Block Krylov subspace methods for functions of matrices

Frommer, Andreas; Lund, Kathryn; Szyld, Daniel B.

doi:10.1553/etna_vol47s100

Cited by 46 publications

(92 citation statements)

References 37 publications

(82 reference statements)

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“…The basic idea of the Krylov subspace approach is to project the exponential of a large matrix onto a relatively small Krylov subspace where calculating the exponential is significantly less computationally expensive. Recent progress in computational linear algebra has led to efficient Krylov subspace algorithms such as the EXPOKIT software of Sidje [8], restarted Krylov methods [9][10][11][12], block Krylov subspaces [13][14][15][16], time-parallel methods [17,15], the shift-and-invert acceleration [18][19][20] and the adaptive methods [21,22]. The phipm adaptive method of Niesen and Wright [21] has been shown to be the most efficient option for the problems under consideration [23].…”

Section: Introductionmentioning

confidence: 99%

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Gaudreault¹,

Rainwater²,

Tokman³

2018

Journal of Computational Physics

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Matrix exponentialThis paper presents a new algorithm KIOPS for computing linear combinations of ϕ-functions that appear in exponential integrators. This algorithm is suitable for largescale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori. We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm. Crown

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Section: Introductionmentioning

confidence: 99%

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Gaudreault¹,

Rainwater²,

Tokman³

2018

Journal of Computational Physics

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“…The basis generated by Algorithm 1 can hence be interpreted in a block sense and be related to the block Krylov spaces

E B_{k}^{□} false(A, {\overset{C}{true}}_{1} false)

and

E B_{k}^{□} false(B, {\overset{C}{true}}_{2} false)

(cf. section 2 of Frommer et al). However, in the framework of projection methods for matrix equations, the spans of the columns of

V_{k}

and

W_{k}

are often considered as the projection spaces.…”

Section: Structure Exploiting Krylov Methodsmentioning

confidence: 98%

Krylov methods for low‐rank commuting generalized Sylvester equations

Jarlebring

Mele

Palitta

et al. 2018

Numerical Linear Algebra App

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Summary We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator Π with a particular structure. More precisely, the commutators of the matrix coefficients of the operator Π and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low‐rank approximability of this problem, that is, the solution to this matrix equation can be approximated with a low‐rank matrix. Projection methods have successfully been used to solve other matrix equations with low‐rank approximability. We propose a new projection method for this class of matrix equations. The choice of the subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low‐rank commutators. We illustrate the effectiveness of our method by solving large‐scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.

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“…. , q ℓ ]) in the theorem are to be understood in the block-wise sense following the framework defined in [18]. In particular, span{q 1 , .…”

Section: Krylov and Extended Krylov Subspace Based Approximationsmentioning

confidence: 99%

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

Kürschner¹

2019

etna

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The low-rank alternating directions implicit (LR-ADI) iteration is a frequently employed method for efficiently computing low-rank approximate solutions of large-scale Lyapunov equations. In order to achieve a rapid error reduction, the iteration requires shift parameters whose selection and generation is often a difficult task, especially for nonsymmetric coefficients in the Lyapunov equation. This article represents a follow up of Benner et al. [ETNA, 43 (2014[ETNA, 43 ( -2015, pp. 142-162] and investigates self-generating shift parameters based on a minimization principle for the Lyapunov residual norm. Since the involved objective functions are too expensive to evaluate and, hence, intractable, compressed objective functions are introduced which are efficiently constructed from the available data generated by the LR-ADI iteration. Several numerical experiments indicate that these residual minimizing shifts using approximated objective functions outperform existing precomputed and dynamic shift parameter selection techniques, although their generation is more involved.

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Block Krylov subspace methods for functions of matrices

Cited by 46 publications

References 37 publications

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Krylov methods for low‐rank commuting generalized Sylvester equations

Approximate residual-minimizing shift parameters for the low-rank ADI iteration

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