R. S. Varga, Factorization and normalized iterative methods, in Boundary Problems in Di#erential Equations, R. E. Langer, ed., 1960, pp. 121--142.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....any initial vector x . Therefore we have T # ) 1. 23) 6 2. 2 Convergence for H matrices Before proceeding with the convergence analysis of (21) we prove an important result concerning the matrices M i defined in (15) A splitting A = M i N i is called regular if M 1 i O and N O [14]; it is called H compatible if ; see [5] be an H matrix and let the matrices M i be of the form (15) Then, A = M i i , i = 1, p, are H compatible splittings. Proof. First, from the definition of notice that ## i A# i #A## i since the diagonal of the matrix ....

....on the local solvers to guarantee convergence of the additive Schwarz methods are the following O and (32) i = 1, p. 33) We note that condition (32) is satisfied automatically if A i is an H matrix. This occurs, e.g. if A i is an incomplete factorization of A i [9] [14]. Condition (33) is equivalent to having the splitting A i = A i ( A i A i ) be H compatible. Note also that under the conditions (32) 33) since we have O, we conclude that # is a regular splitting. These conditions also provide us with the counterpart to Theorem 1. ....

Varga, R. S. (1960), Factorization and normalized iterative methods, in R. E. Langer, editor, Boundary Problems in Di#erential Equations, pp. 121--142, The University of Wisconsin Press, Madison.

....of these methods were in the 1960 s Buleev [27] and Oliphant [81, 82] These authors proposed them as methods for solving discretizations of partial differential equations. The incomplete factorization methods fall in the more general class of matrix splitting techniques considered by Varga [100, 101], who provided a convergence analysis for Stieltjes and M matrices. Evans [59] was believed to be the first to introduce the term preconditioning and he also considered the use of sparse LU factors as preconditioners. Methods of this type were of particular interest to researchers in the field of ....

R.S. Varga. Factorization and normalized iterative methods. In R.E. Langer, editor, Boundary problems in differential equations, pages 121--142. Univ. of Wisconsin Press, Madison, 1960.

....as block versions of them. The other notable example is incomplete factorizations :i Li Ui where the nonzeros of the factors are in the locations of the nonzeros of Ai, and in particular ILU(0) 30] In these cases, the inequality (3. 10) holds, or equivalently, we have (weak) regular splittings [30,41]. 4. Comparison theorem This section can be read independently of the rest of the paper. Theorem 4.1 Let A O. Let A iV] iV M N be two weak regular splittings such that (4.1) M M . Let w 0 be such that w A e for some e O. Then, 4.2) ivIl ;Vll. If the inequality in (4.1) is ....

Varga, R.S. (1960) Factorization and normalized iterative methods. In Langer, r.e. (ed) Boundary Problems in Differential Equations, pages 121 142, Madison. The University of Wisconsin Press.

....of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 26] In these cases, the inequality (3. 10) holds, or equivalently, we have (weak) regular splittings [26] [36]. 4. Comparison Theorem. This section can be read independently of the rest of the paper. N = M Gamma N be two weak regular splittings such that : 4.1) Let w 0 be such that w = A e for some e 0. Then, k M Nkw kM Nkw : 4.2) If the inequality in (4.1) is strict, then, ....

Richard S. Varga. Factorization and normalized iterative methods. In Rudolph E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Madison, 1960. The University of Wisconsin Press.

....# # =0 (B 1 i,# C i,# ) # B 1 i,# = A 1 i,# . Since A i,# is an M matrix, many of the standard iterative methods indeed represent regular splittings (and thus weak nonnegative splittings of either type) Jacobi, Gauss Seidel and their block variants, and also several ILU splittings; see [28, 39]. With these observations it should be obvious that we can now establish a theory for inexact restricted additive Schwarz methods by following the lines of the previous 476 ANDREAS FROMMER AND DANIEL B. SZYLD sections. In particular, analogous to Theorem 4.4 we get that if A is an M matrix and ....

R. S. Varga, Factorization and normalized iterative methods, in Boundary Problems in Differential Equations, R. E. Langer, ed., The University of Wisconsin Press, Madison, WI, 1960, pp. 121--142.

