We present a general framework for preconditioning Hermitian positive definite linear systems based on the Bregman log determinant divergence. This divergence provides a measure of discrepancy between a preconditioner and a target matrix. Given an approximate factorisation of a target matrix, the proposed framework tells us how to construct a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition. This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values. We describe a procedure for determining these \emph{Bregman directions} and prove that preconditioners constructed in this way are minimisers of the aforementioned divergence. Finally, we demonstrate using several numerical examples how the proposed preconditioner performs in terms of convergence of the preconditioned conjugate gradient method (PCG). For the examples we consider, an incomplete Cholesky preconditioner can be greatly improved in this way, and in some cases only a modest low-rank compensation term is required to obtain a considerable improvement in convergence. We also consider matrices arising from interior point methods for linear programming that do not admit such an incomplete factorisation by default, and present a robust incomplete Cholesky preconditioner based on the proposed methodology. The results highlight that the choice of truncation is critical for ill-conditioned matrices. We show numerous examples where PCG converges to a small tolerance by using the proposed preconditioner, whereas PCG with a SVD-based preconditioner fails to do so.

翻译：我们提出了一种基于Bregman对数行列式散度的Hermitian正定线性系统预条件通用框架。该散度衡量了预条件子与目标矩阵之间的差异程度。给定目标矩阵的近似分解，该框架指导我们如何构造通常为不定性的分解误差的低秩近似。由此得到的预条件子是近似分解给出的Hermitian正定矩阵与低秩矩阵之和。值得注意的是，该低秩项通常并非通过截断奇异值分解获得。该框架引出了一种新的截断方法，其中主方向不基于奇异值的大小。我们描述了确定这些Bregman方向的过程，并证明以此方式构造的预条件子是最小化前述散度的最优解。最后，通过多个数值算例展示了所提预条件子在预条件共轭梯度法（PCG）收敛性能方面的表现。在所考虑的算例中，不完全Cholesky预条件子可通过该方法显著改进，且在某些情况下仅需中等程度的低秩补偿项即可大幅提升收敛性。我们还考虑了线性规划内点法中默认无法进行此类不完全分解的矩阵，并基于所提方法提出了一种稳健的不完全Cholesky预条件子。结果表明，对于病态矩阵，截断方式的选择至关重要。我们展示了多个算例，其中使用所提预条件子的PCG能够收敛至小容差，而基于SVD的预条件子的PCG则无法实现收敛。