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Wim Vanroose - One of the best experts on this subject based on the ideXlab platform.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    IEEE Transactions on Parallel and Distributed Systems, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method, p($l$l)-CG, outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus “hiding” global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by $\ell$l three-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    arXiv: Numerical Analysis, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

Siegfried Cools - One of the best experts on this subject based on the ideXlab platform.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    IEEE Transactions on Parallel and Distributed Systems, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method, p($l$l)-CG, outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus “hiding” global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by $\ell$l three-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    arXiv: Numerical Analysis, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

Jamilu Sabiu - One of the best experts on this subject based on the ideXlab platform.

  • a dai liao Conjugate Gradient Method via modified secant equation for system of nonlinear equations
    Arabian Journal of Mathematics, 2020
    Co-Authors: Mohammed Yusuf Waziri, K Ahmed, Jamilu Sabiu
    Abstract:

    In this paper, we propose a Dai–Liao (DL) Conjugate Gradient Method for solving large-scale system of nonlinear equations. The Method incorporates an extended secant equation developed from modified secant equations proposed by Zhang et al. (J Optim Theory Appl 102(1):147–157, 1999) and Wei et al. (Appl Math Comput 175(2):1156–1188, 2006) in the DL approach. It is shown that the proposed scheme satisfies the sufficient descent condition. The global convergence of the Method is established under mild conditions, and computational experiments on some benchmark test problems show that the Method is efficient and robust.

Jeffrey Cornelis - One of the best experts on this subject based on the ideXlab platform.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    IEEE Transactions on Parallel and Distributed Systems, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method, p($l$l)-CG, outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus “hiding” global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by $\ell$l three-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

  • numerically stable recurrence relations for the communication hiding pipelined Conjugate Gradient Method
    arXiv: Numerical Analysis, 2019
    Co-Authors: Siegfried Cools, Jeffrey Cornelis, Wim Vanroose
    Abstract:

    Pipelined Krylov subspace Methods (also referred to as communication-hiding Methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient Method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient Method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace Methods.

Reza Ghanbari - One of the best experts on this subject based on the ideXlab platform.

  • a new three term Conjugate Gradient Method with descent direction for unconstrained optimization
    Mathematical Modelling and Analysis, 2016
    Co-Authors: Xiaoliang Dong, Hongwei Liu, Saman Babaiekafaki, Reza Ghanbari
    Abstract:

    In this paper, we propose a three–term PRP–type Conjugate Gradient Method which always satisfies the sufficient descent condition independently of line searches employed. An important property of our Method is that its direction is closest to the direction of the Newton Method or satisfies conjugacy condition as the iterations evolve. In addition, under mild condition, we prove global convergence properties of the proposed Method. Numerical comparison illustrates that our proposed Method is efficient for solving the optimization problems.

  • a descent extension of the polak ribiere polyak Conjugate Gradient Method
    Computers & Mathematics With Applications, 2014
    Co-Authors: Saman Babaiekafaki, Reza Ghanbari
    Abstract:

    Based on an eigenvalue analysis, a descent class of two-parameter extension of the Conjugate Gradient Method proposed by Polak and Ribiere (1969), and Polyak (1969) is suggested. It is interesting that the one-parameter class of descent Conjugate Gradient Methods proposed by Yu et?al. (2008) is a member of the suggested class. Global convergence analysis for the Methods of the suggested class is made briefly. Preliminary numerical results are reported; they demonstrate proper choices for the parameters of the suggested class of the Conjugate Gradient Methods that may lead to a promising computational performance.

  • the dai liao nonlinear Conjugate Gradient Method with optimal parameter choices
    European Journal of Operational Research, 2014
    Co-Authors: Saman Babaiekafaki, Reza Ghanbari
    Abstract:

    Minimizing two different upper bounds of the matrix which generates search directions of the nonlinear Conjugate Gradient Method proposed by Dai and Liao, two modified Conjugate Gradient Methods are proposed. Under proper conditions, it is briefly shown that the Methods are globally convergent when the line search fulfills the strong Wolfe conditions. Numerical comparisons between the implementations of the proposed Methods and the Conjugate Gradient Methods proposed by Hager and Zhang, and Dai and Kou, are made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed Methods in the sense of the performance profile introduced by Dolan and More.