P. N. Swarztrauber, The methods of cyclic reduction, fourier analysis and the facr algorithm for the discrete solution of poisson's equation on a rectangle, SIAM Review, Vol. 19, No. 3, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the stream based algorithm must scale up better than the second approach because of a smaller communication overhead. 4.4. Comparison with Other Methods Fourier analysis cyclic reduction (FACR) solvers encompass a class of methods for the solution of Poisson s equation on regular grids [12, 24, 25]. In two dimensional problems, one dimensional FFTs are applied to decoupte the equations into independent triangular systems. Cyclic reduction and Gaussian elimination (or another set of one dimensional FFTs and inverse FFTs) are used to solve the linear systems. In the FACR(t) algorithm, t ....

P.N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Re,: 19, 491 (1977).

....with GMRES(5) approximate cyclic reduction preconditioning. 2 The Cyclic Reduction Principle We recall the classical method of cyclic reduction. This method can be used, for example, for solving a linear system with a tridiagonal matrix or with a special block tridiagonal matrix (cf. [10,13,24]) We explain the cyclic reduction principle by considering an n n linear system with a tridiagonal matrix: Ax = b; A = 2 6 6 6 6 6 6 4 a 1 b 1 c 1 a 2 b 2 ; b n 1 c n 1 an 3 7 7 7 7 7 7 5 ; a i 6= 0 for all i : 2) Reordering the unknowns based on an ....

Swarztrauber, P.N.: The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Review 19 (1977) 490-501

....that the stream based algorithm must scale up better than the second approach because of a smaller communication overhead. 13 4. 4 Comparison with Other Methods Fourier Analysis Cyclic Reduction (FACR) solvers encompass a class of methods for the solution of Poisson s equation on regular grids [12, 24, 25]. In two dimensional problems, one dimensional FFTs are applied to decouple the equations into independent triangular systems. Cyclic reduction, Gaussian elimination (or another set of one dimensional FFTs and inverse FFTs) are used to solve the linear systems. In the FACR(#) algorithm, # ....

P. N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review, 19 (1977), pp. 491 -- 501.

....attractive properties as preconditioner. The most common choice is to take a matrix from a separable PDE. A system involving such a matrix can be solved with various so called fast solvers , such as FFT methods, cyclic reduction, or the generalized marching algorithm (see Dorr [75] Swarztrauber [195], Bank [25] and Bank and Rose [27] As a simplest example, any elliptic operator can be preconditioned with the Poisson operator, giving the iterative method Gamma Delta(u n 1 Gamma un ) Gamma(Lu n Gamma f) In Concus and Golub [59] a transformation of this method is considered to speed ....

P. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev., 19 (1977), pp. 490--501.

....to mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation. Subject Classifications : 65N10, 65F30 1. Introduction Capacitance matrix methods are numerical techniques aimed at bringing to bear the efficiency of fast elliptic solvers (cf. [6, 22]) on elliptic boundary value problems to which these fast solvers would otherwise not be applicable because the underlying problem is not separable. Fast elliptic solvers can be viewed as algorithms which efficiently apply a discrete Green s function belonging to a separable differential operator ....

P. N. Swarztrauber, The methods of cyclic reduction, fourier analysis and the facr algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev. 19 (1977), 490--501.

....second order algorithm. Of equal importance is its parallel nature and its readiness for parallel and distributed computers. Solving Helmholtz equations is a key issue of scientific computing. Intensive research has been done in recent years in the field to develop efficient numerical methods [16]. In the late sixties, Hockney [4] and Bunemann [1] developed fast direct methods for elliptic equations on rectangular uniform meshs. These methods take advantage of the special block structure of the resulting system of finite difference discretizations and reduce the number of computations ....

