A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

Zhong, Hong-Xiu; Gu, Xian-Ming

doi:10.1016/j.camwa.2019.03.017

Cited by 8 publications

(4 citation statements)

References 43 publications

(46 reference statements)

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“…Finally we note that a large number of tailored Krylov methods is proposed to deal with shifted systems that require preconditioning and we only refer to [5,14,24,25,29].…”

Section: Methodsmentioning

confidence: 99%

A FEM for an optimal control problem subject to the fractional Laplace equation

Dohr

Kahle

Rogovs

et al. 2019

Calcolo

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We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed.Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

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“…Finally we note that a large number of tailored Krylov methods is proposed to deal with shifted systems that require preconditioning and we only refer to [5,14,24,25,29].…”

Section: Methodsmentioning

confidence: 99%

A FEM for an optimal control problem subject to the fractional Laplace equation

Dohr

Kahle

Rogovs

et al. 2019

Calcolo

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show abstract

“…If the matrix A is large and sparse, such that the mapping C n v → Av ∈ C n is available as an efficient subroutine, then natural choices of methods for the computations (4), (6,7) are iterative methods based on Krylov subspaces, in particular because of their shift invariance. Successful examples are restarted GMRES [15], [31], BiCGStab [14], restarted FOM [25]. Particularly tailored for the IRKA algorithm and the systems (6,7) is the preconditioned BiCG [1].…”

Section: Solution Methodsmentioning

confidence: 99%

Parallel Solver for Shifted Systems in a Hybrid CPU--GPU Framework

Bosner¹,

Bujanović²,

Drmač³

2018

SIAM J. Sci. Comput.

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This paper proposes a combination of a hybrid CPU-GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems (A − σI)X = B with large number of complex shifts σ and multiple right-hand sides. Such problems often appear e.g. in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as well as when computing pseudospectra and structured pseudospectra, or solving large linear systems of ordinary differential equations. The proposed algorithm first jointly reduces the general full n × n matrix A and the n × m full right-hand side matrix B to the controller Hessenberg canonical form that facilitates efficient solution: A is transformed to a so-called m-Hessenberg form and B is made uppertriangular. This is implemented as blocked highly parallel CPU-GPU hybrid algorithm; individual blocks are reduced by the CPU, and the necessary updates of the rest of the matrix are split among the cores of the CPU and the GPU. To enhance parallelization, the reduction and the updates are overlapped. In the next phase, the reduced m-Hessenberg-triangular systems are solved entirely on the GPU, with shifts divided into batches. The benefits of such load distribution are demonstrated by numerical experiments. In particular, we show that our proposed implementation provides an excellent basis for efficient implementations of computational methods in systems and control theory, from evaluation of transfer function to the interpolatory model reduction. *

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“…Table 6 describes some details of these QCD matrices. It should be also mentioned that the values of κ c are considered as suggested in [56]. Table 7 reports the number of matrix-vector products and CPU time when m = 120, k = 2, s = 3 and l = 3.…”

Section: Examplementioning

confidence: 99%

On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems

et al. 2020

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The problem of shifted linear systems is an important and challenging issue in a number of research applications. Krylov subspace methods are effective techniques for different kinds of this problem due to their advantages in large and sparse matrix problems. In this paper, two new block projection methods based on respectively block FOM and block GMRES are introduced for solving sequences of shifted linear systems. We first express the original problem explicitly by a sequence of Sylvester matrix equations whose coefficient matrices are obtained from the shifted linear systems. Then, we show the restarted shifted block FOM (rsh-BFOM) method and derive some of its properties. We also present a framework for the restarted shifted block GMRES (rsh-BGMRES) method. In this regard, we describe two variants of rsh-BGMRES, including: 1) rsh-BGMRES with an unshifted base system that applies a fixed unshifted base system and 2) rsh-BGMRES with a variable shifted base system in which the base block system can change after restart. Furthermore, we consider the use of deflation techniques for improving the performance of the rsh-BFOM and rsh-BGMRES methods. Finally, some numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

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A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

Cited by 8 publications

References 43 publications

A FEM for an optimal control problem subject to the fractional Laplace equation

A FEM for an optimal control problem subject to the fractional Laplace equation

Parallel Solver for Shifted Systems in a Hybrid CPU--GPU Framework

On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems

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