W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. B.G. Teubner Stuttgart, second edition, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....us a natural graph associated with the coarse grid nodes. We will call it coarse grid as an analogy to partial differential equations. Recalling the well known techniques of constructing good preconditioners for the conjugate gradient method applied to symmetric positive definite systems, e.g. [13, 17, 26], we should choose P and Z such that A Gamma1 6 M (1) 6 (1) A Gamma1 (7) with as small as possible and 0. Clearly 1 is the condition number of M (1) A, i.e. the ratio of the largest by the smallest eigenvalue of M (1) A and thus (1) 1 would be optimal. The ....

....in the preconditioned case one has to replace A by (M (1) 1=2 A(M (1) 1=2 . We will discuss the construction of P; Z with minimal (1) in the next subsection. For discretized elliptic partial differential equation one can construct optimal preconditioners using multigrid methods [17]. In order to obtain a similar preconditioner augmented with a suitably chosen coarse grid correction, consider the use of LL in a linear iteration scheme with initial guess x (0) 2 R n . The iteration scheme [29] for the solution of Ax = b has the form x (k 1) x (k) LL (b ....

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W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. B.G. Teubner Stuttgart, second edition, 1993.

....involved. This is the well known trade off between surface quality and computational complexity. 6 Efficient computation The computation of the vertex positions in a fair net requires the solution of the large sparse system (3. 8) Such systems are most efficiently solved by iterative algorithms [Hac91]. Due to the geometric background of the problem, it is possible to find very good starting values for the iteration. Every step of the iterative scheme can be considered as a smoothing step taking the vertices closer to the fair solution of the optimization problem. Typically for such low pass ....

W. Hackbusch, Iterative Losung großer schwach besetzter Gleichungssysteme, Teubner Verlag 1991, Stuttgart

....(L U) see for example section 5.3) In case of two blocks m = 2, there is another very simple way to prove a robust convergence. But this approach can only be applied in case of a slight modification of the Gauss Seidel iteration. This is the symmetric version of the Gauss Seidel iteration (see [30]) Symmetric 2 block Gauss Seidel iteration: S i;sm (w) C i;sm w b i where C i;sm : D Gamma U) Gamma1 L(D Gamma L) Gamma1 U and (2.39) b i = D Gamma U) Gamma1 D(D Gamma L) Gamma1 g i : The symmetric 2 block Gauss Seidel iteration consists of a subspace correction with ....

....we get for the eigenvalue min defined by the norm (2.3) min = min u2Vn a(u; u) kjujk 2 n min u2Vn a(u; u) kjujk 2 n;Q 7 2 2 p 3 Gamma1 : 4. 19) Estimation of max : For the estimation of the upper bound we use a technique which can be found in several articles (see [30]) Assume that u 2 Vn and w i 2 V i such that u = P n i=1 w i . Then we get u = n X i=1 (R odd;x i (w i ) R even;x i (w i ) where R even;x 1 (w 1 ) 0. By the discrete Cauchy Schwarz inequality, we get 2n Gamma1 X i=1 jc i j 2 (2n Gamma 1) 2n Gamma1 X i=1 c 2 i : ....

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W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

.... we get for the eigenvalue min defined by the norm (3) min = min u2Vn a(u; u) kjujk 2 n min u2Vn a(u; u) kjujk 2 n;Q 7 2 2 p 3 Gamma1 : 16) Estimation of max : For the estimation of the upper bound we use a technique which can be found in several articles (see [8]) Assume that u 2 V n and w i 2 V i such that u = P n i=1 w i . Then we get u = n X i=1 (R odd;x i (w i ) R even;x i (w i ) where R even;x 1 (w 1 ) 0. By the discrete Cauchy Schwarz inequality, we get 2n Gamma1 X i=1 jc i j 2 (2n Gamma 1) 2n Gamma1 X i=1 c 2 i : ....