Swarztrauber, P. N. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Review 19, 3 (1977), 490--501. 25

....a Fourier decomposition of A in (1.4) which means that blocks of the vector u can be explicitly expressed as a transformation of blocks of a vector y containing the solutions to a set of tridiagonal linear systems. This idea is the basis of some fast direct solvers for linear systems of this type [8]. Here, however, we take advantage of the fact that as these tridiagonal systems are of Toeplitz form, they can be solved analytically via a set of three term recurrence relations: the construction and solution of these equations is described in section 3. The result of this process is an exact ....

Swarztrauber, P.N. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review 19 (1977) 490-501. 21

....variant of the Cylic Reduction method (PSCR method) and it was described in [3] The basic idea of this method was originally introduced in [7] and further generalized in [1] for example. This method is closely related to the classical block cyclic reduction method studied, for example, in [5], but instead of using the matrix polynomial factorization the so called partial solution technique is employed. Due to this, it is possible to construct general radix q methods which are computationally cheaper than the standard radix two cyclic reduction solvers. We use the radix four ....

Swarztrauber, P. N.: The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Review 19 (1977) 490--501

.... 2 log n; 1) or T (log k n; n p ) CS454: November 9, 1993 [ 5 ] Some rapid elliptic solvers ffl Matrix Decomposition ffl Block Cyclic Reduction ffl FACR(l) Fourier Analysis, Cyclic Reduction) ffl Marching Methods ffl Multigrid Methods Hockney [Hoc65, Hoc70] Sweet [Swe77] Swarztrauber [Swa74a, Swa77,Swa84], Buzbee, Goluband Nielson [BGN70a] Widlund [Wid72] Parallelization and vectorization for FFT based algorithms: Sameh, Chen and Kuck [SCK76] Parallelized BCR considered only recently [GS89b, GS89a, Swe88, SS89] We will look at some of these methods and will discuss parallel algorithms derived ....

P. N. Swarztrauber. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Review, 19(3):490--501, July 1977. Also derives the optimal value for l in uniprocessor FACR.

....for the solution, which all share the property that the rate of convergence is independent of the dimension of the problem. In each iterative step two second order boundary value problems must be solved. For the case of a rectangular region so called fast Poisson solvers can be applied, 7] and [22]. In section 5 we present some results from preliminary numerical experiments. The results were all obtained on a simulated data set. 2 A Scattering Model Now we will briefly discuss the mathematical inverse problem to be resolved in order to recover the ground topography height function from ....

P.N. Swarztrauber The Methods of Cyclic Reduction, Fourier Analysis and the FACR Algorithm for the Discrete Solution of Poisson's Equation on a Rectangle. SIAM Review 19, pp. 490-501, 1977.

....tridiagonal structure. For problems of this type resulting from the discretization of separable elliptic boundary value problems on rectangles or circles, so called fast solvers can solve the linear system in O(N 2 log N ) arithmetic operations, N being the number of mesh points in one direction [3, 10]. However, due to the radiation boundary condition, these techniques cannot be applied to our problem in any obvious way. Thus, we attempt a domain decomposition approach which separates the interior unknowns from the boundary unknowns, solves these two problems separately in an efficient way, and ....

P. N. Swarztrauber The method of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19 pp.490501 (1977)

....n rectangular grid with with a constant less than ten. A combination of the two algorithms due to R. Hockney known as FACR reduces this complexity further to O(mn log log n) All these techniques are described in Buzbee, Golub and Nielson [22] Some survey articles for fast Poisson solvers are [32, 22, 102, 101, 62]. A library of fast Poisson solver software for two and three dimensional problems for rectangular, circular and spherical geometries can be found on Netlib in the software package named Fishpak which was developed by P. Swarztrauber and R. Sweet. 1.1.2 Origins of the Capacitance Matrix Method ....

