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Hiroshi Niki - One of the best experts on this subject based on the ideXlab platform.

  • The preconditioned Gauss-Seidel Method faster than the SOR Method
    Journal of Computational and Applied Mathematics, 2008
    Co-Authors: Hiroshi Niki, Toshiyuki Kohno, Munenori Morimoto
    Abstract:

    In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative Methods for consistent linear systems, Linear Algebra Appl. 154-156 (1991) 123-143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss-Seidel Method for Z-matrices, Linear Algebra Appl. 267 (1997) 113-123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative Method with the preconditioner (I+S"m"a"x), J. Comput. Appl. Math. 145 (2002) 373-378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration Method for Z-matrices, J. Comput. Appl. Math. 75 (1996) 87-97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel Method for linear systems, Internat. J. Comput. Math. 51(1994)119-125 [10]]. Since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row of matrix A. In the present paper, in order to deal with this drawback, we propose a new preconditioner. In addition, the convergence and comparison theorems of the proposed Method are established. Simple numerical examples are also given, and we show that the convergence rate of the proposed Method is better than that of the optimum SOR.

  • The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel Method
    Journal of Computational and Applied Mathematics, 2004
    Co-Authors: Hiroshi Niki, Munenori Morimoto, Kyouji Harada, Michio Sakakihara
    Abstract:

    Several preconditioned iterative Methods reported in the literature have been used for improving the convergence rate of the Gauss-Seidel Method. In this article, on the basis of nonnegative matrix, comparisons between some splittings for such preconditioned matrices are derived. Simple numerical examples are also given.

  • The survey of preconditioners used for accelerating the rate of convergence in the Gauss–Seidel Method
    Journal of Computational and Applied Mathematics, 2004
    Co-Authors: Hiroshi Niki, Munenori Morimoto, Kyouji Harada, Michio Sakakihara
    Abstract:

    AbstractSeveral preconditioned iterative Methods reported in the literature have been used for improving the convergence rate of the Gauss–Seidel Method. In this article, on the basis of nonnegative matrix, comparisons between some splittings for such preconditioned matrices are derived. Simple numerical examples are also given

  • A generalization of the adaptive Gauss-Seidel Method for Z-matrices
    International Journal of Computer Mathematics, 1997
    Co-Authors: Hisashi Kotakemori, Hiroshi Niki, Naotaka Okamoto
    Abstract:

    We proposed an accelerated Adaptive Gauss-Seidel Method with the type (I+βU) as preconditioner in [3]. In this paper we generalize this Method and prove a convergence theorem. We show some numerical examples.

  • Improving the modified Gauss-Seidel Method for Z-matrices
    Linear Algebra and its Applications, 1997
    Co-Authors: Toshiyuki Kohno, Hiroshi Niki, Hisashi Kotakemori, Masataka Usui
    Abstract:

    Abstract In 1991 A. D. Gunawardena et al. reported that the convergence rate of the Gauss-Seidel Method with a preconditioning matrix I + S is superior to that of the basic iterative Method. In this paper, we use the preconditioning matrix I + S(α). If a coefficient matrix A is an irreducibly diagonally dominant Z-matrix, then [I + S(α)]A is also a strictly diagonally dominant Z-matrix. It is shown that the proposed Method is also superior to other iterative Methods.

Sanguthevar Rajasekaran - One of the best experts on this subject based on the ideXlab platform.

  • ISCC - Fast GPU algorithms for implementing the red-black Gauss-Seidel Method for Solving Partial Differential Equations
    2013 IEEE Symposium on Computers and Communications (ISCC), 2013
    Co-Authors: Mahmoud Elmaghrbay, Reda A. Ammar, Sanguthevar Rajasekaran
    Abstract:

    Solving Partial Differential Equations (PDEs) is very important in many areas. Since PDE solvers take very long time for numerous applications of interest, we need efficient parallel implementations. An attractive parallel computing platform that is widely used at present is the Graphics Processing Unit (GPU). In this paper we present an efficient technique that uses the red-black Gauss-Seidel Method to solve PDEs. This technique allows the efficient use of the relatively larger register file available in each Streaming Multiprocessor (SM), as well as the shared memory. It also allows the communication between the threads of a block. We employ the red-black Gauss-Seidel Method, in this paper, to solve the 2D steady state heat conduction problem on two different GPUs. An overall speedup of 484 relative to the CPU sequential implementation is achieved. A speedup of about 2.6 relative to Foster's GPU implementation on the same GPUs is also achieved.

