Preconditioner Design via the Bregman Divergence

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 of the form of a Hermitian positive definite matrix plus a low-rank matrix. For this choice of structure, the generalised eigenvalues of the preconditioned matrix are easily calculated, and we show under which conditions the preconditioner 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.