Nested Krylov Methods for Shifted Linear Systems

and {ωk}k=1 ∈ C is a sequence of n distinct shifts. For example, shifted linear systems arise in model order reduction as well as in the geophysical exploration of both acoustic and elastic waves. In our application, we focus on wave propagation through elastic media in a frequency-domain formulation. This formulation has specific advantages when modeling visco-elastic effects. In order to improve the imaging of the earth crust, so-called full waveform inversion is computed which is an optimization problem at multiple wave frequencies. Therefore, the grid size must be small enough to describe the wave, which for high frequencies results in very large shifted linear systems of the form (1). In principle, a sequence of shifted systems (1) can be solved almost at the cost of a single solve using so-called shifted Krylov methods. These methods exploit the property that Krylov subspaces are invariant under arbitrary diagonal shifts ω to the matrix A, i.e.,