We study a preconditioner for a Hermitian positive definite linear system, which is obtained as the solution of a matrix nearness problem based on the Bregman log determinant divergence. The preconditioner is on the form of a Hermitian positive definite matrix plus a low-rank matrix. For this choice of structure, the generalised eigenvalues of the preconditioned system are easily calculated, and we show that the preconditioner is optimal in the sense that it minimises the $\ell_2$ condition number of the preconditioned matrix. We develop practical numerical approximations of the preconditioner based on the randomised singular value decomposition (SVD) and the Nystr\"om approximation and provide corresponding approximation results. Furthermore, we prove that the Nystr\"om approximation is in fact also a matrix approximation in a range-restricted Bregman divergence and establish several connections between this divergence and matrix nearness problems in different measures. Numerical examples are provided to support the theoretical results.

翻译：我们研究了一种用于Hermite正定线性系统的预条件子，该预条件子通过基于Bregman对数行列式散度的矩阵逼近问题求解得到。该预条件子具有Hermite正定矩阵加低秩矩阵的形式。针对这种结构选择，预条件系统的广义特征值易于计算，并且我们证明了该预条件子在最小化预条件矩阵的$\ell_2$条件数意义下是最优的。我们基于随机奇异值分解（SVD）和Nyström近似开发了预条件子的实用数值近似方法，并给出了相应的近似结果。此外，我们证明了Nyström近似实际上也是一种在限定值域Bregman散度意义下的矩阵近似，并建立了该散度与不同度量下矩阵逼近问题之间的若干联系。最后提供了数值算例以支持理论结果。