....A i; In this section, we consider the case were the subdomain problems are solved approximatively or, in other words, inexactly. We represent this fact by using an approximation A i; of the matrix A i; In practice one uses for example an incomplete factorization of A i; see, e.g. 18] [25]. As in [14] suppose that the inexact solves are such that the splittings A i; A i; A i; A i; are weak regular splittings (21) for i = 1; p; or that A i; is an M matrix and A i; A i; i = 1; p: 22) Note that (22) implies (21) The incomplete factorizations ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Dierential Equations, pages 121-142, Madison, 1960. The University of Wisconsin Press.

....of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 33] In these cases, the inequality (37) holds, or equivalently, we have (weak) regular splittings [33] [47]. For examples of splittings for which the inequality (39) holds see [35] Another situation worth mentioning where (39) holds is when A i is semidefinite and the inexact solver is definite. This process is usually called regularization; see, e.g. 14] 30] In [20] it is shown that the damped ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Madison, 1960. The University of Wisconsin Press.

.... ) B Gamma1 i;ffi = A Gamma1 i;ffi : Since A i;ffi is an M matrix, many of the standard iterative methods indeed represent regular splittings (and thus weak nonnegative splittings of either type) Jacobi, Gauss Seidel and their block variants, but also several ILU splittings, see [25, 33]. With these observations it should be obvious that we can now establish a theory for inexact restricted additive Schwarz methods by following the lines of the previous sections. In particular, in analogy to Theorem 4.4 we get that if A is an M matrix and (27) is satisfied, inexact RAS is ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Madison, 1960. The University of Wisconsin Press.

....Gauss Seidel methods have motivated important developments in the theory of matrix linear algebra. In particular, relevant properties for M matrices, introduced by Ostrowski in 1937 [126] were uncovered and convergence results for so called regular splittings, introduced by Varga (1960) [179] were established. A cornerstone in the convergence theory was the theorem of Stein Rosenberg (1948) 160] which proved relations between the asymptotic rates of convergence for the successive overrelaxation methods, including the Gauss Seidel method, and the Gauss Jacobi method. The concept of ....

....of preconditioning for the CG method. 8.1 Incomplete Factorizations Preconditioning as we know it today refers mostly to approximate or incomplete factorizations of the coefficient matrix. Some of the early publications on such factorizations that are often cited include Buleev [34] Varga [179]. and Oliphant [123] Later in the 1960s a few other procedures were developed specifically for matrices arising from finite difference approximations to elliptic operators, these include the work by Dupont, Kendall, and Rachford [57] In 1977 Meijerink and Van der Vorst introduced the more ....

R.S. Varga. Factorization and normalized iterative methods. In R.E. Langer, editor, Boundary problems in differential equations, pages 121--142. Univ. of Wisconsin Press, Madison, 1960.

....goes back to Ostrowski [30] see also, e.g. Neumaier [25] Definition 2.2. Let A # IR nn . The representation A = M N is called a splitting if M is nonsingular. It is called a convergent splitting if #(M 1 N) 1. A splitting A = M N is called (a) regular if M 1 # O and N # O [39], 40] b) weak regular if M 1 # O and M 1 N # O [1] 29] c) H splitting if #M# N is an M matrix [14] and ETNA Kent State University etna mcs.kent.edu 28 Parallel, synchronous and asynchronous two stage multisplitting methods (d) H compatible splitting if #A# = #M# N ....

Richard S. Varga, Factorization and normalized iterative methods, in Boundary Problems in Di#erential Equations, Rudolph E. Langer, ed., The University of Wisconsin Press, Madison, 1960, pp. 121--142..

....of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 29] In these cases, the inequality (31) holds, or equivalently, we have (weak) regular splittings [29] [41]. For examples of splittings for which the inequality (33) holds see [31] Another situation worth mentioning where (33) holds is when A i is semidefinite and the inexact solver is definite. This process is usually called regularization; see, e.g. 13] 27] In [19] it is shown that the damped ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Madison, 1960. The University of Wisconsin Press.

....based on matrix structure The original incomplete factorizations were developed for solving finite difference equations for elliptic partial differential equations. For these problems, the structure of the incomplete triangular factors was chosen based on the structure of the gridpoint operators [33, 111, 112, 145] (see also the review [39] and the resulting structure of the error matrix E. In most cases, the gridpoint operator 2.4. INCOMPLETE LU PRECONDITIONERS 29 was a five point stencil, and the stencil for the lower (upper) triangular factor was chosen to have the same pattern as the lower (upper) ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Santa Barbara, CA, 1960. University of Wisconsin Press.

