Additive Schwarz with Variable Weights

Greif, Chen; Rees, Tyrone; Szyld, Daniel B.

doi:10.1007/978-3-319-05789-7_75

Cited by 9 publications

(10 citation statements)

References 9 publications

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“…Multipreconditioning significantly improves convergence as has already been observed Gosselet et al, 2015;Greif et al, 2014;Spillane, 2016) and as will be illustrated in the numerical result section. The drawback is that a dense matrix i 2 R N N must be factorized at each iteration and that N search directions per iteration need to be stored.…”

Section: Preliminariessupporting

confidence: 64%

“…In this work, we have implemented the MPCG Greif et al, 2014) algorithm for Restricted Additive Schwarz. We have observed very good convergence on test cases with known difficulties (heterogeneities and almost incompressible behaviour).…”

Section: Discussionmentioning

confidence: 99%

“…This was further studied in Greif et al (2014) which considers Additive Schwarz preconditioners in the Multipreconditioned GMRES (MPGMRES) Greif et al (2016) setting. It is an iterative linear solver, adapted from the Preconditioned Conjugate Gradient (PCG) algorithm , which can be used in cases where several preconditioners are available or the usual preconditioner is a sum of operators.…”

Section: Nicole Spillanementioning

confidence: 99%

See 2 more Smart Citations

Domain Decomposition Methods in Science and Engineering XXIII

Lee¹,

Cai²,

Keyes³

et al. 2017

Lecture Notes in Computational Science and Engineering

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Section: Preliminariessupporting

confidence: 64%

Section: Discussionmentioning

confidence: 99%

Section: Nicole Spillanementioning

confidence: 99%

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Domain Decomposition Methods in Science and Engineering XXIII

Lee¹,

Cai²,

Keyes³

et al. 2017

Lecture Notes in Computational Science and Engineering

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“…Remark Although our numerical results (Section 5) point to the fact that S‐FETI performs perfectly well, we mention the more recent multipreconditioned Generalized Minimal Residual (GMRES) algorithm where, at the cost of saving more directions at each iteration, the error can be minimized over the larger subspace

\sum_{s = 1}^{N} {scriptK}_{i}^{N} ((), boldF, boldP {bold \tilde{B}}^{(s)} {boldS}^{(s)} {boldr}_{0})

, with the multi‐Krylov subspace defined by

K_{i}^{N} (F, x) : = \{p (. . ., P {\overset{B}{true}}^{(s)} S^{(s)} {\overset{B}{true}}^{{(s)}^{T}} F, . . .) x; \begin{matrix} p is a polynomial in N variables \\ of degree at most i - 1 \end{matrix}\},

N being the number of subdomains. In , multiple preconditioned GMRES is applied to the Additive Schwarz domain decomposition technique.…”

Section: Simultaneous Fetimentioning

confidence: 99%

“…N being the number of subdomains. In [48], multiple preconditioned GMRES is applied to the Additive Schwarz domain decomposition technique.…”

Section: Remarkmentioning

confidence: 99%

Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions

Gosselet

Rixen

Roux

et al. 2015

Numerical Meth Engineering

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Domain Decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities. In particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner. Recently a robust approach, named FETI-Geneo, was proposed where troublesome modes are precomputed and deflated from the interface problem. The cost of the FETI-Geneo is however high. We propose in this paper techniques that share similar ideas with FETI-Geneo but where no pre-processing is needed and that can be easily and efficiently implemented as an alternative to standard Domain Decomposition methods. In the block iterative approaches presented in this paper, the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition. Those search directions originate either from the domain-wise preconditioner (in the Simultaneous FETI method) or from the block structure of the right-hand side of the interface problem (Block FETI). We show on 2D structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities, even when the interface is jagged or when the domains have a bad aspect ratio. The Simultaneous FETI was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular FETI method.

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Algebraic Adaptive Multipreconditioning Applied to Restricted Additive Schwarz

Spillane

2017

Lecture Notes in Computational Science and Engineering

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In 2006 the Multipreconditioned Conjugate Gradient (MPCG) algorithm was introduced by Bridson and Greif [4]. It is an iterative linear solver, adapted from the Preconditioned Conjugate Gradient (PCG) algorithm [22], which can be used in cases where several preconditioners are available or the usual preconditioner is a sum of operators. In [4] it was already pointed out that Domain Decomposition algorithms are ideal candidates to benefit from MPCG. This was further studied in [13] which considers Additive Schwarz preconditioners in the Multipreconditioned GMRES (MPGMRES) [14] setting. In 1997, Rixen had proposed in his thesis [21] the Simultaneous FETI algorithm which turns out to be MPCG applied to FETI. The algorithm is more extensively studied in [12] where its interpretation as an MPCG solver is made explicit. The idea behind MPCG is that if at a given iteration N preconditioners are applied to the residual, then the space spanned by all of these directions is a better minimization space than the one spanned by their sum. This can significantly reduce the number of iterations needed to achieve convergence, as we will observe in Section 3, but comes at the cost of loosing the short recurrence property in CG. This means that at each iteration the new search directions must be orthogonalized against all previous ones. For this reason, in [25] it was proposed to make MPCG into an Adaptive MPCG (AMPCG) algorithm where, at a given iteration, only the contributions that will accelerate convergence are kept, and all others are added into a global contribution (as they would be in classical PCG). This works very well for FETI and BDD but the theory in that article does not apply to Additive Schwarz. Indeed, the assumption is made that the smallest eigenvalue of the (globally) preconditioned operator is known. The test (called the τ-test), which chooses at each iteration which contributions should be kept, heavily relies on it. More precisely, the quantity that is examined by the τ-test can be related

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Additive Schwarz with Variable Weights

Cited by 9 publications

References 9 publications

Domain Decomposition Methods in Science and Engineering XXIII

Domain Decomposition Methods in Science and Engineering XXIII

Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions

Algebraic Adaptive Multipreconditioning Applied to Restricted Additive Schwarz

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