A breakdown-free block conjugate gradient method

Ji, Hao; Li, Yaohang

doi:10.1007/s10543-016-0631-z

Cited by 24 publications

(19 citation statements)

References 40 publications

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“…Unlike CG algorithm a breakdown would occur if Z k+1 AZ k+1 is singular, i.e., Z k+1 is not full rank. Although rare this situation can happen in practice and several variants have been developed in order to handle this case [25,19,14,32]. Overall, both Orthomin and Orthodir generate P k+1 such that…”

Section: Rr N°9190mentioning

confidence: 99%

“…As explained previously Orthomin version of ECG can break down. There exist several methods to overcome this issue and in the following we recall the Breakdown-Free block CG method defined in [25]. Starting from the original algorithm of O'Leary [32], the authors propose to perform a rank-revealing QR decomposition of Z k+1 and then drop its null part before A-orthonormalizing it.…”

Section: Curing Breakdowns In Orthominmentioning

confidence: 99%

“…Its implementation is very easy using the LAPACK routine pstrf (Cholesky with pivoting) for computing (R, π) at line 2. Following [25] we use the default tolerance of pstrf for detecting exact rank deficiency of Z k+1 .…”

Section: Cost Of Dynamic Reduction Of Ecgmentioning

confidence: 99%

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Scalable Linear Solvers Based on Enlarged Krylov Subspaces with Dynamic Reduction of Search Directions

Grigori¹,

Tissot²

2019

SIAM J. Sci. Comput.

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Krylov methods are widely used for solving large sparse linear systems of equations. On distributed architectures, their performance is limited by the communication needed at each iteration of the algorithm. In this paper, we study the use of so-called enlarged Krylov subspaces for reducing the number of iterations, and therefore the overall communication, of Krylov methods. In particular, we consider a reformulation of the Conjugate Gradient method using these enlarged Krylov subspaces: the enlarged Conjugate Gradient method. We present the parallel design of two variants of the enlarged Conjugate Gradient method as well as their corresponding dynamic versions where the number of search directions is dynamically reduced during the iterations. For a linear elasticity problem with heterogeneous coefficients, using a block Jacobi preconditioner, we show that this implementation scales up to 16, 384 cores, and is up to 5.7 times faster than the PETSc implementation of PCG.

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Section: Rr N°9190mentioning

confidence: 99%

Section: Curing Breakdowns In Orthominmentioning

confidence: 99%

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Scalable Linear Solvers Based on Enlarged Krylov Subspaces with Dynamic Reduction of Search Directions

Grigori¹,

Tissot²

2019

SIAM J. Sci. Comput.

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“…Under the orthogonality condition (13), the value of parameters α k and β k in the recurrence relations (12) can be obtained by solving the following equations:…”

Section: Theoremmentioning

confidence: 99%

“…To make it fair, all algorithms compared in this example are preconditioned; they are the preconditioned version sof the same six algorithms as in Example 1 and are denoted by PBCOCG, PBCOCR, BFPBCOCG, BFPBCOCR, PBCOCG_rq, and PBCOCR_rq, respectively. The preconditioning strategy we used is IC (3) in [13]. IC(3) produces LL T for a complex symmetric A, and if L is nonsingular, then we can use LL T as a preconditioner.…”

mentioning

confidence: 99%

A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Zhong

Zhang

2019

Symmetry

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The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

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