  Factorization of large dense matrices are ubiquitous in engineering and data
science applications, e.g. preconditioners for iterative boundary integral
solvers, frontal matrices in sparse multifrontal solvers, and computing the
determinant of covariance matrices. HSS and $\mathcal{H}^2$-matrices are
hierarchical low-rank matrix formats that can reduce the complexity of
factorizing such dense matrices from $\mathcal{O}(N^3)$ to $\mathcal{O}(N)$.
For HSS matrices, it is possible to remove the dependency on the trailing
matrices during Cholesky/LU factorization, which results in a highly parallel
algorithm. However, the weak admissibility of HSS causes the rank of
off-diagonal blocks to grow for 3-D problems, and the method is no longer
$\mathcal{O}(N)$. On the other hand, the strong admissibility of
$\mathcal{H}^2$-matrices allows it to handle 3-D problems in $\mathcal{O}(N)$,
but introduces a dependency on the trailing matrices. In the present work, we
pre-compute the fill-ins and integrate them into the shared basis, which allows
us to remove the dependency on trailing-matrices even for
$\mathcal{H}^2$-matrices. Comparisons with a block low-rank factorization code
LORAPO showed a maximum speed up of 4,700x for a 3-D problem with complex
geometry.