....Corollary 1 Assume that a(u; v) R Omega hru; rvi and apply the cg iteration to the preconditioned equation system (2) Then, 16 cg iterations reduce the error in the energy norm q a(u; u) by more than 0:233 independent of the number n of levels. Proof. Apply Theorem 3 and Satz 9.4.12. in [8]. End of proof. ....

W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

....iteration for (19) is an ordinary lexicographic SOR iteration for the matrix F 01 from lemma 3.1. Due to its special form, F 01 is ordered consistently. Lemma 3.3 carries statements of 3. 5 concerning the eigenvalues of J(F lex ) to the eigenavalues of J(F 01 ) The famous Young theorem ([8], Satz 5.6.5) then proves all assertions. Remark 3.7. F is the discrete operator of the 5 point stencil for the Laplacian with Neumann conditions on Gamma D . For Gamma out the boundary condition reads Gamma 2 p x 2 Gamma p = r:h:s (51) Using the Poisson equation for the pressure we ....

Hackbusch, W. Iterative Losung großer schwachbesetzter Gleichungssysteme

....p 0 = 0, pn = pn Gamma1 S(p Gamma pn Gamma1 ) 12) Then lim n 1 pn = p for p 2 PM and kp Gamma pn k 2 fl n kpk 2 (13) where fl = maxfj1 Gamma Aj; j1 Gamma Bjg 1. Since Sp depends only on the samples p(t i ) this is indeed a reconstruction from the samples only. Proof. [11, 24, 39, 42, 43] Since hSp; pi = P r i=1 jp(t i )j 2 by (10) and (11) S is a positive operator on PM and thus invertible. By (8) one obtains (1 Gamma B)kpk 2 2 h(Id Gamma S)p; pi (1 Gamma A)kpk 2 2 (14) and consequently kp Gamma pn k 2 = k(Id Gamma S) p Gamma pn Gamma1 )k 2 flkp Gamma pn Gamma1 ....

.... (1 Gamma A)kpk 2 2 (14) and consequently kp Gamma pn k 2 = k(Id Gamma S) p Gamma pn Gamma1 )k 2 flkp Gamma pn Gamma1 k 2 : fl n kpk 2 : ut In terms of linear algebra this is the Richardson iteration for Sp = q with q given, and it comes with all its advantages and disadvantages [25, 24, 42]. The frame algorithm or Richardson iteration of Lemma 2 yields decent convergence only if explicit estimates for the constants A and B can be derived, and furthermore the optimal convergence factor B GammaA B A for the Richardson iteration is small. For a convergence analysis see [39, ....

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Hackbusch, W. (1991): Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner Studienbucher

....explicit estimates on the frame bounds to perform the algorithm. A thorough discussion of improved reconstruction methods for ill conditioned frames must include conjugate gradient methods and other adaptive procedures. For this we may refer to the standard references in numerical analysis, e.g. [19, 16, 15]. Our point here is to establish a connection between frame reconstruction and numerical acceleration methods and to bring some welldeveloped numerical tools to the attention of wavelet analysts. Proposition 4 Let fe n ; n 2 ZZg H be a frame with frame bounds A; B 0 and ae = B0A B A and S ....

W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

....matrices for the error so the spectrum of these matrices determines convergence of the block Jacobi and Gauss Seidel methods. Because E Jac is weakly two cyclic, we have q ae(A Gamma1 1 B 1 A Gamma1 2 B 2 ) q ae(A Gamma1 2 B 2 A Gamma1 1 B 1 ) ae(E Jac ) see for example [Hac91, pp. 123 127] From (A.6) and (A.4) it now follows that ae(E Jac ) 2 = ae(E GS ) in other words: the block Gauss Seidel algorithm converges exactly twice as fast as the block Jacobi algorithm. Our convergence analysis is based on an explicit expression for the Fourier transformed matrices ....

