  The hierarchical interpolative factorization for elliptic partial
differential equations is a fast algorithm for approximate sparse matrix
inversion in linear or quasilinear time. Its accuracy can degrade, however,
when applied to strongly ill-conditioned problems. Here, we propose a simple
modification that can significantly improve the accuracy at no additional
asymptotic cost: applying a block Jacobi preconditioner before each level of
skeletonization. This dramatically limits the impact of the underlying system
conditioning and enables the construction of robust and highly efficient
preconditioners even at quite modest compression tolerances. Numerical examples
demonstrate the performance of the new approach.
