Shift-invert rational Krylov method for evolution equations

Abstract

In order to obtain the numerical solution of evolution equations which arise in various fields of science and technology, the computation of matrix functions called \(\phi\)-functions is required. This paper proposes a new method called the shift-invert rational Krylov method for the computation of matrix \(\phi\)-functions. This method efficiently computes the matrix \(\phi\)-functions and allows the appropriate parameters to be simply determined. References B. Beckermann and L. Reichel, Error estimation and evaluation of matrix functions via the Faber transform. SIAM Journal on Numerical Analysis 47(5) :3849–3883, 2009. doi:10.1137/080741744 M. Crouzeix, Numerical range and functional calculus in Hilbert space. Journal of Functional Analysis 244 :668–690, 2007. doi:10.1016/j.jfa.2006.10.013 T. Gockler, Rational Krylov subspace methods for \(\phi \)-functions in exponential integrators. Karlsruher Instituts fur Technologie , 2014, Ph.D. thesis. http://d-nb.info/1060425408/34 S. Guttel, Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection. GAMM-Mitteilungen 38(1) :8–31, 2013. doi:10.1002/gamm.201310002 Y. Hashimoto and T. Nodera, Inexact shift-invert Arnoldi method for evolution equations. ANZIAM Journal 58(E) :E1–E27, 2016. doi:10.21914/anziamj.v58i0.10766 M. Hochbruck and A. Ostermann, Exponential integrators. Acta Numerica 19 :209–286, 2010. doi:10.1017/S0962492910000048