HILUCSI: Simple, Robust, and Fast Multilevel ILU with Mixed Symmetric and Unsymmetric Processing
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Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large-scale sparse linear systems. Its multilevel variants, such as those in ILUPACK and ARMS, have been shown to be more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques. However, multilevel ILU still lacked robustness and efficiency for some large-scale, nearly or partially symmetric, ill-conditioned, or indefinite systems, which often arise from systems of partial differential equations (PDEs). In this work, we address this issue by introducing HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping. HILUCSI differs from the state-of-the-art multilevel ILU in three aspects. First, to strike a balance between robustness and efficiency, HILUCSI leverages some proven techniques for symmetric and unsymmetric matrices at the top and lower levels, respectively, for nearly or partially symmetric systems. Second, to simplify the treatment of indefinite systems, HILUCSI statically defers tiny diagonal entries to the next level during preprocessing instead of relying on dynamic partial or Bunch- Kauffman pivoting. Third, to improve the effectiveness for large-scale problems from PDEs, HILUCSI introduces a scalability-oriented dropping in conjunction with a modified inverse-based dropping. We demonstrate the robustness and efficiency of HILUCSI for benchmark problems from a wide range of applications against ILUPACK, the supernodal ILUTP in SuperLU, and multithreaded direct solvers in PARDISO and MUMPS.
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