Seminar in Numerical Analysis: Mike Botchev (University of Twente)
We review some recent advances in Krylov subspace methods to compute an action of the matrix exponential on a given vector. In particular, we briefly discuss residual-based and shift-and-invert Krylov subspace methods in the context of space-discretized 3D Maxwell's equations. In our limited experience, a conventional time stepping, where actions of the matrix exponential have to be repeatedly computed at every time step, are usually inefficient. We therefore discuss an alternative approach, based on block Krylov subspaces, where just a couple evaluations of the matrix exponential suffice to solve the problem for the whole time interval.
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