W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Theorem 3 also applies to the prolongation operators used in [17] The algorithms presented there are modifications of BBOX which are aimed to solve highly anisotropic equations without using line relaxation. When X i is considerably smaller than all the other X j s it is suggested in [35] and [20] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In [35] the same coarse grid is used for both corrections, whereas in the Frequency Decomposition Multigrid (FDM) method of [20] the coarse ....

....in [35] and [20] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In [35] the same coarse grid is used for both corrections, whereas in the Frequency Decomposition Multigrid (FDM) method of [20] the coarse grid used for the latter correction is shifted so that odd numbered variables are used in the ith spatial direction. This appears to be a considerable improvement (see Table 5 below) Here we consider the Child of FDM (CFDM) of [17] for which 2 coarse grid corrections are used. For ....

[Article contains additional citation context not shown here]

Hackbusch, W.: The Frequency Decomposition Multigrid Method, Part 1: Application to anisotropic equations. Numer. Math. 56 (1989), 229-245.

....into the algorithm through a specific choice of the underlying subspace splitting which can be determined easily from the coefficients, and does not increase the complexity of the algorithm. In this respect, our approach is different from Hackbusch s Frequency Decomposition Multigrid Method [Ha2, Ha3] and its recent wavelet variation [RWZ, RZ] where one subspace splitting serves all coefficient choices, at the cost of additional but unnecessary work (cf. Ha2] The theoretical justification of our solvers is given for the additive methods only (see Sections 2 and 3) though the multiplicative ....

....complexity of the algorithm. In this respect, our approach is different from Hackbusch s Frequency Decomposition Multigrid Method [Ha2, Ha3] and its recent wavelet variation [RWZ, RZ] where one subspace splitting serves all coefficient choices, at the cost of additional but unnecessary work (cf. [Ha2]) The theoretical justification of our solvers is given for the additive methods only (see Sections 2 and 3) though the multiplicative variants of the algorithms based on the same subspace splittings work reasonable well, too (c.f. Xu, Ys, GO2] 2 The remainder of this paper is outlined as ....

W. Hackbusch, The frequency decomposition multigrid method. Part I: application to anisotropic equations. Numer. Mathematik 56 (1989), 229-245.

....that contains additionally the basis functions of all grids that result from semirefinement steps with respect to both coordinate directions. Furthermore, the full and the sparse grid case [ is considered. For the level oriented methods, we obtain MG algorithms similar to that of Hackbusch [5] [6], Naik and van Rosendale [9] Moulder [8] and Wesseling [ For the point oriented method we observe a substantial improvement of the error reduction properties for the resulting algorithm in comparison to the case using E k only. We obtain a really fast algorithm with a convergence rate of about ....

, The frequency decomposition multi-grid method, part I: applications to anisotropic equations, Numerische Mathematik, Vol.---, 1989.

.... robust interpolation methods for multigrid; see also Kettler and Meijerink [67] Along this line were also the black box multigrid method by Dendy [39, 40] and matrix dependent approaches by de Zeeuw [115] and Reusken [85, 86] Other related approaches include frequency decomposition by Hackbusch [60], and filtering decomposition by Wittum [108, 109] The purely algebraic methods, on the other end of the spectrum, were first proposed by Brandt, McCormick and Ruge [25] and then popularized by Ruge and Stuben [88] These is a recent resurgence of interest in AMG and other multigrid algorithms ....

.... converges slowly when the coefficients exhibit anisotropy [59] large jumps in discontinuity [1, 19, 39, 40] or large oscillations [48, 73, 95] Special techniques such as line Gauss Seidel [19] semi coarsening [41, 42, 90] algebraic multigrid [14, 25, 85, 88, 93] frequency decomposition [43, 60, 95], and homogenization [48, 73] are used to handle some of these cases. In the next sections, we survey the state of the art of each individual multigrid components and discuss how they bring insight into the design of robust multigrid methods. 2. Interpolation. Sophisticated designs of ....

[Article contains additional citation context not shown here]

W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.

....problems, there are yet many issues of multigrid for solving self adjoint problems which are either less known, not understood, or just left behind. Multigrid methods can be extremely slow for PDEs whose coefficients are discontinuous [1, 46, 86] oscillatory [54, 93, 128] or anisotropic [74], although they have been proved successful for smooth coefficient problems. Besides, multigrid methods for unstructured grid computations are not easy to design. The main issue is that there is no natural choice of coarse grids [31, 70, 96, 90] Another is how to define interpolation between ....

W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.

....of the scheme for Equation (2) is the same for points i and i e( i) With these assumptions, a simple representation of P is available, with which the two level analysis in Section 3 below is easier to implement. We also consider the Frequency Decomposition Multigrid (FDM) method of [8] for the solution of highly anisotropic equations. When X i is considerably smaller than all the other X j s it is suggested in [15] and [8] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In ....

