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Computable upper error bounds for Krylov approximations to matrix exponentials and associated φ -functions.

Tobias Jawecki1, Winfried Auzinger1, Othmar Koch2

  • 11Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10/E101, 1040 Vienna, Austria.

BIT. Numerical Mathematics
|March 13, 2020
PubMed
Summary

A new, rigorous upper bound for Krylov approximations of the matrix exponential is developed. This error estimate is computationally efficient and asymptotically correct for time-stepping methods, improving accuracy in numerical simulations.

Keywords:
A posteriori error estimationKrylov approximationMatrix exponentialUpper bound

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Area of Science:

  • Numerical Analysis
  • Scientific Computing
  • Matrix Computations

Background:

  • Krylov subspace methods are widely used for approximating the matrix exponential.
  • Existing error estimates for these approximations are often asymptotic and lack rigorous guarantees.
  • Accurate and reliable error estimation is crucial for the stability and efficiency of numerical algorithms.

Purpose of the Study:

  • To derive a posteriori error estimates for Krylov approximations of the matrix exponential.
  • To establish these estimates as rigorous upper bounds on the approximation error.
  • To investigate the properties and applications of these bounds, particularly in time-stepping schemes.

Main Methods:

  • Derivation of an a posteriori error estimate based on the defect (residual) of the Krylov approximation.
  • Theoretical analysis to prove the estimate constitutes a rigorous upper bound.
  • Investigation of asymptotic correctness for time-stepping applications as the time step tends to zero.
  • Extension of the methodology to approximations of φ-functions and improved Krylov methods.

Main Results:

  • A novel, rigorous upper bound for Krylov matrix exponential approximations is established.
  • The bound is computationally economical, computable within the Krylov space.
  • The bound is shown to be asymptotically correct for small time steps in time-stepping contexts.
  • The approach is successfully extended to related problems involving φ-functions.

Conclusions:

  • The derived error estimate provides a reliable and efficient way to quantify the accuracy of Krylov approximations.
  • The rigorous upper bound offers significant advantages over existing asymptotic approximations.
  • The findings have direct implications for improving the stability and accuracy of time integration algorithms using matrix exponentials.