Unprojected recycled block Krylov subspace methods for shifted systems

The use of block Krylov subspace methods for computing the solution to a sequence of shifted linear systems using subspace recycling was first proposed in [Soodhalter, SISC 2016], where a recycled shifted block GMRES algorithm (rsbGMRES) was proposed. Such methods use the equivalence of the shifted system to a Sylvester equation and exploit the shift invariance of the block Krylov subspace generated from the Sylvester operator. This avoids the need for initial residuals to span the same subspace and allows for a viable restarted Krylov subspace method with recycling for solving sequences of shifted systems. In this paper we propose to develop these types of methods using unprojected Krylov subspaces. In doing so we show how one can overcome the difficulties associated with developing methods based on projected Krylov subspaces such as rsbGMRES, while also allowing for practical methods to fit within a well known residual projection framework. In addition, unprojected methods are known to be advantageous when the projector is expensive to apply, making them of significant interest for High-Performance Computing applications. We develop an unprojected rsbFOM and unprojected rsbGMRES. We also develop a procedure for extracting shift dependent harmonic Ritz vectors over an augmented block Krylov subspace for shifted systems yielding an approach for selecting a new recycling subspace after each cycle of the algorithm. Numerical experiments demonstrate the effectiveness of our methods.