norm minimization [19] 34] 17] 35] However, all these techniques are considerably more expensive than standard ones based on incomplete factorizations, and furthermore, as acknowledged in the papers just referenced, ....

....is completely automatic, and very little knowledge is required of the problem at hand. This greatly facilitates the use of such techniques in a production environment. Another consequence of this work is that robust but costly preconditioning techniques such as those proposed in [1] 2] 8] [14], 17] 34] lose some of their appeal. These methods have been described as being able to solve difficult problems for which standard, ILU type methods fail. It now appears that such problems can be solved by ILU type methods when combined with the preprocessing techniques used in this paper. ....

R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 168--193.

.... Gamma bjj 2 , where A 2 R m Thetan , x 2 R n and b 2 R m , if f(x) in (1) is taken to be f(x) Ax Gamma b. In this case the BNMS of f(x) amounts to a partition of the matrix A by columns and hence is not a member of the class of row projection methods as described, for example, in [3]. Further, this approach also does not yield the same kind of iterative procedure as described in [12] On the contrary, depending on the means for solving the subproblems, this technique can be seen as an iterative QR algorithm, or as any column blocked iterative scheme of your choice, see eg ....

R. Bramley and A. Sameh, Row Projection Methods for Large Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comput., Vol. 13, No. 1, (1992), pp. 168-193.

....are two kinds of projection methods to solve it: For the consistent systems (M = N ) the linear equation is decomposed into several smaller linear equations by row partitioning . Then x can be solved by iteration methods such as Kaczmarz projection method and Cimmino projection method [3] 4][5]. For the inconsistent systems (M N ) A is decomposed into smaller submatrices by column partitioning . Then x and the residual can be solved by gradient based iteration method [6] Because the whole data matrix is used to compute the gradient, it is non adaptive in nature and the convergence ....

R. Bramley and A. Samem, "Row projection methods for large nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., vol. 13, pp. 168--193, Jan. 1992.

....the GSVD for analyzing and solving generalized linear model regression problem. In the block Kaczmarz and Cimmino algorithms for solving large sparse nonsymmetric linear system, the CSD has found to be a useful tool to analyze the conditioning of the orthogonal row projectors used in the procedure [14]. 5 Algorithms Competitive numerical algorithms have been proposed for computing the CSD and the GSVD. In this section, we shall review the existing serial and parallel algorithms for computing these decompositions, and present some modifications to the algorithms. As we have seen in x2, if the ....

R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, (1992), pp.168--193.

....= b as was noted by Bjork and Elfving [20] and this would correspond to taking directions d k along the columns of the matrix. This class of methods regained interest in the 70s and 80s with the work of Bjork and Elfving [20] Tanabe [159] and later Kamath and Sameh [98] and Bramley and Sameh [21]. However, one dimensional projections methods of a different type, based on very general definitions of norms were very popular in the late 50s. Here, we mention the work of Gastinel among others. Gastinel s approach [74] consisted of defining generating vectors for norms. Consider an arbitrary ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992.

....Cimmino method is a generalization of the Cimmino method [9] Basically, we partition the linear system of equations: Ax = b; 1) where A is a m Theta n matrix, into l subsystems, with l m, such that: 0 B A 1 . A l 1 C A x = 0 B b 1 . b l 1 C A (2) The block method ([6, 2]) computes a set of l row projections, and a combination of these projections is used to build the next approximation to the solution of the linear system. 4 Arioli et al. Now, we formulate the Block Cimmino iteration as: ffi i (k) A i i b i Gamma A i x (k) j (3) x (k 1) ....

R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Scientific and Statistical Computing, 13 (1992), pp. 168--193.

....decomposition [15] However, since the condition number of CC T is the square of the condition number of C, CC T ) 2 (C) and a loss of information can occur when computing CC T , this method should only be used for well conditioned problems. This approach was used by Bramley and Sameh [7] as well as Benzi et al. 4] for the so called Cimmino method, see Section 6. Another approach is the augmented system method where the indefinite linear system I C T C 0 y z = 0 d is solved by means of LDL T decomposition. However, since z = Gamma(C C T ) Gamma1 d ....

