Krylov-subspace recycling via the POD-augmented conjugate-gradient algorithm

This work presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by a non-invariant symmetric-positive-definite matrix. 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 inexact solutions. 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 iterative method that leverages (1) the optimal ordering of POD basis vectors, and (2) well-conditioned reduced matrices. Numerical experiments performed on real-world solid-mechanics problems highlight the benefits of the proposed method over standard approaches for Krylov-subspace recycling.