The solution of large sparse linear systems is required in many scientific fields such as computational fluid dynamics, computational electromagnetism, computational finance, etc. The computation of the solution of these systems is performed with preconditioned iterative methods, which rely on effective preconditioning schemes. A new class of approximate inverses is proposed, namely incomplete inverse matrices, which are computed using a recursive Schur complement-based approach. This class of approximate inverses is based on a priori knowledge of a sparsity pattern. In order to have finer control over the density of the proposed approximate inverse, especially in the case of three-dimensional problems, on-the-fly filtration is used, resulting in substantial reduction in the number of nonzero elements. Implementation details and analysis for computing the proposed scheme are given. Numerical results depicting the effectiveness and applicability of the proposed scheme are also provided.

中文翻译：

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不完全逆矩阵

计算流体动力学、计算电磁学、计算金融学等许多科学领域都需要大型稀疏线性系统的求解。这些系统的解的计算是通过依赖于有效的预处理方案的预处理迭代方法进行的。提出了一类新的近似逆矩阵，即不完全逆矩阵，它们是使用基于递归 Schur 补码的方法计算的。这类近似逆是基于稀疏模式的先验知识。为了更好地控制所提出的近似逆的密度，特别是在三维问题的情况下，使用了动态过滤，从而大大减少了非零元素的数量。给出了计算所提出方案的实现细节和分析。还提供了描述所提出方案的有效性和适用性的数值结果。