....of the Cimmino projection method. With the system matrix decomposed into single rows we have the Cimmino method as originally proposed by Cimmino, see [9] and [32] in 1939. Later, non overlapping block versions have been studied by Elfving [11] Ruiz [29] Arioli et al. 19] and Bramley and Sameh [7] in combination with conjugate gradient acceleration. These approaches differs mainly in the computation of the orthogonal projections as discussed in Section 4.1 and in the partitioning of the system matrix by using different subspace decompositions. Zilli [40] calculates the projections ....

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R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 168--193.

....parallel, the communication scheme, and distribution of tasks among processors. The Cimmino method ( 7] is a row projection method in which the solution of the linear systems of equations is obtained through row projections of the original matrix in given subspaces. In the Block Cimmino method ([2, 5]) the blocks are obtained by dividing the linear system of equations into subsystems. At every iteration, it computes one projection per subsystem and uses these projections to construct an approximation to the solution of the linear system. The Block CG method can also be used to accelerate the ....

....Cimmino method [7] Basically, we partition the linear system of equations: Ax = b; 3.1) where A is a m Theta n matrix, into l subsystems, with l m, such that: 0 B B B B A 1 A 2 . A l 1 C C C C A x = 0 B B B B b 1 b 2 . b l 1 C C C C A (3. 2) The block method ([5, 2]) computes a set of l row projections, and a combination of these projections is used to build the next approximation to the solution of the linear system. Now, we formulate the Block Cimmino iteration as: ffi i (k) A i b i Gamma P R(A i T ) x (k) 3.3) A i i b i Gamma ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing 1992, 13, pp 168-193.

.... with a row projection linear solver of Cimmino type, particularly suited for parallel computation (truncated or Inexact versions, in the sense of [5] In the recent years there has been a growing of interest in the row projection methods for solving large sparse nonsymmetric linear systems [1] [2], 15] 16] Several numerical experiences have shown that they may be competitive with other iterative solvers, in particular when the Conjugate Gradient (CG) acceleration is employed. In [16] and [17] some preliminary numerical results in solving large sparse systems of nonlinear equations by ....

Bramley R., and Sameh, A., Row projections methods for large nonsymmetric linear systems, Siam J. Sci. Stat. Comput. 13 (1), 168-193, 1992.

....including a 64 processor GC Power Plus machine. For general systems, one could apply a block Jacobi preconditioning to the normal equations which would result in the block Cimmino algorithm [13] A similar relationship exists between a block SOR preconditioning and the block Kaczmarz algorithm [35]. Block preconditioning for symmetric systems is discussed in [50] in [51] incomplete factorizations are used within the diagonal blocks. Attempts have been made to preorder matrices to put large entries into the diagonal blocks so that the inverse of the matrix is well approximated by the block ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing, 13:168--193, 1992.

....setting, where problems on subdomains are solved by the direct method but the interaction between the subproblems is handled by an iterative technique. A related example of this is the work on block projection methods like Block Cimmino (Arioli, Duff, Noailles and Ruiz 1992) or Block Kacmarz (Bramley and Sameh 1992). Block preconditioning for symmetric systems is discussed by Concus, Golub and Meurant (1985) and Concus and Meurant (1986) use incomplete factorizations within the diagonal blocks. Attempts have been made to preorder matrices to put large entries into the diagonal blocks so that the inverse of ....

Bramley, R. and Sameh, A. (1992), `Row projection methods for large nonsymmetric linear systems', SIAM J. Scientific and Statistical Computing 13, 168--193.

....to an approximate solution restricted to subspaces spanned by the rows of the system matrix. These corrections can be performed in succession, as in the Kaczmarz method, or in parallel, as in the Cimmino method. Extensions to these methods to non overlapping block rows have been studied, cf. [5, 1, 3, 9, 2], and shown remarkable robustness and potential for parallelism. In this note we focus on extensions to general subspaces, that may overlap, and discuss how the incorporation of weighting schemes can improve convergence significantly. 2. Subspace Correction Methods. For a finite dimensional ....

