J. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., New York, 1976, Academic Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bandwidth. In a Grid environment, the significantly higher overheads can completely incapacitate these methods. Block iterative methods increase granularity by hav ing each processor apply the same fine grain operations on a block of vectors, thus increasing the computation communication ratio [3, 13]. As a side benefit, block operations also demonstrate better cache performance. However, the granularity increase is usually marginal, and because block methods increase the total number of floating point operations, the benefits from such implementations of these algorithms are limited. In ....

P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. Academic Press, New York, NY, 1976.

....development of new methods with better properties in the presence of bad aspect ratios. Because we are concerned with self adjoint, V elliptic problems, which yield linear systems with symmetric, positive definite matrices, the basic solution method is the preconditioned conjugate gradient method [10 12] and we concentrate on the design of effective preconditioners. We give a theoretical analysis of condition numbers for high aspect ratios, and present a method with better convergence properties in such a case. The bound on the condition number still depends (weakly) on the aspect ratio, due to ....

P. Conms, G.H. Golub and D.P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic differential equations, in: J.R. Bunch and D.J. Rose, eds., Sparse Matrix Computations (Academic Press, New York, 1976) 309-332.

....bandwidth. In a Grid environment, the significantly higher overheads can completely incapacitate these methods. Block iterative methods increase granularity by having each processor apply the same fine grain operations on a block of vectors, thus increasing the computation communication ratio [3, 13]. As a side benefit, block operations also demonstrate better cache performance. However, the granularity increase is usually marginal, and because block methods increase the total number of floating point operations, the benefits from such implementations of these algorithms are limited. In ....

P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. Academic Press, New York, NY, 1976.

....solution of a symmetric positive definite system of linear equations. Gaussian elimination and other equivalent methods for solving systems of linear equations consist of obtaining an LU decomposition of the system matrix, followed by the solution of two triangular systems of equations (see [ 10] [ 1 4], and [l 9] for a complete treatment of this subject) Also, considerations of sparsity are paramount in the practical implementation of Gaussian elimination. We present, in the end of this section, a basic implementation of a sparse symmetric Gaussian elimination procedure. Consider system (2.1) ....

P. CONCUS, G.H. GOLUB AND D.P. O'LEARY, 1976. A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations, in J.R. Bunch and D.J. Rose (eds.), Sparse Matrix Computations, Academic Press, New York, pp. 309-332.

.... When the Lanczos algorithm is applied to a Hermitian matrix, the resulting eigenvalue estimates are Ritz values (B = I) The relationship between the conjugate gradient method and the Lanczos algorithm has been described in a variety of papers and used to obtain eigenvalue estimates (cf. [CGO76]) However, in the general case it is the roots of the orthogonal polynomials not the residual polynomials that yield B Ritz values (c.f. AMS90] For many implementations the distinction is minimal. For example, consider the original algorithm of Hestenes and Stiefel (CGHS) HeSt52] Here A is ....

....(r 0 ; A) 2. 41) and the roots of the residual polynomials are in fact the Ritz values (B = I) of A with respect to K i (r 0 ; A) while the roots of the orthogonal polynomials are B Ritz values (B = A) of A with respect to K i (r 0 ; A) In the preconditioned conjugate gradient algorithm (PCG) [CGO76] the matrices A and C are assumed to be HPD with B = A. Under these hypotheses, CA is B self adjoint positive definite, that is, hBCAx; yi = hBx; CAyi; 2.42) for every x; y 6= 0 2 C n and hBCAx;xi hBx; xi 0 (2.43) for every x 2 C n . While the roots of the residual polynomials ....

P. Concus, G. H. Golub and D. P. O`Leary, A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, 1976.

....i can be expressed as a polynomial in A of degree i Gamma 1, acting on r 0 . The minimization interpretation makes it possible to bound the error for CG by replacing the CG polynomial by easier to analyze polynomials, for instance a Chebyshev polynomial. This leads to the well known upper bound [102, 48, 45, 4] kx i Gamma xkA p 2 p Gamma 1 p 1 i kx 0 Gamma xkA ; 5) for symmetric positive definite matrices, in which = max (A) min (A) This upper bound describes well the convergence behavior for matrices A of which the eigenvalues are distributed rather homogeneously. For more ....

....in the course of the process. Proofs for superlinear convergence had been given already in the early fifties [104, 92] but these did not reveal that the superlinear behavior may take place in early phases of the iteration process; they were qualitative rather than quantitative. Concus et al. [45] related this convergence behavior to the Krylov subspace approximations, by stating that the extremal eigenvalues are approximated especially well (by the Ritz values corresponding to the Krylov subspace) as CG proceeds, the iteration then behaving as if the corresponding eigenvectors are not ....

[Article contains additional citation context not shown here]

P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In James R. Bunch and Donald J. Rose, editors, Sparse Matrix Computations, pages 309--332, New York, 1976. Academic Press.

....m 0 (8.23) Since they are orthogonal with respect to the weight function w(x) 1 Gamma x 2 ) Gamma1=2 , they can be computed using the three term recursion: Y 0 (x) 1; Y 1 (x) x; Ym 1 (x) 2xYm Gamma Ym Gamma1 ; m 1: 8. 24) 5 Convergence at a rate that increases per iteration [21, 58]. 148 SEARCHING FOR FAST SOLUTIONS FOR RADIOSITY SYSTEMS x Y m (x) degree m=3 degree m=4 degree m=5 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 Figure 8.1: Monic Chebyshev polynomials. Most importantly, Chebyshev polynomials ....