.... 1 X =0 (B 1 i; C i; B 1 i; A 1 i; Since A i; is an M matrix, many of the standard iterative methods indeed represent regular splittings (and thus weak nonnegative splittings of either type) Jacobi, Gauss Seidel and their block variants, but also several ILU splittings, see [25, 33]. With these observations it should be obvious that we can now establish a theory for inexact restricted additive Schwarz methods by following the lines of the previous sections. In particular, in analogy to Theorem 4.4 we get that if A is an M matrix and (27) is satis ed, inexact RAS is ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Dierential Equations, pages 121-142, Madison, 1960. The University of Wisconsin Press.

....of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 29] In these cases, the inequality (31) holds, or equivalently, we have (weak) regular splittings [29] [41]. For examples of splittings for which the inequality (33) holds see [31] Another situation worth mentioning where (33) holds is when A i is semide nite and the inexact solver is de nite. This process is usually called regularization; see, e.g. 13] 27] In [19] it is shown that the damped ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Dierential Equations, pages 121-142, Madison, 1960. The University of Wisconsin Press.

....Gauss Seidel methods have motivated important developments in the theory of matrix linear algebra. In particular, relevant properties for M matrices, introduced by Ostrowski in 1937 [122] were uncovered and convergence results for so called regular splittings, introduced by Varga (1960) [171] were established. A cornerstone in the convergence theory was the theorem of Stein Rosenberg (1948) 155] which proved relations between the asymptotic rates of convergence for the successive overrelaxation methods, including the Gauss Seidel method, and the Gauss Jacobi method. The concept of ....

....of preconditioning for the CG method. 8.1 Incomplete Factorizations Preconditioning as we know it today refers mostly to approximate or incomplete factorizations of the coefficient matrix. Some of the early publications on such factorizations that are often cited include Buleev [30] Varga [171]. and Oliphant [119] Later in the 1960s a few other procedures were developed specifically for matrices arising from finite difference approximations to elliptic operators, these include the work by Dupont, Kendall, and Rachford [53] In 1977 Meijerink and Van der Vorst introduced the more ....

R.S. Varga. Factorization and normalized iterative methods. In R.E. Langer, editor, Boundary problems in differential equations, pages 121--142. Univ. of Wisconsin Press, Madison, 1960.

....for i = 1 to q M i x (i) k = Nx k Gamma1 b) i) 3) In the standard Block Jacobi method, the blocks in the diagonal are chosen as M i = A ii . Other choices of the blocks M i , such as changing some entries in A ii , can also be considered without any change in the analysis; cf. Varga [25], 26] To implement Algorithm 1.1 on a parallel computer, suppose for example that we have q processors. During iteration k the processors can solve the q equations (3) simultaneously, since they are independent from each other. Before starting the next iteration k 1, however, all results from ....

....to Ostrowski [21] see also e.g. 19] ut Definition 1.3. Let A 2 R n Thetan . The representation A = M Gamma N is called a splitting if M is nonsingular. It is called a convergent splitting if ae(M Gamma1 N) 1. A splitting A = M Gamma N is called (a) regular if M Gamma1 O and N O [25], 26] b) weak regular if M Gamma1 O and M Gamma1 N O [2] 20] c) H splitting if hMi Gamma jN j is an M matrix [12] and (d) H compatible splitting if hAi = hMi Gamma jN j [12] Lemma 1.4. Let A = M Gamma N be a splitting. a) If the splitting is weak regular, then ae(M Gamma1 ....

Varga, R.S. (1960): Factorization and normalized iterative methods. In: Langer, R.E., ed., Boundary Problems in Differential Equations, pp. 121--142. Madison, The University of Wisconsin Press

....based on matrix structure The original incomplete factorizations were developed for solving finite difference equations for elliptic partial differential equations. For these problems, the structure of the incomplete triangular factors was chosen based on the structure of the gridpoint operators [7, 28, 29, 39] (see also the review [9] and the resulting structure of the error matrix E. In most cases, the gridpoint operator was a five point stencil, and the stencil for the lower (upper) triangular factor was chosen to have the same pattern as the lower (upper) triangular part of the original stencil. ....