P. N. Swarztrauber. The methods of cyclic reduction, fourier analysis and the facr algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Rev., 19:490--501, 1977. BIBLIOGRAPHY 138

.... operators have trigonometric eigenvectors then the fast Fourier transform can be used [Hym51] This is the route which has lead to fast Poisson solvers [Hoc65] see also [FGH 74, Ban78, SS79] It can be combined with cyclic reduction [Bun69] to get even better complexity bounds [Swa77] An even lower complexity bound is obtained by multigrid methods [BPWX91] When the partial differential equation is separable, but the boundary conditions are given on more general regions than correctly oriented rectangles, then the extension of the fast methods by embedding the region into a ....

Paul N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review 19 (1977), 490--501.

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P. N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review, 19 (1977), pp. 490--501.

....results of a few numerical experiments. 2 Cyclic reduction for a tridiagonal matrix We recall the classical method of cyclic reduction. This method can be used, for example, for solving a linear system with a tridiagonal matrix or with a special block tridiagonal matrix (cf. 17] 20] 30] [32]) We explain the cyclic reduction principle by considering an n n linear system with a tridiagonal matrix: Ax = b; A = 2 6 6 6 6 6 6 6 4 a 1 b 1 c 1 a 2 b 2 ; b n 1 c n 1 a n 3 7 7 7 7 7 7 7 5 ; a i 6= 0 for all i : 2) Reordering the unknowns based on ....

....This small system is then solved and the previously eliminated (red) unknowns are found by a simple back substitution process. Note that cyclic reduction is equivalent to Gaussian elimination applied to a permuted system of equations and that di erent implementations are possible (cf. 17] [32]) When solving a system as in (2) with cyclic reduction, one usually adapts the righthand side in the reduction phase. For example, in the rst reduction step the original system is transformed to 2 6 4 S bb ; A rb A rr 3 7 5 Px = 2 6 4 I A br A 1 rr ; I 3 7 5 Pb : 5) In such a ....

P.N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review, 19 (1977), pp. 490-501.

....thirty years starting from 1965, when Hockney [5] described the Fourier analysis method. In both [5] and [6] another kind of method, called cyclic reduction was discussed. The original cyclic reduction approach is unstable, but being combined with Fourier analysis in the FACR(l) algorithm [6] [12], the unstability poses no serious restriction since in problems of practical size, typically less than five steps of reduction process needs to be performed. Stable, so called Buneman s variants of cyclic reduction methods were introduced in [3] making the cyclic reduction a robust technique. ....

....Buneman s variants of cyclic reduction methods were introduced in [3] making the cyclic reduction a robust technique. The arithmetical complexity of Buneman s method is O(N log N) floating point operations, and for the FACR(l) algorithm with optimal choice of l it is O(N log log N) operations [12]. In mid 80 s, another O(N log N) method, utilizing the so called partial solution techniques [2] 9] was introduced in [14] and further developed in [8] In [10] this method was generalized to the case of symmetric separable block band matrices, and it was also named Divide Conquer method ....

P. N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review, 19 (1977), pp. 490--501.

No context found.

P. N. Swarztrauber, The methods of cyclic reduction, fourier analysis and the facr algorithm for the discrete solution of poisson's equation on a rectangle, SIAM Review, Vol. 19, No. 3, 1977.

No context found.

P. Swarztrauber, \The method of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle", SIAM Rev., 19 (1977), pp. 490-501.

No context found.

Paul N. Swartztrauber. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Posson's equation of a rectangle. SIAM Review, 19(3):490--501, 1977.

No context found.

P. Swarztrauber, The method of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev., 19 (1977), pp. 490-501.

No context found.

P. N. SWARZTRAUBER, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev., 19 (1977), pp. 490--501.

No context found.

P. N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Review, 19 (1977), pp. 490--501.

No context found.

P. N. Swarztrauber. The methods of cyclic reduction, fourier analysis and the FACR algorithm for the discrete solution of poisson equation on a rectangle. SIAM Review, 19:490--501, 1977.

No context found.

Paul N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson equation on a rectangle, J. Computational Phys. 15 (1974), 46-54.

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