R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 168--193.

....slowly. It is therefore important to precondition the system, in order to improve the condition number. This has been investigated to some degree in [6] where an incomplete LU factorization as well as its modified form are used. Other preconditioned forms of CGNE are the row projection methods [3], where a block Jacobi or block SSOR preconditioning is used. These methods turn out to be very robust in many cases, however, they often converge slowly. One can maintain the minimization property by choosing the direction vector as a linear combination of the residual vector and k previous ....

R. Bramley, A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13: 168-193 (1992).

....this matrix can be expressed as a sum of projections onto the the null spaces of the blocks of rows. We exploit this feature to obtain an efficient parallel algorithm for the solution of such systems. Several parallel algorithms have been proposed over the years to solve general linear systems [2, 10, 1, 4, 6, 11]. Our approach is similar to the one described in [7] and more closely related to the reduced system approach proposed in [8] The paper is organized as follows: in x2 we present a technique to obtain the reduced system. We describe the balanced method to solve the reduced system in x3. In x4 we ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992.

....may be developed for systems that are not banded. 3. 2 Row Projection Methods Row projection methods are a general class of iterative algorithms that compute successive approximations to the solution by projecting the current estimate for solution onto subspaces defined by block rows (see e.g. [7,8]) The orthogonal projection of a vector x onto the subspace of the ith block row E i is P i x = E T i (E i E T i ) Gamma1 E i x. The Kaczmarz projection method uses a strategy of successive projections onto these subspaces giving the following iteration: x k 1 = 2 4 p Y j=1 (I ....

R. Bramley and A. H. Sameh, "Row projection methods for large nonsymmetric linear systems", SIAM J. Sci. Stat. Comput., 13 (1992) 168--193.

....[2] proposed using CG acceleration of the basic RP schemes, and C. Kamath and A. Sameh [9] first proposed and developed methods that allow parallelism in the block tridiagonal case. They also examined various methods of computing the projections for the Kaczmarz scheme. R. Bramley and A. Sameh [4] provided a detailed analysis of the effects of different row partitionings for the seven point difference operator, and a theoretical comparison with CG on the normal equations. Standard RP methods for solving Ax = f begin by partitioning A 2 N ThetaN into m block rows: A T = A 1 ; A 2 ; ....

....definite and so the method of conjugate gradients (CG) can be applied to solve it. Note that each CG iteration will require 2m Gamma 1 orthogonal projections to be computed. It is possible to reduce this number to 2m Gamma 2, by starting with the initial estimate x 0 in a certain affine set; see [4] for details. 1.2. The Cimmino Row Projection Method. The second basic RP method can be derived as a preconditioner for the CG algorithm. Premultiply the system Ax = b by A = A 1 (A T 1 A 1 ) Gamma1 ; A 2 (A T 2 A 2 ) Gamma1 ; Am (A T m Am ) Gamma1 ] to obtain (P 1 P ....

[Article contains additional citation context not shown here]

R. Bramley and A. Sameh, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 168--193.

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R. Bramley, A. Sameh, Row projection methods for large nonsymmetric linear systems, CSRD Tech. Rept. 957 (revised), Center for Supercomputing Research and Development, Univ. Illinois - Urbana, (1990).

....product. Such computation requires projections onto null spaces of the block rows, and can be computed concurrently. We exploit this feature to obtain an efficient parallel formulation of our algorithm. Several parallel algorithms have been proposed over the years to solve general linear systems [1,2,4,6,10,11]. Our approach is similar to the one described in [7] and more closely related to the reduced system approach proposed in [8] The paper is organized as follows: in Sec. 2 we present a technique to obtain the reduced system. We describe the balanced method to solve the reduced system in Sec. 3. ....

R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992.

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R. Bramley and A. Sameh (1992), Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 13 (1), 168--193.

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R. Bramley and A. Sameh. Row projection methods for large nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 13:168--193, 1992.

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Comp. 8, 186--190. Bramley, R. and Sameh, A. (1992), `Row projection methods for large nonsymmetric linear systems', SIAM J. Scientific and Statistical Computing 13, 168--193.

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