W. L. Briggs, V. E. Henson, and S. F. McCormick. A multigrid tutorial: second edition. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....array as argument and broadcasts the element value to all processes of the active process group. Typically it is used with a scalar section to broadcast an element of a general array to all members of the active process group, as here: int [ a = new int [ x, y] intn=3 Adlib.broadcast(a [[10, 10]] The second form of broadcast( just takes an ordinary Java value as the source. This value should be defined on the process or group of processes identified by root. It is broadcast to all members of the active process group. 53 4.3.2 Reductions Reduction operations take one or more ....

....The particular solver was adapted from an existing Fortran program (called PDE2) taken from the Genesis parallel benchmark suite [5] The whole of this program has been ported to HPJava (it is about 800 lines) but in this section we will only consider a few critical routines. The Multigrid [10] method is a fast algorithm for solution of linear and nonlinear problems using restrict and interpolate operations between current grids (fine grid) and restricted grids (coarse grid) As applied to basic relaxation methods for PDEs, it hugely accelerates elimination of the residual by ....

William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial. The Society for Industrial and Applied Mathematics (SIAM), 2000.

.... wave tomography [25] For ODT, a two resolution wavelet decomposition was used to speed inversion of a problem linearized with a Born approximation [26] Multigrid methods are a special class of multiresolution algorithms which work by recursively operating on the data at different resolutions [27], 28] 29] 30] Multigrid algorithms originally attracted interest for numerical analysis to facilitate the computation of PDE solvers by effectively removing smooth error components which are not damped in some fixed grid relaxation schemes. This advantage of the multigrid methods has been ....

....typically increases by a factor of 8 each time the resolution is doubled. Solving problems at fine resolution also tends to slow convergence. For example, many fixed grid algorithms such as ICD effectively eliminate error at high spatial frequencies, but low frequency errors are damped slowly [27], 10] C. Multigrid Inversion Algorithm In this section, we derive the basic multigrid inversion algorithm for solving the optimization of (5) Let x denote the finest grid image, and let x be a coarse resolution representation of x with a grid sampling period of 2 times the finest ....

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W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd Ed., Society for Industrial and Applied Mathematics, Philadelphia, 2000.

....the most efficient algorithms for the numerical solution of that type of equation. This chapter will only provide a brief introduction to the essential ideas of the multigrid concept. A more detailed description of multigrid methods and the mathematical background can be found in [Bra84, Hac85, BHM00] 3.1.1 Model Problem and Discretization This work focuses on boundary value problems of second order elliptic partial differential equations. The general form of these PDEs in a two dimensional domain is: Au xx 2Bu xy Cu yy Du x Eu y Fu G = 0 (3.1) AC B 0; 8(x; y) x 0 x 1 ....

....Idea The representation of the problem domain with a grid leads to a discretization error since a grid usually will not represent a continuous domain correctly. Using a finer grid with a smaller grid distance will reduce the error involved with the discretization. However, it can be shown [BHM00] that the reduction of low frequency error parts decreases with decreasing grid spacing h. With decreasing h the number of grid points and consequently the amount of work to be done for one iteration will increase by O(h ) i.e. finer grids will require more work for one iteration and the ....

W.L. Briggs, V.E. Henson, and S.F. McCormick. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, USA, second edition, 2000.

....gradient method that is preconditioned by one multigrid (MG) 0.8 0.6 0.4 Figure 7.2 Optimal state (z) h 1 32) V cycle with one red black Gaug Seidel iteration as presmoother and one adjoint red black Gaug Seidel iteration as postsmoother. Standard references on multigrid methods include [23, 72, 73, 145]. Our semismooth Newton methods with MG preconditioned conjugate gradient solver of the Newton systems belong to the class of Newton multilevel methods [44] For other multigrid approaches to variational inequalities we refer to [21, 82, 83, 99, tOO, tot] For the solution of the semismooth ....

W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second ed., 2000.

....of degeneracy is still visible for most of the stationary subdivision schemes. Also, for the interpolatory schemes, the irregularities become more noticeable at extraordinary vertices. Variational methods for subdivision modeling are related to the multigrid method in the numerical analysis [52, 97, 18]. The methods consist of two factors. E: the energy or fairness functional. M: the discretized domain grid. Basically, a variational approach try to solve the minimization problem for a given continuous functional over a domain. We discretize the domain into a grid M, and the variational ....

