Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

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

doi:10.1137/19m1255847

Cited by 19 publications

(24 citation statements)

References 33 publications

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“…There are several possible definitions for the subspace K m (A, R 0 ). In this article, we consider the classical definition [3,4,5,8,9]; i.e., let X 0 be an arbitrary initial approximation to the solution of the system (1) and R 0 := B − AX 0 its associated initial residual; let S ⊆ R s×s be a vector subspace containing the identity I s and closed under matrix multiplication and conjugate transposition, then the mth block Krylov subspace (corresponding to A, R 0 of system (1), and S) is defined as K S m (A, R 0 ) := span S {R 0 , AR 0 , . .…”

Section: Block-polynomial Formulation Of Block Cgmentioning

confidence: 99%

A posteriori superlinear convergence bounds for block conjugate gradient

Schaerer¹,

Szyld²,

Torres³

2021

Preprint

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In this paper, we extend to the block case, the a posteriori bound showing superlinear convergence of Conjugate Gradients developed in [J. Comput. Applied Math., 48 (1993), pp. 327-341], that is, we obtain similar bounds, but now for block Conjugate Gradients. We also present a series of computational experiments, illustrating the validity of the bound developed here, as well as the bound from [SIAM Review, 47 (2005), pp. 247-272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity, and how this onset depends on the eigenvalue distribution and the block size.

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Section: Block-polynomial Formulation Of Block Cgmentioning

confidence: 99%

A posteriori superlinear convergence bounds for block conjugate gradient

Schaerer¹,

Szyld²,

Torres³

2021

Preprint

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“…In both cases, a colinear relationship between r m (0) and r m (t) is obtained, 1 allowing them to perform restarts for shifted systems from a single Krylov space at each new cycle. Theorem 4.1 from Frommer et al [46] comprises both results by formulating a cospatial relationship for block Krylov methods. We state here just the result for FOM.…”

Section: Quadrature-based Restarting For Cauchy-stieltjes Functionsmentioning

confidence: 99%

“…Differential equations on four-dimensional computational domains may arise, for example, from space-time formulations of time-dependent PDEs. Usually, such a formulation does not lead to matrix functions, but there is an important exception: in lattice QCD the solutions of linear systems involving the matrix square root or the matrix sign function are needed [16,46,67,104]. Except for toy examples, it is a futile attempt to apply sparse direct solvers to such matrices.…”

Section: Differential Equationsmentioning

confidence: 99%

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Limited‐memory polynomial methods for large‐scale matrix functions

Güttel,

Kressner,

Lund

2020

GAMM-Mitteilungen

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Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited‐memory methods for the approximation of the action of a large‐scale matrix function on a vector. Emphasis is put on polynomial methods, whose memory requirements are known or prescribed a priori. Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail. An overview of existing software is also given, as well as a discussion of challenging open problems.

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“…Thanks to the equivalence of our tensor function definition with the f ( A ) B problem, we have a number of options for computing

f (𝒜) * ℬ

, such as AlMohy, Frommer, Frommer, and Simoncini . We focus on adapting the block Krylov subspace methods (KSMs) from Frommer et al, and in light of the so‐called “curse of dimensionality,” we also present a computational complexity analysis for this algorithm in the tensor function context.…”

Section: Introductionmentioning

confidence: 99%

The tensor t‐function: A definition for functions of third‐order tensors

Lund

2020

Numerical Linear Algebra App

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A definition for functions of multidimensional arrays is presented. The definition is valid for third-order tensors in the tensor t-product formalism, which regards third-order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third-order tensors and computed for a small-scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided. K E Y W O R D Sblock circulant matrices, matrix functions, multidimensional arrays, network analysis, tensor t-product, tensors

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Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

Cited by 19 publications

References 33 publications

A posteriori superlinear convergence bounds for block conjugate gradient

A posteriori superlinear convergence bounds for block conjugate gradient

Limited‐memory polynomial methods for large‐scale matrix functions

The tensor t‐function: A definition for functions of third‐order tensors

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