Rational Krylov Approximation of Matrix Functions and Applications

Some problems in scientific computing, like the forward simulation of electromagnetic waves in geophysical prospecting, can be
solved via approximation of f(A)b, the action of a large matrix function f(A) onto a vector b. Iterative methods based on rational Krylov
spaces are powerful tools for these computations, and the choice of parameters in these methods is an active area of research.
We provide an overview of different approaches for obtaining optimal parameters, with an emphasis on the exponential and resolvent function, and the square root. We will discuss applications of the rational Arnoldi method for iteratively generating near-optimal absorbing boundary layers for indefinite Helmholtz problems, and for rational least squares vector fitting.