Residual and Restarting in Krylov Subspace Evaluation of the $\varphi$ Function
Mikhail Aleksandrovich Botchev, Leonid Aronovich Knizhnerman, Eugene E. Tyrtyshnikov · SIAM Journal on Scientific Computing · 2021
An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains, and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. We analyze residual convergence and prove, for matrices with numerical range in the stable complex half-plane, that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDEs, in particular, diffusion and convection-diffusion problems.