  Krylov subspace recycling is a powerful tool for solving long series of
large, sparse linear systems that change slowly. In PDE constrained shape
optimization, these appear naturally, as hundreds or more optimization steps
are needed with only small changes in the geometry. In this setting, however,
applying Krylov subspace recycling can be difficult. As the geometry evolves,
so does the finite element mesh, especially if re-meshing is needed. As a
result, the number of algebraic degrees of freedom in the system may change
from one optimization step to the next, and with it the size of the finite
element system matrix. Changes in the mesh also lead to structural changes in
the matrices. In the case of remeshing, even if the geometry changes only a
little, the corresponding mesh might differ substantially from the previous
one. This prevents any straightforward mapping of the approximate invariant
subspace of the linear system matrix (the focus of recycling in this paper)
from one step to the next; similar problems arise for other selected subspaces.
We present an algorithm for general meshes to map an approximate invariant
subspace of the system matrix for the previous optimization step to an
approximate invariant subspace of the system matrix for the current
optimization step. We exploit the map from coefficient vectors to finite
element functions on the mesh combined with function approximation on the
finite element mesh. In addition, we develop a straightforward warm-start
adaptation of the Krylov-Schur algorithm [G.W. Stewart, SIAM J. Matrix Anal.
Appl. 23, 2001] to improve the approximate invariant subspace at the start of a
new optimization step if needed. We demonstrate the effectiveness of our
approach numerically with several proof of concept studies for a specific
meshing technique.