....direct methods can be expensive in computation time and memory requirement. The multigrid method try to solve the system hierarchically. Using a sequence of grid # which becomes finer over the level #, and the corresponding matrix E # , the solution can be computed in the following three steps [18]. 1. Prediction: Compute an initial guess p # at level #, by computing p # = S # 1 p # . S # 1 is a matrix that maps vectors from # 1 to # . It is called a prolongation operator . The simplest way is using piecewise linear interpolation. 2. Smoothing: Improve the quality of the ....

W. L. Briggs, V. E. Henson, and S. F. McCormick. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, July 2000.

....The Taskspaces framework is tested for two scientific computing problems. In the first problem a linear system of algebraic equations deriving from finite di#erence discretization of a Laplace partial di#erential equation on a square two dimensional domain is solved by the iterative Jacobi method [3]. The square domain with regular Cartesian mesh is divided into n square subdomains of equal size that are assigned to n tasks. Every task stores the unknowns that are part of its assigned subdomain, and every unknown is updated using its nearest neighbors. In every iteration step every task ....

.... because it has complexity O(N ) with N the number of unknowns, but testing the scalability of the Jacobi algorithm is relevant because fixed numbers of iterations of similar algorithms are used as building blocks in multigrid methods for linear systems that achieve optimal O(N) complexity [3]. The second problem is Lattice Boltzmann simulation of the Navier Stokes equations for metal foam applications. In particular we have implemented the D2Q9 LBM algorithm as described in [18] This is a real life scientific computing problem that demands very high resolutions and high numbers of ....

W. L. Briggs, V. E. Henson, and S. F. McCormick. A multigrid tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000.

....ill conditioned [18] 21] 12] Hundreds of iterations may be required per solution for large problems. Multigrid methods, or more generally, multilevel methods, are known to be the most efficient iterative techniques in the solution of elliptic partial differential equations (PDE s) 1] 9] [2] due to their fast convergence. However, multilevel methods are not well developed for first kind integral equations [12] defined over complicated geometries, as is our case here. In this paper, we address this void by developing a multigrid iterative solver, and then integrating it with ....

....the coarsegrid problem, since # O i # O g 4. This motivates our development of a two grid method (TGM) in which the problem is solved iteratively at level with the help of direct solution at level two principal algorithmic components, analogous to TGM for PDE s [1] 9] [2], are the smoothing operator and the intergrid transfer, or restriction prolongation, operators. In our TGM iteration for solving O d 2 c O d 0 c , the error in the th iterate, 2 t c , is smoothed by carefully solving a series of local problems. This first stage is typically called ....

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W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....developing algorithms at multiple resolutions. In particular, MR algorithms o#er the promise of computational e#ciency. This can be seen in a variety of methods for the solution of large systems of equations (e.g. representing discretizations of partial di#erential equations) Multigrid methods [44, 45, 109, 190, 319] represent one class of examples, in which coarser (and hence computationally simpler) versions of a problem are used to guide (and thus accelerate) the solution of finer versions, with finer versions used in turn to correct for coarsening or aliasing errors in the coarser versions. Multipole ....

....[319] represents what to the author s knowledge is the first thorough examination of the application of multigrid methods to image processing computer vision problems. Full multigrid methods, such as those used in [319] and discussed in much more depth in references devoted to the subject such as [44, 45], involve both coarse to fine and fine to coarse operations in an iterative algorithmic structure. The idea in the coarse to fine step is essentially the same as for the methods described in the preceding paragraph: We interpolate a coarser approximation of the estimate to the next finer scale to ....

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W. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....successfully eliminated the oscillatory portion of the error, which it eliminates rapidly, but is unable to effectively treat the smooth portion of the error. This is precisely the difficulty that multigrid methods were devised to overcome. At the heart of multigrid is the coarse grid correction [14]. Many common relaxation iterative relaxation methods for solving a linear problem Au = f have the property that the relaxation effectively eliminates the high frequency (oscillatory) components of the error but leave the low frequency (smooth) components essentially unaffected. However, because ....

....information regarding the performance of the method, comparing convergence rates for various choices of parameters. We find that for this problem we are able to obtain convergence rates that are similar to those obtained on the linear elliptic model problems for which multigrid is best known ([14], 16] 17] Data for the one dimensional problem are not shown, however, they are very similar to the two dimensional case. Dimension p( I Fine grid Average V cycle size convergence factor 2 2xy 2 63 x 63 0.051 5 0.050 8 0.078 2 2sin(2rx)sin(ry) 2 63 x 63 0.060 5 0.063 8 0.104 2 x y 2 63 ....