....with which the two level analysis in Section 3 below is easier to implement. We also consider the Frequency Decomposition Multigrid (FDM) method of [8] for the solution of highly anisotropic equations. When X i is considerably smaller than all the other X j s it is suggested in [15] and [8] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In [15] the same coarse grid is used for both corrections, whereas in [8] the coarse grid used for the latter correction is shifted so that odd ....

[Article contains additional citation context not shown here]

Hackbusch, W.: The Frequency Decomposition Multigrid Method, Part 1: Application to anisotropic equations. Numer. Math. 56 (1989), 229-245.

....Theorem 3 also applies to the prolongation operators used in [17] The algorithms presented there are modifications of BBOX which are aimed to solve highly anisotropic equations without using line relaxation. When X i is considerably smaller than all the other X j s it is suggested in [35] and [20] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In [35] the same coarse grid is used for both corrections, whereas in the Frequency Decomposition Multigrid (FDM) method of [20] the coarse ....

....in [35] and [20] to use two coarse grid corrections: one uses the standard bilinear prolongation and the other uses bilinear like prolongation but with alternating signs. In [35] the same coarse grid is used for both corrections, whereas in the Frequency Decomposition Multigrid (FDM) method of [20] the coarse grid used for the latter correction is shifted so that odd numbered variables are used in the ith spatial direction. This appears to be a considerable improvement (see Table 5 below) Here we consider the Child of FDM (CFDM) of [17] for which 2 d coarse grid corrections are used. ....

[Article contains additional citation context not shown here]

Hackbusch, W.: The Frequency Decomposition Multigrid Method, Part 1: Application to anisotropic equations. Numer. Math. 56 (1989), 229-245.

....[20] have large jumps [1, 5, 10, 11] or are highly oscillatory [17, 26, 34] standard multigrid methods will converge very slowly. Special techniques such as line Gauss Seidel block smoothing [5] semi coarsening [12, 13, 32] algebraic multigrid [6, 28, 30, 33] frequency decomposition [14, 21, 34], and homogenization [17, 26] are used to handle some of these cases. In this paper, we study the design of multigrid methods from the energy minimization point of view, which gives powerful insight into the design of robust multigrid methods. The success of multigrid hinges on the choice of the ....

W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.

.... of the standard multigrid methods applying to PDE s whose coefficients are anisotropic [16] discontinuous [1, 6, 9] or oscillatory [12, 31] Special techniques, for instance, block smoothing [6] semi coarsening [10, 30] matrix dependent interpolations [24, 26] frequency decomposition [17, 11, 31], homogenization [12] were introduced to handle some of these cases. In this paper, we shall study multigrid methods from the energy minimization and the approximation point of views which hopefully gives an insight to a robust multigrid algorithm. The success of multigrid hinges on the choice of ....

W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.

....[26] developed a variant of Ta asan s method using standard smoothers. Ta asan s and Hackbusch s methods are both referred to as robust multigrid, which adds confusion (and heated discussions) to the field. More recently, Hackbusch s method has been referred to as frequency decomposition multigrid [27]. Frederickson and McBryan [21] using an approach different from that of Gannon and Van Rosendale, also investigated ways to keep all of the processors busy on a massively parallel single instruction, multiple data (SIMD) machine. They used standard interpolation and projection methods and an ....

, The frequency decomposition multigrid method, part I: Application to anisotropic equations, Numer. Math., 56 (1989), pp. 229--245.

No context found.

, The frequency decomposition multi-grid method, part i: applications to anisotropic equations, Numerische Mathematik, Vol.---, 1989.

....could be formed directly instead of variationally. However, in the special case of V j s corresponding to (0; 0) and ( 9] follows the methodology just described. A variant of Brandt and Ta asan s method is the frequency decomposition multigrid method, developed independently by Hackbusch [10]. To describe this method, let us assume doubly periodic boundary conditions and suppose that the finest grid is the collection of points Omega M shown in Fig. 1. Subdivide Omega M into the four sets f Omega M Gamma1 k;l ; k = 0; 1; l = 0; 1g as shown. Define I k;l : Omega M Gamma1 ....

....grids is obvious since each point communicates only with its nearest left and right neighbors. Communication in y is efficient since each point communicates with bottom and top neighbors a power of two distant, and with the immediate left and right neighbors of these points. Like the methods in [11, 15, 10], the methods here keep all the processors busy on every grid level, and again this busyness is actually a disadvantage when the number of points per processor exceeds one (vp ratio greater than one) for then the virtual processors are kept busy on every level as well. In the method of [4] work ....

W. Hackbusch, The frequency decomposition multigrid method, part I: application to anisotropic equations, Numer. Math. 56(1989), pp. 229-245.

No context found.

W. Hackbusch. The frequency decomposition multigrid method, part I: Application to anisotropic equations. Numer. Math., 56:229--245, 1989.

No context found.

Hackbusch, W.: The Frequency Decomposition Multigrid Method, Part I: Application to Anisotropic Equations. Numer. Math., 56, 1989, pp. 229-245.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC