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Jongho Park - One of the best experts on this subject based on the ideXlab platform.

M A Aloy - One of the best experts on this subject based on the ideXlab platform.

  • on the equivalence between the scheduled relaxation Jacobi Method and richardson s non stationary Method
    Journal of Computational Physics, 2017
    Co-Authors: Jose E Adsuara, I Corderocarrion, P Cerdaduran, Vassilios Mewes, M A Aloy
    Abstract:

    Abstract The Scheduled Relaxation Jacobi (SRJ) Method is an extension of the classical Jacobi iterative Method to solve linear systems of equations ( A u = b ) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of P relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of M consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's Method where the matrix of the system of equations is preconditioned multiplying it by D = diag ( A ) . Our Method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix A (or the corresponding iteration matrix of the underlying weighted Jacobi scheme) is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis of the continuous elliptic operator. This set of weights is also the optimal one for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. The amplification factor of the Method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter ω in the Successive Overrelaxation (SOR) Method in some cases. Finally, we demonstrate with practical examples that our Method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed (e.g., a 9- or 17-points discretization). This is of interest since the former discretizations do not yield consistently ordered A matrices and, hence, the theory of Young cannot be used to predict the optimal value of the SOR parameter. Furthermore, the optimal SRJ schemes deduced here are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation.

  • scheduled relaxation Jacobi Method
    Journal of Computational Physics, 2016
    Co-Authors: Jose E Adsuara, I Corderocarrion, P Cerdaduran, M A Aloy
    Abstract:

    Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of Methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ Methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficiency in the reduction of the residual increases with the number of levels employed in the algorithm. Applying the original Methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal SRJ schemes, as the mixed (non-linear) algebraic-differential system of equations from which they result becomes notably stiff. Here we present a new Methodology for obtaining the parameters of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ schemes with up to 15 levels and resolutions of up to 215 points per dimension, allowing for acceleration factors larger than several hundreds with respect to the Jacobi Method for typical resolutions and, in some high resolution cases, close to 1000. Most of the success in finding SRJ optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs. We compute new optimal parameters of the Scheduled Relaxation Jacobi Method.The new parameters are calculated for SRJ schemes with P = 6 to P = 15 levels.We reduce the stiffness in the computation of optimal SRJ parameters analytically.We provide a grid of optimal parameters for different P and numerical resolutions.We benchmark SRJ Methods against other algorithms to solve linear systems.

Chang-ock Lee - One of the best experts on this subject based on the ideXlab platform.

Jose E Adsuara - One of the best experts on this subject based on the ideXlab platform.

  • on the equivalence between the scheduled relaxation Jacobi Method and richardson s non stationary Method
    Journal of Computational Physics, 2017
    Co-Authors: Jose E Adsuara, I Corderocarrion, P Cerdaduran, Vassilios Mewes, M A Aloy
    Abstract:

    Abstract The Scheduled Relaxation Jacobi (SRJ) Method is an extension of the classical Jacobi iterative Method to solve linear systems of equations ( A u = b ) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of P relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of M consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's Method where the matrix of the system of equations is preconditioned multiplying it by D = diag ( A ) . Our Method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix A (or the corresponding iteration matrix of the underlying weighted Jacobi scheme) is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis of the continuous elliptic operator. This set of weights is also the optimal one for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. The amplification factor of the Method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter ω in the Successive Overrelaxation (SOR) Method in some cases. Finally, we demonstrate with practical examples that our Method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed (e.g., a 9- or 17-points discretization). This is of interest since the former discretizations do not yield consistently ordered A matrices and, hence, the theory of Young cannot be used to predict the optimal value of the SOR parameter. Furthermore, the optimal SRJ schemes deduced here are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation.

