SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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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.

Organized by         Universidad de Valladolid     IMUVA