  A hierarchical solver is proposed for solving sparse ill-conditioned linear
systems in parallel. The solver is based on a modification of the LoRaSp
method, but employs a deferred-compression technique, which provably reduces
the approximation error and significantly improves efficiency. Moreover, the
deferred-compression technique introduces minimal overhead and does not affect
parallelism. As a result, the new solver achieves linear computational
complexity under mild assumptions and excellent parallel scalability. To
demonstrate the performance of the new solver, we focus on applying it to solve
sparse linear systems arising from ice sheet modeling. The strong anisotropic
phenomena associated with the thin structure of ice sheets creates serious
challenges for existing solvers. To address the anisotropy, we additionally
developed a customized partitioning scheme for the solver, which captures the
strong-coupling direction accurately. In general, the partitioning can be
computed algebraically with existing software packages, and thus the new solver
is generalizable for solving other sparse linear systems. Our results show that
ice sheet problems of about 300 million degrees of freedom have been solved in
just a few minutes using a thousand processors.
