Show full list of Invited Contributions to Minisymposia Show talk context (MS14)

__A Parallel Rational Krylov Subspace Method for the Approximation of $\varphi$-Functions in Exponential Integrators__

T. Göckler and V. Grimm

**Abstract**

The efficient approximation of the matrix $\varphi$-functions is an important task in the application of exponential integrators. Recent advances have shown that rational Krylov methods have a great advantage over standard Krylov methods for large matrices $A$ with a huge field-of-values in the left complex half-plane.
We consider the approximation of $\varphi(A)v$ in the space
$
\text{span}\left\{(z_{-m}I-A)^{-1}v,\ldots,(z_mI-A)^{-1}v\right\}
$
with equidistant poles on the line $\text{Re}(z)=\gamma>0$. It is possible to solve the occurring linear systems in parallel by using a suitable parallel implementation. In this way, we achieve a significant speed-up compared to a serial implementation.
We present error bounds that predict a uniform convergence. This is a fundamental property for the successful application of this parallel rational Krylov method in exponential integrators. The advantages and efficiency of our method are illustrated by several numerical experiments.