Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large-scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large-scale saddle-point problems, which often arise from systems of PDEs. We introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping. As a multilevel preconditioner, HILUCSI statically and dynamically permutes individual rows and columns to the next level for deferred factorization. Unlike ILUPACK, HILUCSI applies symmetric preprocessing techniques at the top levels but always uses unsymmetric preprocessing and unsymmetric factorization at the coarser levels. The deferring combined with mixed preprocessing enabled a unified treatment for nearly or partially symmetric systems and simplified the implementation by avoiding mixed $1\times 1$ and $2\times 2$ pivots for symmetric indefinite systems. We show that this combination improves robustness for indefinite systems without compromising efficiency. Furthermore, to enable superior efficiency for large-scale systems with millions or more unknowns, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping. We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier–Stokes equations. We also compare its performance with ILUPACK and the supernodal ILUTP in SuperLU.

中文翻译：

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HILUCSI：用于 PDE 中大规模鞍点问题的简单、稳健且快速的多级 ILU

不完全分解是 Krylov 子空间方法的一种广泛使用的预处理技术，用于解决大规模稀疏线性系统。它的多级变体（例如 ILUPACK）对于许多对称或非对称线性系统比传统的单级不完全 LU（或 ILU）技术更稳健。然而，对于一些通常由偏微分方程系统产生的大规模鞍点问题，以前的多级 ILU 技术仍然缺乏鲁棒性和效率。我们介绍了HILUCSI，或具有面向可扩展性和基于逆的丢弃的分层不完全 LU-Crout. 作为多级预处理器，HILUCSI 静态和动态地将单个行和列排列到下一个级别以进行延迟分解。与 ILUPACK 不同，HILUCSI 在顶层应用对称预处理技术，但始终在较粗的级别使用非对称预处理和非对称分解。延迟与混合预处理相结合，可以对几乎或部分对称的系统进行统一处理，并通过避免混合来简化实现$1\times 1$ 和 $2\times 2$对称不定系统的枢轴。我们表明，这种组合在不影响效率的情况下提高了不确定系统的鲁棒性。此外，为了为具有数百万或更多未知数的大规模系统提供卓越的效率，HILUCSI 引入了面向可扩展性的丢弃以及基于逆丢弃的变体。我们证明了 HILUCSI 对数十个基准问题的有效性，包括来自泊松方程、斯托克斯方程和 Navier-Stokes 方程的混合公式的问题。我们还将其性能与 ILUPACK 和 SuperLU 中的超节点 ILUTP 进行了比较。