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....of the coarse space operator A , showing that A = P h A . Thus we may define P 2h = Combining this with the observation that B 2h = P 2h , we find that the standard variational conditions of multigrid are satisfied by this discretization [5]. 6.2 Relaxation Following the principles of PML, we select sets of m subspaces S whose unions equal S , respectively. The most obvious choice is to select m = N and S , that is, each subspace is the span of an individual strip pixel. Lemma 2. Let S for = 1 : N , ....

W. L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....step in AFCM and deserves special attention in its numerical implementation. Because the difference equation (7) is space varying, the gain field cannot be found using standard frequency domain filters. The equation could be solved iteratively using the Jacobi or Gauss Seidel schemes [30, 31], but these methods take a large number of iterations to converge. In [19, 27] this equation was solved using a standard multigrid algorithm at each iteration of AFCM (for a general overview of multigrid algorithms, see [17] or [31] For 2 D images, this approach is sufficiently fast, but for ....

....solved iteratively using the Jacobi or Gauss Seidel schemes [30, 31] but these methods take a large number of iterations to converge. In [19, 27] this equation was solved using a standard multigrid algorithm at each iteration of AFCM (for a general overview of multigrid algorithms, see [17] or [31]) For 2 D images, this approach is sufficiently fast, but for large 3 D images, execution times can grow to several hours. We now describe a modified multigrid algorithm that yields significantly faster overall execution time without loss of accuracy. Its premise is that during early iterations ....

[Article contains additional citation context not shown here]

W.L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, 1987.

....K can be decomposed into K = D Gamma L Gamma U , where D is the diagonal of K , and GammaL and GammaU are the strictly lower and upper triangular parts of K . If we define the Gauss Seidel iterative matrix PG by PG = D Gamma L) Gamma1 U , we can express Gauss Seidel iterative method as ([2]) u n 1 PG u n (D Gamma L) Gamma1 f: 1) We update the components of u in ascending order. Components of the new approximation are used as soon as they are computed. In other words, we solve the j th equation for u j using new approximations for components 1; 2; j Gamma 1. To ....

William L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

....multi level methods currently in use is multi grid. Multi grid recursively generates increasingly dense approximations to the exact continuous flow using a nested sequence of domain grids. This method provides exceptionally good convergence rates and has found many successful applications [2]. While multi grid solvers are very efficient compared to standard iterative solvers, they remain too costly to permit interactive modeling in many situations. One alternate technique that has proven very useful in surface modeling is subdivision. Subdivision schemes have been used to model a ....

....[30] provide background material on several iterative smoothing methods. Unfortunately, these iterative methods tend to converge very slowly for large problems. Multi grid solvers converge much more rapidly and provide an efficient method for recursively solving large systems of linear equations [2]. In multi grid, the domain grid T is replaced by a sequence of nested grids T 0 # T 1 # : # T n . Similarly, the discrete entities D, u and b are replaced by corresponding entities D k , u k and b k ,defined over the associated domain grid T k . Consequently, the problem now consists of ....

W. L. Briggs: A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 1987.

....have been used only occasionally [25] For other vision applications, several previous optimization approaches rely on coarse versions of a cost function. Similar techniques are employed by multi grid algorithms, which have rst been developed for the solution of partial di erential equations [26]. Multi grid methods have been adopted to a broad range of optimization problems with locally interacting variables including image processing tasks [27, 28, 29] They rely on incremental coarse grid corrections and, therefore, on continuous optimization variables. Similar in spirit but ....

W. Briggs and S. McCormick, \Introduction," in Multigrid Methods (S. McCormick, ed.), Frontiers in Applied Mathematics, ch. 1, pp. 1-31, Society for Industrial and Applied Mathematics, 1987.

....We developed an algorithm for solving this system based on graphs fG 0 ; G 1 ; Gn g. This gives us a framework for a multilevel preconditioner for solving Lt i = r i . Multilevel iterative methods have proven to be effective for solving systems of linear equations arising from PDE s [4, 23]. We adapted the multilevel idea of iterative methods for developing a preconditioner for the graph Laplacian system. While the traditional multilevel approach for solving elliptic PDE s obtains smaller problems by coarsening the spatial discretization, our Preconditioned Davidson Algorithm ....

W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.

