  We present a method for updating certain hierarchical factorizations for
solving linear integral equations with elliptic kernels. In particular, given a
factorization corresponding to some initial geometry or material parameters, we
can locally perturb the geometry or coefficients and update the initial
factorization to reflect this change with asymptotic complexity that is
polylogarithmic in the total number of unknowns and linear in the number of
perturbed unknowns. We apply our method to the recursive skeletonization
factorization and hierarchical interpolative factorization and demonstrate
scaling results for a number of different 2D problem setups.
