Abstract:

Sequences of sparse linear systems arise in many applications, including iterative methods for PDE-constrained optimization. In such cases, inexact solves are often su cient to guarantee convergence, wherein a forcing sequence defines modest solver tolerances such that the computed solutions satisfy. Krylov-subspace recycling methods accelerate convergence for such problems by reusing (truncated) information generated during the solution of previous linear systems. In particular, they search for solutions in the subspace Km + Y, where the subspace Y contains information from previous solves and `augments'' the Krylov subspace Km. Existing recycling techniques (e.g., de ation, optimal truncation) are tailored to improve convergence toward exact solutions; as a result, they do not always realize convergence acceleration when inexact solutions are sought. Instead, we propose a recycling technique inspired by model reduction that aims to e ciently compute inexact solutions. In particular, we construct Y by a goal-oriented proper orthogonal decomposition (POD) of previous search directions, where the POD inner product enables eficient computation of solutions in Y.