....ill conditioned [18] 21] 12] Hundreds of iterations may be required per solution for large problems. Multigrid methods, or more generally, multilevel methods, are known to be the most efficient iterative techniques in the solution of elliptic partial differential equations (PDE s) 1] 9] [2] due to their fast convergence. However, multilevel methods are not well developed for first kind integral equations [12] defined over complicated geometries, as is our case here. In this paper, we address this void by developing a multigrid iterative solver, and then integrating it with ....

....problem, since # O i 1 # O g 4. This motivates our development of a two grid method (TGM) in which the problem is solved iteratively at level b with the help of direct solution at level bj 1 . The two principal algorithmic components, analogous to TGM for PDE s [1] 9] [2], are the smoothing operator and the intergrid transfer, or restriction prolongation, operators. In our TGM iteration for solving 1 c O d 2 c O d 0 c O d , the error in the r th iterate, 2 t c O d , is smoothed by carefully solving a series of local problems. This first stage is ....

[Article contains additional citation context not shown here]

W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....V cycle. step in AFCM and deserves special attention in its numerical implementation. Because the difference equation (6) is space varying, the gain field cannot be found using standard frequency domain filters. The equation could be solved iteratively using the Jacobi or Gauss Seidel schemes [4], but these methods take a large number of iterations to converge. In [11, 10] this equation was solved using a standard multigrid algorithm at each iteration of AFCM (for a general overview of multigrid algorithms, see [17] or [4] For 2 D images, this approach is sufficiently fast, but for ....

....be solved iteratively using the Jacobi or Gauss Seidel schemes [4] but these methods take a large number of iterations to converge. In [11, 10] this equation was solved using a standard multigrid algorithm at each iteration of AFCM (for a general overview of multigrid algorithms, see [17] or [4]) For 2 D images, this approach is sufficiently fast, but for large 3 D images, execution times can grow to several hours. We now describe a modified multigrid algorithm that yields significantly faster overall execution time without loss of accuracy. Its premise is that during early iterations ....

[Article contains additional citation context not shown here]

Briggs, W.: A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 1987.

....on the logarithm of the number of nodes. Finite element methods or Galerkin methods [110, 93] depending on the type of basis functions, are usually more accurate than finite difference methods, but require more costly function evaluations. Multigrid or multilevel methods of Brandt [10] see also [11, 89]) can also used, in conjunction with the finite element method or with other methods, in order to reduce the necessary number of nodes by successive use of fine and coarse methods to enhance accuracy beyond the accuracy of such grids when used only as single grids. Akian, Chancelier and Quadrat ....

W. L. Briggs, Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987.

....the SOR Benchmark This section details building a model for a Successive Over Relaxation (SOR) benchmark, a member of the Regular SPMD application class. In the SOR implementation, the application is divided into red and black phases, with communication and computation alternating for each [Bri87] This repeats for a predefined number of iterations, as described in Section 2.E.1. For this application, like the GA, we implemented the code in three dedicated settings. 3.C.1 Top level Model for SOR Given the four task structure of the SOR code, possible top level models for this ....

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Lancaster Press, 1987.

....truncated Fourier series u h on the mesh. Usually u h is even more accurate than h , since the higher modes are damped by L 1 . It may be worthwhile to compare our method with some of the many other techniques available for this problem. The advantage of our method over multigrid methods [1] (which are equally ecient for a given grid size but less accurate) is its spectral accuracy, while the advantage over standard spectral methods [3] which are equally accurate but less ecient for a given grid size) is its eciency. 3 Numerical results Our numerical results use d = 2 dimensions ....

....L and L exactly, then solved the problem numerically and calculated the error in and u. The variable coecients of L were constructed from six (M 1) 2 term Fourier cosine series F s (x) M X k 1 ;k 2 =0 F k cos(2 k 1 x 1 ) cos(2 k 2 x 2 ) 5 with coecients F k generated randomly on [ 1,1] for each s = 1 through 6. Since we want L elliptic, we generated a 2 by 2 upper triangular matrix F with entries F 1 , F 2 and F 3 , and set (a ij ) I F T F where I is the 2 by 2 identity matrix. Thus a 11 = 1 F 2 1 , a 12 = 2F 1 F 2 , a 21 = 0, and a 22 = 1 F 2 2 F 2 3 . The ....

W. L. Briggs. A multigrid tutorial. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987.

....than O(M 3 x M 3 y ) the computational complexity of inversion. The e#cient solution of such sparse and banded linear estimation problems is well studied in [42] Many methods exist to solve these types of problems, including multigrid methods, subspace methods and elimination methods [43, 44]. In this thesis we will use the preconditioned conjugate gradient method [42] although alternative solution methods may also be useful. The derivative operator, Dp , can be any of the discrete derivative operators described in the discussion of discretization. In practice, for 2D images, the ....

W. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia PA, 1987.

....so that we arrive at a cascade of iterative methods. For practical experience with this type of preconditioning see [60, 64, 80] Multilevel methods have become a powerful tool for the solution of partial differential equations. The technique can be used as preconditioner. For details we refer to [5, 8, 9, 13, 14, 38, 39, 40, 53]. A special multilevel technique tuned for preconditioning cg methods is DARE [65, 66, 85, 86] 5.5. Preconditioned orthogonalization methods. In this section we focus on step dependent preconditioners because constant preconditioners are obtained as special case. A preconditioned ....

W.L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

....both for computational and for storage reasons, given the sheer size of the error covariance matrices involved. A critical aspect of these estimation problems is the requirement that estimation error statistics be computed. This necessity precludes the use of accelerated methods such as multigrid [11], which do not supply such error statistics, or the fast Fourier transform (FFT) which requires spatially stationary prior models and spatially regular measurement patterns, requirements that cannot be met in many applications including most, if not all, remote sensing problems. For these ....

....for compu63 tational and for storage reasons, given the sheer size of the error covariance matrices involved. A critical aspect of these estimation problems is the requirement that estimation error statistics be computed. This necessity precludes the use of accelerated methods such as multigrid [11], which do not supply such error statistics, or the FFT, which requires spatially stationary prior models and spatially regular measurement patterns, requirements that cannot be met in many applications including most, if not all, remote sensing problems. For these reasons, it is clear that there ....

[Article contains additional citation context not shown here]

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....by T i and the corresponding energy matrices by E i , then the multigrid method attempts to find a sequence of vectors p i satisfying the equation E i p i = b i (1) for increasing i, i.e. for increasingly dense grids. Typically, the solution process at level i consists of three steps (see Briggs [1] for more details) 1. Prediction: Compute an initial guess p 0 i to the exact solution p i . This initial guess is derived from the solution p i,1 on the next coarser grid using some type of linear prediction function. In terms of matrix notation, this prediction step can be written as p 0 ....

W. L. Briggs: A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 1987.

....parameter on filter performance and convergence time of CG is needed. It is possible that some of the noise surrounding outlying data points, for example in Figure 3.20 (p. 83) might be due to too small a parameter . ffl It may be possible to adopt multigrid methods (see for example [47]) to solve the least squares equations. Multigrid methods are generally much faster than CG for large problems. ffl Convergence of the CG method might be quicker if a succession of regularization parameters were used, starting with large values and ending with small values. Larger values of the ....

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

....on the order of discretization error in a total amount of work equal to about 16 relaxation sweeps on the finest grid. Theoretical foundation for the classical equations of linear elasticity can be found in [2, 3, 4, 13] A description of basic multigrid algorithms and principles can be found in [5]. This paper is organized as follows. In Section 2, we introduce notation and define the spaces on which we pose our formulations. Two FOSLS approaches are introduced in Section 3. In Section 4, we describe the multigrid algorithm and, in Section 5, we present the results of various numerical ....

....= 1; 4 : 4.2. Multigrid (MG) The multigrid algorithm adopted in this paper is most easily described in the present context as a functional minimization process. Here we describe a simple two level procedure that provides the basis for V, W, and FMG cycle algorithms in the usual way. See [5] for more detail on multigrid methods. Suppose we are given a current approximation V h 2 V h to the solution of the minimization problem V = argminfG(W; f ) W 2 Vg : 4.1) Here, G = G 1 or G 2 and the minimization is to be taken over the corresponding subspace V of [H 1 ( Omega Gamma5 4 ....

W. L. Briggs, A multigrid tutorial, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1987.

....acceptable efficiency, by varying the density of the represented points depending on the local characteristics of the problem. Two adaptive discretization methods that have been previously reported are adaptive stepsize control [Press et al. 86; Gear 71] and multigrid methods [Brandt 77; Briggs 87] 5.1 Step Size Refinement First we overview the stages of the step size refinement algorithm. Each stage is discussed in more detail in the following subsections, which are each titled with the name of the stage given in this overview: 1. Qualitative simulation: Simulate the model with the QSIM ....

