A study of defect-based error estimates for the Krylov approximation of $\varphi$-functions
Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential ${\rm e}^{tA}v$, is extended to the case of associated $\varphi$-functions (which occur within the class of exponential integrators). In particular, a~posteriori erro...
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| Format: | Journal Article |
| Language: | English |
| Published: |
31-01-2020
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| Subjects: | |
| Online Access: | Get full text |
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| Summary: | Prior recent work, devoted to the study of polynomial Krylov techniques for
the approximation of the action of the matrix exponential ${\rm e}^{tA}v$, is
extended to the case of associated $\varphi$-functions (which occur within the
class of exponential integrators). In particular, a~posteriori error bounds and
estimates, based on the notion of the defect (residual) of the Krylov
approximation are considered. Computable error bounds and estimates are
discussed and analyzed. This includes a new error bound which favorably
compares to existing error bounds in specific cases. The accuracy of various
error bounds is characterized in relation to corresponding Ritz values of $A$.
Ritz values yield properties of the spectrum of $A$ (specific properties are
known a~priori, e.g. for Hermitian or skew-Hermitian matrices) in relation to
the actual starting vector $v$ and can be computed. This gives theoretical
results together with criteria to quantify the achieved accuracy on the run.
For other existing error estimates the reliability and performance is studied
by similar techniques. Effects of finite precision (floating point arithmetic)
are also taken into account. |
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| DOI: | 10.48550/arxiv.2001.11922 |