A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions

Eiermann, Michael; Ernst, Oliver G.

doi:10.1137/050633846

Cited by 127 publications

(165 citation statements)

References 29 publications

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“…The few theoretical results concerning the convergence of the restarted Arnoldi method for matrix functions available in the literature [2,10] are based on approximation theory and make use of bounds for the error of interpolating polynomials for certain classes of analytic functions, using the connection between Krylov subspace methods and polynomial interpolation (as explained in, e.g., [12,30]). Here we take a different approach, using the intimate relation between Arnoldi's method for matrix functions and FOM for families of shifted linear systems [34], as well as a similar relation with the "shifted GMRES" method from [15].…”

Section: 4)mentioning

confidence: 99%

“…To overcome this problem a number of restarting approaches have been proposed in the literature, where-similarly to the techniques for linear systems-after a certain number of iterations the Arnoldi basis is discarded and a new Arnoldi cycle is started to approximate the error of the last iterate, cf. [1,2,10,11,16,23,35]. While much work has been devoted to tuning the methods towards numerical stability and efficiency, there are only few theoretical results concerning the convergence of these methods.…”

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confidence: 99%

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Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices

Frommer

Güttel²,

Schweitzer³

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. To approximate f (A)b-the action of a matrix function on a vector-by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating f on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contains important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices A and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guarantees convergence for the same classes of functions and any restart length if A is not necessarily Hermitian but positive real.

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Section: 4)mentioning

confidence: 99%

mentioning

confidence: 99%

Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices

Frommer

Güttel²,

Schweitzer³

2014

SIAM J. Matrix Anal. & Appl.

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“…For particularly challenging problems, however, an unacceptably large approximation space may be required to obtain a sastisfactory approximation. This difficulty has lead to the study of enhancement techniques that aim either at enriching the approximation space or at making the overall procedure less expensive [4], [16], [18], [22], [33], [39], [40], [41], [46], [54].…”

mentioning

confidence: 99%

A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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Abstract. For large square matrices A and functions f , the numerical approximation of the action of f (A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f (A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

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“…Due to the equivalence with the Arnoldi method, the algorithm may suffer from the typical disadvantages of the Arnoldi method, for instance, the fact that the computation time per iteration increases with the iteration number. The standard approach to resolve this issue is by using restarting [7], which we leave for future work. We note that the technique we have presented is in principle applicable also to ODEs with several perturbation variables.…”

Section: An a Posteriori Error Estimate For The Krylov Approximationmentioning

confidence: 99%

Krylov Approximation of Linear ODEs with Polynomial Parameterization

Koskela

Jarlebring

Hochstenbach³

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. We propose a new numerical method to solve linear ordinary differential equations of the type ∂u ∂t (t, ε) = A(ε) u(t, ε), where A : C → C n×n is a matrix polynomial with large and sparse matrix coefficients. The algorithm computes an explicit parameterization of approximations of u(t, ε) such that approximations for many different values of ε and t can be obtained with a very small additional computational effort. The derivation of the algorithm is based on a reformulation of the parameterization as a linear parameter-free ordinary differential equation and on approximating the product of the matrix exponential and a vector with a Krylov method. The Krylov approximation is generated with Arnoldi's method and the structure of the coefficient matrix turns out to be independent of the truncation parameter so that it can also be interpreted as Arnoldi's method applied to an infinite dimensional matrix. We prove the superlinear convergence of the algorithm and provide a posteriori error estimates to be used as termination criteria. The behavior of the algorithm is illustrated with examples stemming from spatial discretizations of partial differential equations.

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A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions

Cited by 127 publications

References 29 publications

Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices

Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices

A new investigation of the extended Krylov subspace method for matrix function evaluations

Krylov Approximation of Linear ODEs with Polynomial Parameterization

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