W. L. Briggs, A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 1987.

....We developed an algorithm for solving this system based on graphs fG 0 ; G 1 ; Gn g. This gives us a framework for a multilevel preconditioner for solving Lt i = r i . Multilevel iterative methods have proven to be effective for solving systems of linear equations arising from PDE s [5, 23]. While the traditional multilevel approach for solving elliptic PDE s obtains smaller problems by coarsening the spatial discretization, our Preconditioned Davidson Algorithm implements a preconditioner which uses fG 0 ; G 1 ; Gn g as its multilevel framework. This preconditioner can be ....

W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

....Fourier series u h on the mesh. Usually u h is even more accurate than oe h , since the higher modes are damped by L Gamma1 . It may be worthwhile to compare our method with some of the many other techniques available for this problem. The advantage of our method over multigrid methods [1] (which are equally efficient for a given grid size but less accurate) is its spectral accuracy, while the advantage over standard spectral methods [3] which are equally accurate but less efficient for a given grid size) is its efficiency. 3 Numerical results Our numerical results use d = 2 ....

....L and L exactly, then solved the problem numerically and calculated the error in oe and u. The variable coefficients of L were constructed from six (M 1) 2 term Fourier cosine series F s (x) M X k1 ;k2 =0 F k cos(2 k 1 x 1 ) cos(2 k 2 x 2 ) with coefficients F k generated randomly on [ 1,1] for each s = 1 through 6. Since we want L elliptic, we generated a 2 by 2 upper triangular matrix F with entries F 1 , F 2 and F 3 , and set (a ij ) I F T F where I is the 2 by 2 identity matrix. Thus a 11 = 1 F 2 1 , a 12 = 2F 1 F 2 , a 21 = 0, and a 22 = 1 F 2 2 F 2 3 . The ....

W. L. Briggs. A multigrid tutorial. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1987.

....of charges or boundary conditions. Unfortunately, local refinement introduces several added issues like matching the solution at grid interfaces. These will be examined in more depth. A familiarity with multigrid methods for solving linear problems is assumed; a good reference is Briggs [1]. Ph.D. candidate in mechanical engineering, U.C. Berkeley, supported by the Computational Science Graduate Fellowship Program of the Department of Energy. Email: martin euler.ME.berkeley.edu y Graduate student in physics, U.C. Berkeley, supported by ONR AASERT N10001494 1 1033 cells that ....

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....P S h S 2h = i P S 2h S h j T . Combining this with the observation that B 2h = A 2h A 2h = P S 2h S h A h A h P S h S 2h = P S 2h S h B h P S h S 2h , we find that the standard variational conditions of multigrid are satisfied by this discretization [5]. 6.2 Relaxation Following the principles of PML, we select sets of m subspaces S h and T h whose unions equal S h and T h , respectively. The most obvious choice is to select m = N and S h = span n h o , that is, each subspace is the span of an individual strip pixel. ....

W. L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 1987.

....in strips [Fig96] It was written in C with PVM for the heterogeneous cluster of machines in the PCL as the GA code was. 6. 1 Structural Model for SOR In our implementation, the application is divided into red and black phases, with communication and computation alternating for each [Bri87] This repeats for a predefined number of iterations. The data decomposition for this code is depicted in Figure 5. P1 P2 P3 Figure 5. SOR Application A structural model for this application might be: ExTime = Max [RedComp i RedComm i BlackComp i BlackComm i ] where ffl Max : A ....

William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Lancaster Press, 1987.

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W. L. Briggs, V. E. Henson, and S. F. McCormick. A multigrid tutorial: second edition. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000.

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William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

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William L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.

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William L. Briggs, Van Emden Henson, and Steve F. McCormick. A multigrid tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000.

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William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial. The Society for Industrial and Applied Mathematics (SIAM), 2000.

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W. L. Briggs, V. E. Henson, and S. F. McCormick. A multigrid tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2000.

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W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

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W. L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987.

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W. L. Briggs. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, Philadelphia, 1987.

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William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial. The Society for Industrial and Applied Mathematics (SIAM), 2000.

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W. L. Briggs, A Multigrid Tutorial. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1987.

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W. L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania (1987).

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W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial, 2nd Ed., Society for Industrial and Applied Mathematics, Philadelphia, 2000.

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W.L. Briggs, V.E. Henson, and S.F. McCormick. A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 2000.

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W. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 1987.

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W. Briggs, A multigrid tutorial, Society for Industrial and Applied Mathematics, 1987.

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W. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 1987.

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W. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 1987.

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