Rodríguez Ferran Antonio - One of the best experts on this subject based on the ideXlab platform.

  • The block Gauss-Seidel Method in sound transmission problems
    'World Scientific Pub Co Pte Lt', 2010
    Co-Authors: Poblet-puig Jordi, Rodríguez Ferran Antonio
    Abstract:

    Sound transmission through partitions can be modeled as an acoustic fluid–elastic structure interaction problem. The block Gauss–Seidel iterative Method is used in order to solve the finite element linear system of equations. The blocks are defined, respecting the fluid and structural domains. The convergence criterion is analyzed and interpreted in physical terms by means of simple one-dimensional problems. This analysis highlights the negative influence on the convergence of a strong degree of coupling between the acoustic domains and the structure. A selective coupling strategy has been developed and applied to problems with strong coupling (e.g. double walls).Peer Reviewe

  • The block Gauss-Seidel Method in sound transmission problems
    2009
    Co-Authors: Poblet-puig Jordi, Rodríguez Ferran Antonio
    Abstract:

    Sound transmission through partitions can be modelled as an acoustic fluid-elastic structure interaction problem. The block Gauss-Seidel iterative Method is used in order to solve the finite element linear system of equations. The blocks are defined in a natural way, respecting the fluid and structural domains. The convergence criterion (spectral radius of iteration matrix smaller than one) is analysed and interpreted in physical terms by means of simple one-dimensional problems. This analysis highlights the negative influence on the convergence of a strong degree of coupling between the acoustic domains. A selective coupling strategy has been developed and successfully applied to problems with strong coupling (i.e. sound transmission through double walls)

Fabio Pellacini - One of the best experts on this subject based on the ideXlab platform.

  • Vivace: a practical Gauss-Seidel Method for stable soft body dynamics
    ACM Transactions on Graphics, 2016
    Co-Authors: Marco Fratarcangeli, Valentina Tibaldo, Fabio Pellacini
    Abstract:

    The solution of large sparse systems of linear constraints is at the base of most interactive solvers for physically-based animation of soft body dynamics. We focus on applications with hard and tight per-frame resource budgets, such as video games, where the solution of soft body dynamics needs to be computed in a few milliseconds. Linear iterative Methods are preferred in these cases since they provide approximate solutions within a given error tolerance and in a short amount of time. We present a parallel randomized Gauss-Seidel Method which can be effectively employed to enable the animation of 3D soft objects discretized as large and irregular triangular or tetrahedral meshes. At the beginning of each frame, we partition the set of equations governing the system using a randomized graph coloring algorithm. The unknowns in the equations belonging to the same partition are independent of each other. Then, all the equations belonging to the same partition are solved at the same time in parallel. Our algorithm runs completely on the GPU and can support changes in the constraints topology. We tested our Method as a solver for soft body dynamics within the Projective Dynamics and Position Based Dynamics frameworks. We show how the algorithmic simplicity of this iterative strategy enables great numerical stability and fast convergence speed, which are essential features for physically based animations with fixed and small hard time budgets. Compared to the state of the art, we found our Method to be faster and scale better while providing stabler solutions for very small time budgets.

Mahmoud Elmaghrbay - One of the best experts on this subject based on the ideXlab platform.

  • ISCC - Fast GPU algorithms for implementing the red-black Gauss-Seidel Method for Solving Partial Differential Equations
    2013 IEEE Symposium on Computers and Communications (ISCC), 2013
    Co-Authors: Mahmoud Elmaghrbay, Reda A. Ammar, Sanguthevar Rajasekaran
    Abstract:

    Solving Partial Differential Equations (PDEs) is very important in many areas. Since PDE solvers take very long time for numerous applications of interest, we need efficient parallel implementations. An attractive parallel computing platform that is widely used at present is the Graphics Processing Unit (GPU). In this paper we present an efficient technique that uses the red-black Gauss-Seidel Method to solve PDEs. This technique allows the efficient use of the relatively larger register file available in each Streaming Multiprocessor (SM), as well as the shared memory. It also allows the communication between the threads of a block. We employ the red-black Gauss-Seidel Method, in this paper, to solve the 2D steady state heat conduction problem on two different GPUs. An overall speedup of 484 relative to the CPU sequential implementation is achieved. A speedup of about 2.6 relative to Foster's GPU implementation on the same GPUs is also achieved.