  • scheduled relaxation Jacobi Method
    Journal of Computational Physics, 2016
    Co-Authors: Jose E Adsuara, I Corderocarrion, P Cerdaduran, M A Aloy
    Abstract:

    Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of Methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ Methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficiency in the reduction of the residual increases with the number of levels employed in the algorithm. Applying the original Methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal SRJ schemes, as the mixed (non-linear) algebraic-differential system of equations from which they result becomes notably stiff. Here we present a new Methodology for obtaining the parameters of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ schemes with up to 15 levels and resolutions of up to 215 points per dimension, allowing for acceleration factors larger than several hundreds with respect to the Jacobi Method for typical resolutions and, in some high resolution cases, close to 1000. Most of the success in finding SRJ optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs. We compute new optimal parameters of the Scheduled Relaxation Jacobi Method.The new parameters are calculated for SRJ schemes with P = 6 to P = 15 levels.We reduce the stiffness in the computation of optimal SRJ parameters analytically.We provide a grid of optimal parameters for different P and numerical resolutions.We benchmark SRJ Methods against other algorithms to solve linear systems.

Vjeran Hari - One of the best experts on this subject based on the ideXlab platform.

  • Jacobi Method for symmetric 4 4 matrices converges for every cyclic pivot strategy
    Numerical Algorithms, 2018
    Co-Authors: Erna Begovic Kovac, Vjeran Hari
    Abstract:

    The paper studies the global convergence of the Jacobi Method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A [t+3]) ≤ γ S(A [t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A [t] stands for the matrix obtained from A after t full cycles of the Jacobi Method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi Method.

  • Jacobi Method for symmetric 4 times4 matrices converges for every cyclic pivot strategy
    arXiv: Numerical Analysis, 2017
    Co-Authors: Erna Begovic, Vjeran Hari
    Abstract:

    The paper studies the global convergence of the Jacobi Method for symmetric matrices of size $4$. We prove global convergence for all $720$ cyclic pivot strategies. Precisely, we show that inequality $S(A^{[t+3]})\leq\gamma S(A^{[t]})$, $t\geq1$, holds with the constant $\gamma<1$ that depends neither on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi Method and $S(A)$ is the off-diagonal norm of $A$. We show why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jacobi Method.

  • Jacobi Method for symmetric matrices of order 4 converges for every cyclic pivot strategy
    arXiv: Numerical Analysis, 2017
    Co-Authors: Erna Begovic, Vjeran Hari
    Abstract:

    The paper studies the global convergence of the Jacobi Method for symmetric matrices of order $4$. The global convergence has been proved for all $720$ cyclic pivot strategies. It has been shown that inequality $S(A^{[t+3]})\leq\gamma S(A^{[t]})$, $t\geq1$, holds with the constant $\gamma<1$ that does not depend neither on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi Method and $S(A)$ is the off-norm of $A$. It is shown why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jacobi Method.

  • full block j Jacobi Method for hermitian matrices
    Linear Algebra and its Applications, 2014
    Co-Authors: Vjeran Hari, Sanja Singer, Sasa Singer
    Abstract:

    Abstract The paper considers convergence, accuracy and efficiency of a block J -Jacobi Method. The Method is a proper BLAS 3 generalization of the known Method of Veselic for computing the hyperbolic singular value decomposition of rectangular matrices. At each step, the proposed algorithm diagonalizes the block-pivot submatrix. The convergence is proved for cyclic strategies which are weakly equivalent to the row-cyclic strategy. The relative accuracy is proved under the standard conditions. Numerical tests show improved performance with respect to the block-oriented generalization of the original Method of Veselic. Combined with the Hermitian indefinite factorization, the proposed Method becomes accurate and efficient eigensolver for Hermitian indefinite matrices.

  • accelerating the svd block Jacobi Method
    Computing, 2005
    Co-Authors: Vjeran Hari
    Abstract:

    The paper discusses how to improve performance of the one-sided block-Jacobi algorithm for computing the singular value decomposition of rectangular matrices. In particular, it is shown how cosine-sine decomposition of orthogonal matrices can be used to accelerate the slowest part of the algorithm – updating the block-columns.