HILUCSI: Simple, Robust, and Fast Multilevel ILU for Large-Scale Saddle-Point Problems from PDEs
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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 saddle-point problems, which often arise from systems of partial differential equations (PDEs). In this work, we introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping, which is specifically designed to take advantage of some special features of such systems. HILUCSI differs from the state-of-the-art ILU techniques in two main aspects. First, HILUCSI leverages the near or partial symmetry of the underlying problems and the inherent block structures of multilevel ILU to improve robustness and to simplify the treatment of indefinite systems. Second, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping to improve the efficiency for large-scale problems from PDEs. We demonstrate the effectiveness of HILUCSI for a number of benchmark problems, including those from mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations. We also compare its performance with ILUPACK, the supernodal ILUTP in SuperLU, and multithreaded direct solvers in PARDISO and MUMPS.
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