Scalable Linear Time Dense Direct Solver for 3-D Problems without Trailing Sub-Matrix Dependencies

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$\mathrm{O}(N^{3})$to$\mathrm{O}(N)$. 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$\mathrm{O}(N)$. 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.