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

Daas, Hussam Al; Grigori, Laura; Hénon, Pascal; Ricoux, Philippe

doi:10.1145/3439746

Cited by 11 publications

(46 citation statements)

References 41 publications

(65 reference statements)

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“…For alternative Krylov subspace recycling methods, for a range of applications, and HPC implementations, see [7,8,9,10,11,12,13,14]. A survey of Krylov subspace recycling methods is given in [15].…”

Section: Krylov Subspace Recyclingmentioning

confidence: 99%

Krylov Subspace Recycling for Evolving Structures

Bolten¹,

Sturler²,

Hahn³

2020

Preprint

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Krylov subspace recycling is a powerful tool when solving a long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these series appear naturally, as typically hundreds or thousands of optimization steps are needed with only small changes in the geometry.In this setting, however, applying Krylov subspace recycling can be a difficult task. As the geometry evolves, in general, so does the finite element mesh defined on or representing this geometry, including the numbers of nodes and elements and element connectivity. This is especially the case if re-meshing techniques are used. As a result, the number of algebraic degrees of freedom in the system changes, and in general the linear system matrices resulting from the finite element discretization change size from one optimization step to the next.Changes in the mesh connectivity also lead to structural changes in the matrices. In the case of remeshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. Obviously, this prevents us from any straightforward mapping of the approximate invariant

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Section: Krylov Subspace Recyclingmentioning

confidence: 99%

Krylov Subspace Recycling for Evolving Structures

Bolten¹,

Sturler²,

Hahn³

2020

Preprint

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“…19 The techniques mentioned above were enhanced for nonlinear application problems, for example, in further research. 20,21 As a recently published study, we refer to the paper 22 in which Daas et al introduced a method based on singular value decomposition. Moreover, we note that a block Krylov method can be used together with convergence acceleration methods, although it is a popular technique for a multiple right-hand side problem in itself.…”

Section: Introductionmentioning

confidence: 99%

Convergence acceleration of preconditioned conjugate gradient solver based on error vector sampling for a sequence of linear systems

Iwashita

Ikehara

Fukaya

et al. 2023

Numerical Linear Algebra App

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In this article, we focus on solving a sequence of linear systems that have identical (or similar) coefficient matrices. For this type of problem, we investigate subspace correction (SC) and deflation methods, which use an auxiliary matrix (subspace) to accelerate the convergence of the iterative method. In practical simulations, these acceleration methods typically work well when the range of the auxiliary matrix contains eigenspaces corresponding to small eigenvalues of the coefficient matrix. We develop a new algebraic auxiliary matrix construction method based on error vector sampling in which eigenvectors with small eigenvalues are efficiently identified in the solution process. We use the generated auxiliary matrix for convergence acceleration in the following solution step. Numerical tests confirm that both SC and deflation methods with the auxiliary matrix can accelerate the solution process of the iterative solver. Furthermore, we examine the applicability of our technique to the estimation of the condition number of the coefficient matrix. We also present the algorithm of the preconditioned conjugate gradient method with condition number estimation.

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“…Augmented Krylov Subspace methods showed great promise in handling sequences of linear systems, 27 such as those arising in parametrized PDEs, however, the augmentation of the usual Krylov subspace with data from multiple previous solves led in certain cases to disproportional computational and memory requirements. To alleviate this cost, optimal truncation strategies have been proposed in References 28,29, as well as deflation techniques 30‐32 …”

Section: Introductionmentioning

confidence: 99%

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

Nikolopoulos,

Kalogeris,

Stavroulakis

et al. 2023

Numerical Meth Engineering

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SummaryRecent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost‐efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the ‘exact’ solution of the problem. To address this issue, the present work proposes the use of up‐to‐date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large‐scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD‐2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD‐2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

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

Cited by 11 publications

References 41 publications

Krylov Subspace Recycling for Evolving Structures

Krylov Subspace Recycling for Evolving Structures

Convergence acceleration of preconditioned conjugate gradient solver based on error vector sampling for a sequence of linear systems

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

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