Concus, P., Golub, G., and O'Leary, D. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In 187 Sparse Matrix Computations (New York, 1976), J. Bunch and D. Rose, Eds., Academic Press, pp. 309--322.

....iteration gets going. 6. Step 2: Estimated vs. actual spectrum. Our second step is logically trivial but of great importance in practice. Among other things, it can be viewed as responsible for the phenomenon of superlinear convergence often observed in the later stages of a matrix iteration [4], 55] 79] 83] This is the gap between the estimated spectrum S of the matrix A and its actual spectrum, which we denote by #(A) Symbolically, STEP2) min p#Pn #p# S # min p#Pn #p# #(A) The mathematics of (STEP2) is immediate. If S is di#erent from the spectrum #(A) then in ....

....that the spectrum is not continuous but discrete. Here S = 1, 100] and #(A) 1, 2, 3, 100 . We must point out that given a spectrum #(A) there is no unique choice of a set S that can be used to approximate it. In the example of Fig. 12, for example, suppose we were to take S = 1 # 2 # 3 #[4, 100] instead of S = 1, 100] This would not a#ect the curve of E n (#(A) but it would a#ect how we interpret it. We would now describe some of the downward bend of this curve as caused by the finite n phenomenon of Fig. 10 rather than by the gap between S and #(A) 7. Step 3: Normal vs. nonnormal ....

P. CONCUS,G.H.GOLUB, AND D. P. O'LEARY, A generalized Conjugate Gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, 1976, pp. 309-- 332.

....in x4. 2 The conjugate gradient algorithm In this section we will present several variants of the conjugate gradient algorithm, based on elimination of the A inner product of the search directions. 2. 1 The standard formulation We begin by reviewing the standard conjugate gradient procedure [2, 8] for solving the linear system Ax = b : 2) 2 For simplicity, we assume a zero initial guess, and residual vector r 1 = b, with hx; yi = x t y as the usual inner product. For k = 1; 2; fl k = hr k ; r k i fi k = fl k =fl k Gamma1 (fi 1 = 0) p k = r k fi k p k Gamma1 (p 1 = r 1 ) ....

P. Concus, G. Golub, and D. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In J. Bunch and D. Rose, editors, Sparse Matrix Computations, pages 309-- 322. Academic Press, New York, 1976.

No context found.

P.Concus, G.H.Golub and D.P.O'Leary. A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Dierential Equations. In \Sparse Matrix Computations", ed. J.R.Bunch and D.J.Rose, Academic Press, NY, 1976.

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Concus P, Golub GH, O'Leary DP. A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations. InSpar--- Matr--- Computations, Bunch JR, Rose DJ (eds). Academic Press: New York, 1976; 309 -- 332. Reprinted in Studies inNumer#PC Analysis, Golub GH (ed). MAA Studies in Mathematics, Vol. 24, Math. Assn. of America, 1984; 178 --198.

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P.Concus, G.H.Golub and D.P.O'Leary. A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Dierential Equations. In Sparse Matrix Computations, ed. J.R.Bunch and D.J.Rose, Academic Press, NY, 1976, pp 309-332.

....precision arithmetic was not in agreement with that point of view, they were initially considered with suspicion. Through work of Reid [79] it became clear that these methods, if used as iterative techniques, could be used to advantage for many classes of linear systems. With preconditioning [21], these methods could be made even more efficient. Among the first popular preconditioned methods was ICCG [69, 63] In the period after 1975, we have seen that symmetric positive definite sparse systems were usually solved by preconditioned CG methods, when very large, and by sparse direct ....

....is defined by kyk A = y; y) A j (y; Ay) and we need the positive definiteness of A in order to get a proper innerproduct ( Delta; Delta) A . The local convergence behavior of CG, and especially the occurrence of superlinear convergence, was first explained in a qualitative sense in [21], and later in a quantitative sense in [94] In both papers it was linked to the convergence of eigenvalues (Ritz values) of T i;i towards eigenvalues of A, for increasing i. The global convergence can be bounded with expressions that involve condition numbers, for details see for instance [21, ....

[Article contains additional citation context not shown here]

P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In J. R. Bunch and D. J. Rose, editors, Sparse Matrix Computations. Academic Press, New York, 1976.

No context found.

J. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., New York, 1976, Academic Press.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, NY, 1976, pp. 309--332.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, Sparse Matrix Computations (New York, NY) (J. R. Bunch and D. J. Rose, eds.), Academic Press, New York, NY, 1976, pp. 309--332.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, NY, 1976, pp. 309--332.

No context found.

P. Concus, Gene H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations. In J. R. Bunch and D. J. Rose, editors, Sparse Matrix Computations, pages 309--332. Academic Press, New York, 1976. 24

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P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations. In J. R. Bunch and D. J. Rose, editors, Sparse Matrix Computations, pages 309--332. Academic Press, New York, NY, 1976.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, NY, 1976, pp. 309--332.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, 1976, pp. 309--332.

No context found.

Concus, P., Golub, G., & O'Leary, D., A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations. Proc. Symposium on Sparse Matrix Computations. Bunch, J.R., & Rose, D.J., eds. Academic Press, 1976.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary. A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations. In J. R. Bunch and D. J. Rose, editors, Sparse Matrix Computations. Academic Press, New York, 1976.

No context found.

P. Concus, G. Golub, and D. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial di#erential equations, J. R. Bunch and D. J. Rose, eds., Sparse Matrix Computations, pp. 309--332, Academic Press, 1976.

No context found.

P. Concus, G. H. Golub, and D. P. O'Leary (1976). A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, pp. 309-- 332.

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