A residual based error estimate for Leja interpolation of matrix functions

Kandolf, Peter; Ostermann, Alexander; Rainer, Stefan

doi:10.1016/j.laa.2014.04.023

Cited by 7 publications

(15 citation statements)

References 14 publications

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“…On the other hand, it can also be beneficial to make the error estimate only every couple of iterations rather than in each step to save computational cost, see [3]. A second approach for an a posteriori error estimate for the Leja method based on the computation of a residual can be found in [10]. This procedure can also be used here.…”

Section: Early Termination Criterionmentioning

confidence: 99%

The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

Caliari¹,

Kandolf²,

Ostermann³

et al. 2016

SIAM J. Sci. Comput.

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The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix exponential on a given vector is a typical application. This quantity is required, e.g., in exponential integrators.The Leja method essentially depends on three parameters: the scaling parameter, the location of the interpolation points, and the degree of interpolation. We present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy. Additional aspects that are required for an efficient and reliable implementation are discussed. Numerical examples illustrating the performance of our Matlab code are included.Mathematics Subject Classification (2010): 65F60, 65D05, 65F30

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Section: Early Termination Criterionmentioning

confidence: 99%

The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

Caliari¹,

Kandolf²,

Ostermann³

et al. 2016

SIAM J. Sci. Comput.

Self Cite

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“…we computed the action of the generator exponential on functions by means of Leja polynomial interpolation algorithms [32,33], used in exponential integrators for ODEs.…”

Section: Discussionmentioning

confidence: 99%

“…The problem of computing the action of matrix exponentials on vectors has been studied extensively [48][49][50]. Here, we approximate the semigroup action in (14) using the method of Caliari et al [32] and Kandolf et al [33] which approximates the action of the matrix exponential on vectors via polynomial interpolation at Leja nodes. Specifically, the method in [32,33] approximates e tL b by p d (tL) b where p d is a Newton interpolating polynomial of degree d associated with a sequence {ζ 0 , .…”

Section: Approximation Via Leja Interpolationmentioning

confidence: 99%

“…By employing an interpolating polynomial, the method does not require explicit evaluation of the matrix exponential; this is important from a computational efficiency standpoint since, for instance, e tL will generally be a full matrix even if L is sparse. Moreover, the properties of Leja interpolation nodes ensure that the approximation remains well-conditioned at large d. The properties of Leja sequences also allow for efficient iterative approximation refinement by increasing d until an error tolerance is met, using a stopping criterion derived from an a posteriori error estimate to terminate the iterations [33].…”

Section: Approximation Via Leja Interpolationmentioning

confidence: 99%

“…Next, we examine the effects of the continuous part of the spectrum ofw to the numerically computed eigenfunctions of L. As shown in Fig. 2 and Movie 1, the numerical eigenfunctions of L include functions (e.g., z 29 and z 33 shown here) which are highly localized around individual streamlines in the comoving frame with the vortex center, and also exhibit abrupt changes along those streamlines. As a result, the numerical eigenfunctions of this class also have large Dirichlet energy (see Table 1).…”

Section: Koopman Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

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Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Giannakis

Das

2020

Physica D: Nonlinear Phenomena

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We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron-Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space-time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems.

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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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A residual based error estimate for Leja interpolation of matrix functions

Cited by 7 publications

References 14 publications

The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Limited‐memory polynomial methods for large‐scale matrix functions

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