Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

Jolivet, Pierre; Tournier, Pierre-Henri

doi:10.1109/sc.2016.16

Cited by 15 publications

(14 citation statements)

References 68 publications

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“…This method is a variant of the restarted GMRES in which an abstract subspace can be used as a deflation subspace. Several papers followed based on this work [3,16,12,39,1,2,19].…”

Section: Introductionmentioning

confidence: 99%

Recycling Krylov Subspaces and Truncating Deflation Subspaces for Solving Sequence of Linear Systems

Daas

Grigori

Hénon

et al. 2021

ACM Trans. Math. Softw.

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This article presents deflation strategies related to recycling Krylov subspace methods for solving one or a sequence of linear systems of equations. Besides well-known strategies of deflation, Ritz-, and harmonic Ritz-based deflation, we introduce an Singular Value Decomposition based deflation technique. We consider the recycling in two contexts: recycling the Krylov subspace between the restart cycles and recycling a deflation subspace when the matrix changes in a sequence of linear systems. Numerical experiments on real-life reservoir simulation demonstrate the impact of our proposed strategy.

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“…This method is a variant of the restarted GMRES in which an abstract subspace can be used as a deflation subspace. Several papers followed based on this work [3,16,12,39,1,2,19].…”

Section: Introductionmentioning

confidence: 99%

Recycling Krylov Subspaces and Truncating Deflation Subspaces for Solving Sequence of Linear Systems

Daas

Grigori

Hénon

et al. 2021

ACM Trans. Math. Softw.

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“…Recently, Block Krylov methods are receiving an increasing attention in the HPC field [4,1,27,36,30]. They appear to be well suited for modern computers' architectures with a high level of parallelism because they allow to reduce the number of global synchronizations, while also featuring a higher arithmetic intensity at the cost of some extra computations.…”

Section: Block Krylov Methodsmentioning

confidence: 99%

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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“…Recycling solvers have been implemented in major software/solver libraries, most notably in PETSc [20][21][22]119,120], see [84] for a discussion on performance and applications in elasticity and electromagnetics, Trilinos [23,79,131], and the DLR-TAU library from the German Aerospace Center [154,155], which also detail several challenging applications in CFD.…”

Section: Uses In Practical Applicationsmentioning

confidence: 99%

“…Another important area is nonlinear optimization, such as nonlinear least-squares, for example, in tomography [90,100,111,130] and blind deconvolution [78]. Many applications arise in engineering, such as computational fluid dynamics and nonlinear structural problems [8,74,95,96,124,154,155], acoustics [89,99], and problems from electromagnetics and electrical circuits [49,72,73,84,118,156]. Recycling has found many applications in uncertainty quantification and partial differential equations with stochastic components [46,83].…”

Section: Computational Scientific and Engineering Applicationsmentioning

confidence: 99%

A survey of subspace recycling iterative methods

2020

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This survey concerns subspace recycling methods, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial vectors over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.

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Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

Cited by 15 publications

References 68 publications

Recycling Krylov Subspaces and Truncating Deflation Subspaces for Solving Sequence of Linear Systems

Recycling Krylov Subspaces and Truncating Deflation Subspaces for Solving Sequence of Linear Systems

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

A survey of subspace recycling iterative methods

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