W. Hackbusch. Iterative losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

....is gradually decreased during the process as more and more of the extremal eigenvalues are sufficiently well approximated (for details on this see [80] 5.1. 3 Further references A more formal presentation of CG, as well as many theoretical properties, can be found in the textbook by Hackbusch [39]. A shorter presentation is given 8 in [35] An overview of papers, published in the first 25 years of existence of the method, is given in [34] Vector processing and parallel computing aspects are discussed in [23] and [57] 5.2 MINRES and SYMMLQ: When A is not positive definite, but still ....

W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

....the outer iteration. Concerning the outer iteration, SOR type methods are the state of the art [2, 4, 6, 7, 20] However, the convergence behaviour of such methods can not be guaranteed in general [31] Therefore, more robust iteration schemes have to be developed. Here, techniques for both linear [12] and nonlinear systems [17, 21] can be the starting point. As a first step, cg and GMRES based math. model A math. model B solution of the coupled problem coupling condition discrete model A discrete model B solution of model A solution of model B Figure 2 Partitioned solution methods have been ....

W. Hackbusch, Iterative Losung großer schwachbesetzter Gleichungssysteme, Teubner, Stuttgart, 1991.

....and that for M matrices ILU yields a regular splitting. Moreover, for M matrices we show that the decomposition process for 0 is more stable than for = 0. The only other remark that we found in literature about the existence of the ILU decomposition for 0 of general matrices was in [Hac91], where it was noted that the proof of the existence of ILU 0 given there cannot be generalized to ILU . In Sect. 3, we consider ILU as a smoother. Using results from Sect. 2, for general M matrices A l and general symmetric decomposition patterns we prove the properties (1.6) and (1.7) and, if ....

Hackbusch, W. (1991): Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart

.... R N I again via a norm preserving explicit extension operator b E Omega Gamma from V into W : b E : V W ; including additional linear smoothing operators S k : W k W k , realized via the discrete smoothing operator ( iteration operator of the affine linear smoothing iteration [14]) S I;k : R N I;k R N I;k , k=1; fulfilling k S I;k v h k K I k k v h k K I 8 Phi I v h 2 W k n W k Gamma1 high frequencies ( k S I;k high k K I k 1 k S I;k v h k K I k v h k K I 8 Phi I v h 2 W k Gamma1 ; W k low frequencies ( k S I;k low k K I 1 ; ....

W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart, 1991.

....in (8) Define iteratively p 0 = 0, p n = p n01 S(p 0 p n01 ) 12) Then lim n 1 p n = p for p 2 PM and kp 0 p n k 2 fl n kpk 2 (13) where fl = maxfj1 0 Aj; j1 0 Bjg 1. Since Sp depends only on the samples p(t i ) this is indeed a reconstruction from the samples only. Proof. [11, 23, 39, 42, 43] Since hSp; pi = P r i=1 jp(t i )j 2 by (10) and (11) S is a positive operator on PM and thus invertible. By (8) one obtains (1 0 B)kpk 2 2 h(Id 0 S)p; pi (1 0 A)kpk 2 2 (14) and consequently kp 0 p n k 2 = k(Id 0 S) p 0 p n01 )k 2 flkp 0 p n01 k 2 : fl n kpk 2 : In ....

.... B)kpk 2 2 h(Id 0 S)p; pi (1 0 A)kpk 2 2 (14) and consequently kp 0 p n k 2 = k(Id 0 S) p 0 p n01 )k 2 flkp 0 p n01 k 2 : fl n kpk 2 : In terms of linear algebra this is the Richardson iteration for Sp = q with q given, and it comes with all its advantages and disadvantages [24, 23, 42]. The frame algorithm or Richardson iteration of Lemma 2 yields decent convergence only if ffl explicit estimates for the constants A and B can be derived, and furthermore ffl the optimal convergence factor B0A B A for the Richardson iteration is small. For a convergence analysis see [39, ....

[Article contains additional citation context not shown here]

W. Hackbusch. Iterative Losung großer schwachbesetzter Gleichungssysteme. Teubner Studienbucher, 1991.

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