Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The red/black ordering scheme is often used to increase the parallelism of iterative methods for solving elliptic partial differential equations (PDEs). Hover, the convergence rates are also affected, often adversely.  provides a unified approach, called the two-color Fourier analysis, to study the convergence rates of iterative algorithms for elliptic problems with the red/black ordering. This Fourier tool is used to analyze different types of iterative algorithms, including the successive over-relaxation (SOR) method, symmetric successive over-relaxation (SSOR) method, preconditioned iterative methods with SSOR, ILU, and MILU preconditioners, and multigrid (MG) methods. By comparing the convergence rates of algorithms with the natural and red/black orderings, it is shown that although the red/black ordering does not affect the rate of convergence in the context of SOR and MG methods, it slows down the convergence significantly in the context of SSOR and preconditioned iterative methods. computers. One common approach to obtain parallel iterative algorithms for the solution of partial differential equations (PDEs) is reordering. By reordering,  rearrange the computational sequences to increase the percentage of computations that can be done independently (27). A crucial issue associated with reordering is how the convergence rate of an iterative algorithm is affected by a reordering scheme. The multicolor ordering scheme for grid points provides more parallelism than the natural rowwise or columnwise ordering scheme. It is ll known that by using red and black, two colors to order the grid points in a checkerboard fashion for the 5- point Laplacian,  are able to separate the coupling beten any two red (or black) points so that the values at all red (or black) points can be updated simultaneously. Similarly, four colors are needed to separate the coupling beten grid points of the same color for the 9-point Laplacian (1)-(4),(20),(22),(23). On either vector or parallel computers, an algorithm with the multicolor ordering is always easier to vectorize or parallelize than its naturally ordered counterpart so that such a reordering is attractive for parallel implementation. There are numerous discussions on the implementation of iterative algorithms with the red/black ordering on vector and parallel computers in the literature, for example, in (1),(5),(8),(11),(23),(27),(28), and (33). ,  examine how the convergence rate of an iterative algorithm is affected by the red/black ordering. Our study includes the successive over-relaxation (SOR), symmetric successive over-relaxation (SSOR), ILU, and MILU preconditioners
['C.-C. Jay Kuo', 'Tony F. Chan']
I am writing a research paper on a idea Two-color fourier analysis of iterative algorithms for elliptic problems with red/black ordering
[{"from": "human", "value": "I am writing a research paper on a idea Two-color fourier analysis of iterative algorithms for elliptic problems with red/black ordering"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The red/black ordering scheme is often used to increase the parallelism of iterative methods for solving elliptic partial differential equations (PDEs). Hover, the convergence rates are also affected, often adversely.  provides a unified approach, called the two-color Fourier analysis, to study the convergence rates of iterative algorithms for elliptic problems with the red/black ordering. This Fourier tool is used to analyze different types of iterative algorithms, including the successive over-relaxation (SOR) method, symmetric successive over-relaxation (SSOR) method, preconditioned iterative methods with SSOR, ILU, and MILU preconditioners, and multigrid (MG) methods. By comparing the convergence rates of algorithms with the natural and red/black orderings, it is shown that although the red/black ordering does not affect the rate of convergence in the context of SOR and MG methods, it slows down the convergence significantly in the context of SSOR and preconditioned iterative methods. computers. One common approach to obtain parallel iterative algorithms for the solution of partial differential equations (PDEs) is reordering. By reordering,  rearrange the computational sequences to increase the percentage of computations that can be done independently (27). A crucial issue associated with reordering is how the convergence rate of an iterative algorithm is affected by a reordering scheme. The multicolor ordering scheme for grid points provides more parallelism than the natural rowwise or columnwise ordering scheme. It is ll known that by using red and black, two colors to order the grid points in a checkerboard fashion for the 5- point Laplacian,  are able to separate the coupling beten any two red (or black) points so that the values at all red (or black) points can be updated simultaneously. Similarly, four colors are needed to separate the coupling beten grid points of the same color for the 9-point Laplacian (1)-(4),(20),(22),(23). On either vector or parallel computers, an algorithm with the multicolor ordering is always easier to vectorize or parallelize than its naturally ordered counterpart so that such a reordering is attractive for parallel implementation. There are numerous discussions on the implementation of iterative algorithms with the red/black ordering on vector and parallel computers in the literature, for example, in (1),(5),(8),(11),(23),(27),(28), and (33). ,  examine how the convergence rate of an iterative algorithm is affected by the red/black ordering. Our study includes the successive over-relaxation (SOR), symmetric successive over-relaxation (SSOR), ILU, and MILU preconditioners"}]