We introduce a new class of preconditioners to enable flexible GMRES to find a least-squares solution, and potentially the pseudoinverse solution, of large-scale sparse, asymmetric, singular, and potentially inconsistent systems. We develop the preconditioners based on a new observation that generalized inverses (i.e., $\boldsymbol{A}^{g}\in\{\boldsymbol{G}\mid\boldsymbol{A}\boldsymbol{G}\boldsymbol{A}=\boldsymbol{A}\}$) enable the preconditioned Krylov subspaces (KSP) to converge in a single step. We then compute an approximate generalized inverse (AGI) efficiently using a hybrid incomplete factorization (HIF), which combines multilevel incomplete LU with rank-revealing QR on its final Schur complement. We define the criteria of $\epsilon$-accuracy and stability of AGI to guarantee the convergence of preconditioned GMRES for consistent systems. For inconsistent systems, we fortify HIF with iterative refinement to obtain HIFIR, which allows accurate computations of the null-space vectors. By combining the two techniques, we then obtain a new solver, called PIPIT, for obtaining the pseudoinverse solutions for systems with low-dimensional null spaces. We demonstrate the robustness of HIF and HIFIR and show that they improve both accuracy and efficiency of the prior state of the art by orders of magnitude for systems with up to a million unknowns.

翻译：我们引入了新型的前提条件,使灵活GMRES能够找到最不平方的解决方案,并有可能是大规模分散、不对称、单一和可能不一致的系统(KSP)的假反面解决方案。然后,我们根据一种普遍反向的新观察(即,$\boldsymbol{A ⁇ g ⁇ in ⁇ boldsymbol{G ⁇ mid\boldsymbol{A ⁇ boldsymbol{G ⁇ boldsymbol{A ⁇ boldsymbol{A ⁇ $$$),来开发一个先决条件的前提条件性解决方案。然后,我们用一种混合的不完全因子化(HIF),将多级不完整的LU与最后Schur补充的递增等级QR相结合,将多级的GIF的准确度和低度系统稳定起来,以确保具有一致性的系统的统一性。为了不一致性的系统,我们将HIF的HIF的准确性与两次的硬度组合,我们用静式的硬度的硬度技术来进行硬度的硬度的硬度测试。