W. Auzinger,T. Jawecki, O. Koch:

"Computable upper error bounds for Krylov subspace approximations to matrix exponentials";

Talk: SciCADE 2019, International Conference on Scientific Computation and Differential Equations, Innsbruck; 2019-07-22 - 2019-07-26.

The reliability of well-known a posteriori error estimates for Krylov approximations to the matrix

exponential is discussed. In many cases error estimates constitute upper error bounds and even can

be tightened. Additionally, a new defect-based a posteriori error estimate is introduced. This error

estimate constitutes an upper norm bound on the error without further assumptions and can be

computed during the construction of the Krylov subspace with nearly no computational effort.

The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation

is smaller than a given tolerance. Apart from step size control, the new upper error bound can be

used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the

problem in a single time step with the required accuracy.

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