  This work presents a new Krylov-subspace-recycling method for efficiently
solving sequences of linear systems of equations characterized by varying
right-hand sides and symmetric-positive-definite matrices. As opposed to
typical truncation strategies used in recycling such as deflation, we propose a
truncation method inspired by goal-oriented proper orthogonal decomposition
(POD) from model reduction. This idea is based on the observation that model
reduction aims to compute a low-dimensional subspace that contains an accurate
solution; as such, we expect the proposed method to generate a low-dimensional
subspace that is well suited for computing solutions that can satisfy inexact
tolerances. In particular, we propose specific goal-oriented POD `ingredients'
that align the optimality properties of POD with the objective of
Krylov-subspace recycling. To compute solutions in the resulting `augmented'
POD subspace, we propose a hybrid direct/iterative three-stage method that
leverages 1) the optimal ordering of POD basis vectors, and 2) well-conditioned
reduced matrices. Numerical experiments performed on solid-mechanics problems
highlight te benefits of the proposed method over existing approaches for
Krylov-subspace recycling.