R. S. Varga. Factorization and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Santa Barbara, California, 1960. University of Wisconsin Press.

....of them. The other notable example is incomplete factorizations A i = L i U i where the nonzeros of the factors are in the locations of the nonzeros of A i , and in particular ILU(0) 26] In these cases, the inequality (3. 10) holds, or equivalently, we have (weak) regular splittings [26] [36]. 4. Comparison Theorem. This section can be read independently of the rest of the paper. Theorem 4.1. Let A Gamma1 O. Let A = M Gamma N = M Gamma N be two weak regular splittings such that M Gamma1 M Gamma1 : 4.1) Let w 0 be such that w = A Gamma1 e for some e ....

Richard S. Varga. Factorization and normalized iterative methods. In Rudolph E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142, Madison, 1960. The University of Wisconsin Press.

....of A; efficient, in the sense that the cost of solving Mu = v is much smaller than the cost of solving Au = v, and good in the sense that the convergence rate for the preconditioned iteration is significantly faster than for the unpreconditioned one. Incomplete factorization techniques [24], 17] 20] provide a good preconditioning strategy for solving linear systems with Krylov subspace methods. Usually, however, simply applying this strategy to the full naturally ordered linear system leads to a method with little parallelism. Incomplete factorization is also useful as an ....

R. S. VARGA, Factorizations and normalized iterative methods, in Boundary Problems in Differential Equations, R. E. Langer, ed., Univ. of Wisconsin Press, Madison, WI, 1960, pp. 121--142.

....involved in 9 applying the preconditioner allows the iterative algorithm to easily take advantage of the vector and parallel capabilities of high performance architectures. The second class of methods is based on the incomplete LU or Cholesky factorization of the coefficient matrix A [5,6,8,36,37,42,44,51]. In this class, the preconditioner M = LU , where LU is an incomplete factorization of the matrix A. The term incomplete factorization is used because certain nonzero entries in the L and U factors are discarded during the factorization process, i.e. A = LU Gamma R, where the entries in R ....

R. S. Varga. Factorizations and normalized iterative methods, in Boundary Problems in Differential Equations (edited by R.E. Langer). The University of Wisconsin Press, Madison, Wisconsin, 1960.

....The revival of interest in preconditioning produced the important paper of J. Meijerink and Henk van der Vorst [38] available in preprint form in the early 1970 s. They developed an algorithm for computing an incomplete LU factorization of an M matrix (as did Richard Varga in a 1960 paper [51]) This work inspired the hope of having a library KMP Algorithms 5 of preconditioners that would apply to broad problem classes. Preconditioning was also discussed by Owe Axelsson [2] by Paul Concus, Gene Golub, and Dianne O Leary [10] and for nonlinear problems by Jim Douglas and Todd Dupont ....

R. S. Varga, Factorization and normalized iterative methods, in Boundary Problems in Differential Equations, R. E. Langer, ed., University of Wisconsin Press, Madison, 1960, 8 O'Leary pp. 121--142.

....is to reuse the preconditioner that was computed for one value of the barrier parameter in order to solve systems for several successive values of [3] 21] This reduces the computational work in forming the factorization. An incomplete Cholesky factorization, originally proposed by Varga [34], could be used in place of the Cholesky if density of the matrix factors is too great, but we do not pursue that idea in our implementations. Rather than keeping the preconditioner fixed when changes, though, we can update it by a small rank change, since the normal equations matrix is a ....

Richard S. Varga. Factorization and normalized iterative methods. In Rudolph E. Langer, editor, Boundary Problems in Differential Equations, pages 121--142. University of Wisconsin Press, Madison, 1960.

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R. S. Varga, Factorization and normalized iterative methods, in Boundary Problems in Di#erential Equations, R. E. Langer, ed., 1960, pp. 121--142.

No context found.

R. S. Varga. Factorizations and normalized iterative methods. In R. E. Langer, editor, Boundary Problems in Di#erential Equations, pages 121--142, Madison, WI, 1960. University of Wisconsin Press.

No context found.

Netherlands. Varga, R. S. (1960), Factorizations and normalized iterative methods, in R. E.

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