Stefan Güttel: Rational Krylov Methods for 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 the rational Krylov spaces introduced by A. Ruhe in the 1980s are now very popular 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 (in some sense) optimal parameters, with an emphasis on the exponential and resolvent function, and the square root. If time permits, we will discuss a surprising new application of the rational Arnoldi method for iteratively generating near-optimal absorbing boundary layers for indefinite Helmholtz problems.