## Technical Papers Archive

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

**Authors:** Qianxiang Ma, Sameer Deshmukh, and Rio Yokota (Tokyo Institute of Technology)

**Abstract:** HSS and H^2-matrices are hierarchical low-rank matrix formats that can reduce the complexity of factorizing dense matrices from O(N^3) to O(N). For HSS matrices, it is possible to remove the dependency on the diagonal blocks during Cholesky/LU factorization, which results in a highly parallel algorithm. However, the weak admissibility of HSS limits it’s applicability to simple problems in 1-D and 2-D geometries. On the other hand, the strong admissibility of H^2-matrices allows it to handle actual 3-D problems, but introduces the dependency on the diagonal blocks during the factorization. In the present work, we propose a decoupling of the low-rank basis and the Schur complement basis in H^2-matrices, which allows us to remove the dependency on the diagonal blocks. This results in a highly parallel H^2-matrix factorization. We compare with other scalable approximate dense matrix factorization codes such as Lorapo.

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